Subgroups — #135×(C₃⋊D₃)⋊C₃ #13153481

Order 1728.

993 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×(C₃⋊D₃)⋊C₃	C₂	1	2	yes	maximal, complemented
2	H(3,𝔽₃)×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	#64159⋊C₃	C₂	1	2	yes	maximal, complemented
4	#64128⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₁₆×(C₃⋊D₃)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#64125⋊C₃	C₂	1	2	yes	maximal, complemented
7	SD₁₆×(C₃⋊D₃)⋊C₃	C₂	1	2	yes	maximal, complemented
8	#64121⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	D₈×(C₃⋊D₃)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#64113⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×(C₃⋊D₃)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃⋊D₃×#135	C₃	1	1	yes	maximal, characteristic, complemented
13	C₃×D₃×#135	—	3	1	no	maximal
14	#41×H(3,𝔽₃)	C₂²	1	2	yes	complemented
15	#88722	C₂²	1	2	yes	complemented
16	Q₈×(C₃⋊D₃)⋊C₃	C₂²	1	2	yes	—
17	#3830⋊C₃	—	2	2	no	—
18	#3830⋊C₃	C₂²	1	2	yes	complemented
19	D₄×(C₃⋊D₃)⋊C₃	—	2	2	no	—
20	D₄×(C₃⋊D₃)⋊C₃	C₂²	1	2	yes	—
21	#3828⋊C₃	—	2	2	no	—
22	#3828⋊C₃	C₂²	1	2	yes	complemented
23	C₂×C₄×(C₃⋊D₃)⋊C₃	—	2	2	no	—
24	C₂×C₄×(C₃⋊D₃)⋊C₃	C₂²	1	1	yes	characteristic
25	Q₁₆×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
26	SD₁₆×H(3,𝔽₃)	C₂²	1	2	yes	complemented
27	D₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
29	#3755⋊C₃	C₂²	1	2	yes	complemented
30	#3756⋊C₃	C₂²	1	2	yes	complemented
31	#3753⋊C₃	C₂²	1	2	yes	complemented
32	#3754⋊C₃	C₂²	1	2	yes	complemented
33	C₂×(C₃⋊(C₃⋊C₈))⋊C₃	C₂²	1	1	yes	characteristic, complemented
34	#3745⋊C₃	C₂²	1	1	yes	characteristic, complemented
35	#3744⋊C₃	C₂²	1	2	yes	complemented
36	C₈×(C₃⋊D₃)⋊C₃	C₂²	1	1	yes	characteristic
37	C₈×(C₃⋊D₃)⋊C₃	C₂²	1	1	yes	characteristic
38	C₈×(C₃⋊D₃)⋊C₃	C₂²	1	1	yes	characteristic, complemented
39	C₈×(C₃⋊D₃)⋊C₃	C₂²	1	1	yes	characteristic, complemented
40	#3746⋊C₃	C₂²	1	1	yes	characteristic, complemented
41	#41×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
42	C₃×D₃×#41	—	3	2	no	—
43	C₃²×#135	—	2	1	no	—
44	C₃²×#135	D₃	1	1	yes	characteristic, complemented
45	C₃²×#135	C₂×C₃	1	1	yes	characteristic, complemented
46	#64159	C₂×C₃	1	2	yes	complemented
47	#64128	C₂×C₃	1	1	yes	characteristic, complemented
48	Q₁₆×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
49	#64125	C₂×C₃	1	2	yes	complemented
50	SD₁₆×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
51	#64121	C₂×C₃	1	1	yes	characteristic, complemented
52	D₈×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
53	#64113	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂×C₈×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
55	C₃×#950	—	3	2	no	—
56	C₃×#919	—	3	1	no	—
57	C₃×D₃×Q₁₆	—	3	1	no	—
58	C₃×#916	—	3	2	no	—
59	C₃×D₃×SD₁₆	—	3	2	no	—
60	C₃×#912	—	3	1	no	—
61	C₃×D₃×D₈	—	3	1	no	—
62	C₃×#904	—	3	1	no	—
63	C₂×C₃×D₃×C₈	—	3	1	no	—
64	C₂²×(C₃⋊D₃)⋊C₃	—	2	2	no	—
65	C₂²×(C₃⋊D₃)⋊C₃	D₄	1	1	yes	characteristic
66	Q₈×H(3,𝔽₃)	C₂³	1	2	yes	—
67	D₄×H(3,𝔽₃)	—	2	2	no	—
68	D₄×H(3,𝔽₃)	C₂³	1	2	yes	—
69	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
70	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
71	#638⋊C₃	—	2	2	no	—
72	#638⋊C₃	—	2	2	no	—
73	#638⋊C₃	—	2	2	no	—
74	C₂×(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
75	C₂×(C₃⋊Dic₃)⋊C₃	D₄	1	1	yes	characteristic
76	C₄⋊((C₃⋊D₃)⋊C₃)	—	2	2	no	—
77	C₄⋊((C₃⋊D₃)⋊C₃)	C₂³	1	2	yes	—
78	C₄×(C₃⋊D₃)⋊C₃	—	2	2	no	—
79	C₄×(C₃⋊D₃)⋊C₃	—	2	2	no	—
80	C₄×(C₃⋊D₃)⋊C₃	—	2	2	no	—
81	C₄×(C₃⋊D₃)⋊C₃	D₄	1	1	yes	characteristic
82	C₄×(C₃⋊D₃)⋊C₃	D₄	1	1	yes	characteristic
83	C₄×(C₃⋊D₃)⋊C₃	C₂³	1	1	yes	characteristic
84	C₄×(C₃⋊D₃)⋊C₃	C₂³	1	1	yes	characteristic
85	#634⋊C₃	—	2	2	no	—
86	#634⋊C₃	C₂³	1	2	yes	—
87	C₈×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
88	C₈×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
89	(C₃⋊(C₃⋊C₈))⋊C₃	C₂³	1	1	yes	characteristic
90	(C₃⋊(C₃⋊C₈))⋊C₃	C₂³	1	1	yes	characteristic
91	C₂×C₃×#135	—	9	1	no	—
92	D₃×#135	—	3	1	no	—
93	D₃×#135	—	9	1	no	—
94	C₃²×#41	—	2	2	no	—
95	C₃²×#41	C₂×D₃	1	2	yes	complemented
96	C₃²×#41	C₂²×C₃	1	2	yes	complemented
97	#3832	C₂²×C₃	1	2	yes	complemented
98	Q₈×C₃⋊D₃	C₂²×C₃	1	2	yes	—
99	#3830	—	2	2	no	—
100	#3830	C₂²×C₃	1	2	yes	complemented
101	D₄×C₃⋊D₃	—	2	2	no	—
102	D₄×C₃⋊D₃	C₂²×C₃	1	2	yes	—
103	#3828	—	2	2	no	—
104	#3828	C₂²×C₃	1	2	yes	complemented
105	C₂×C₄×C₃⋊D₃	—	2	2	no	—
106	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
107	C₃×#239	—	3	2	no	—
108	C₃×D₃×Q₈	—	3	2	no	—
109	C₃×#237	—	3	2	no	—
110	C₃×#237	—	6	2	no	—
111	C₃×D₃×D₄	—	3	2	no	—
112	C₃×D₃×D₄	—	6	2	no	—
113	C₃×#235	—	3	2	no	—
114	C₃×#235	—	6	2	no	—
115	C₂×C₃×C₄×D₃	—	3	1	no	—
116	C₂×C₃×C₄×D₃	—	6	2	no	—
117	C₃²×Q₁₆	—	2	1	no	—
118	C₃²×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
119	C₃²×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
120	C₃²×SD₁₆	—	2	2	no	—
121	C₃²×SD₁₆	C₂×D₃	1	2	yes	complemented
122	C₃²×SD₁₆	C₂²×C₃	1	2	yes	complemented
123	C₃²×D₈	—	2	1	no	—
124	C₃²×D₈	C₂×D₃	1	1	yes	characteristic, complemented
125	C₃²×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
126	C₂×C₃²×C₈	—	2	1	no	—
127	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
128	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
129	#3756	C₂²×C₃	1	2	yes	complemented
130	#3755	C₂²×C₃	1	2	yes	complemented
131	#3754	C₂²×C₃	1	2	yes	complemented
132	#3753	C₂²×C₃	1	2	yes	complemented
133	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic, complemented
134	#3746	C₂²×C₃	1	1	yes	characteristic, complemented
135	#3745	C₂²×C₃	1	1	yes	characteristic, complemented
136	#3744	C₂²×C₃	1	2	yes	complemented
137	C₈×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
138	C₈×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
139	C₈×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic, complemented
140	C₈×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic, complemented
141	C₃×#216	—	3	2	no	—
142	C₃×#215	—	3	2	no	—
143	C₃×#214	—	3	2	no	—
144	C₃×#213	—	3	2	no	—
145	C₂×C₃×C₃⋊C₈	—	3	1	no	—
146	C₃×Dic₁₂	—	3	1	no	—
147	C₃×D₂₄	—	3	1	no	—
148	C₃×#204	—	3	2	no	—
149	C₃×D₃×C₈	—	3	1	no	—
150	C₃×D₃×C₈	—	3	1	no	—
151	C₃×D₃×C₈	—	3	1	no	—
152	C₃×D₃×C₈	—	3	1	no	—
153	C₂²×H(3,𝔽₃)	—	2	2	no	—
154	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
155	C₂×(C₃⋊D₃)⋊C₃	—	2	2	no	—
156	C₂×(C₃⋊D₃)⋊C₃	—	2	2	no	—
157	C₂×(C₃⋊D₃)⋊C₃	—	4	4	no	—
158	C₂×(C₃⋊D₃)⋊C₃	C₂×D₄	1	1	yes	characteristic
159	C₂×(C₃⋊D₃)⋊C₃	C₂×D₄	1	1	yes	characteristic
160	C₄×H(3,𝔽₃)	—	2	2	no	—
161	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
162	C₄×H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic
163	(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
164	(C₃⋊Dic₃)⋊C₃	C₂×D₄	1	1	yes	characteristic
165	(C₃⋊Dic₃)⋊C₃	C₂×D₄	1	1	yes	characteristic
166	C₂×C₃×#41	—	9	2	no	—
167	D₃×#41	—	3	2	no	—
168	D₃×#41	—	9	2	no	—
169	C₃×#135	—	3	1	no	—
170	C₃×#135	—	3	1	no	—
171	C₃×#135	—	6	1	no	—
172	C₃×#135	—	9	1	no	—
173	C₃×#135	—	9	1	no	—
174	C₃×#135	—	9	2	no	—
175	C₃×#135	—	9	1	no	—
176	C₃×#135	—	9	2	no	—
177	C₃×#135	C₃×D₃	1	1	yes	characteristic
178	C₂×C₃×Q₁₆	—	9	1	no	—
179	C₂×C₃×SD₁₆	—	9	2	no	—
180	C₂×C₃×D₈	—	9	1	no	—
181	C₂²×C₃×C₈	—	9	1	no	—
182	#950	—	3	2	no	—
183	#950	—	9	2	no	—
184	#919	—	3	1	no	—
185	#919	—	9	1	no	—
186	D₃×Q₁₆	—	3	1	no	—
187	D₃×Q₁₆	—	9	1	no	—
188	#916	—	3	2	no	—
189	#916	—	9	2	no	—
190	D₃×SD₁₆	—	3	2	no	—
191	D₃×SD₁₆	—	9	2	no	—
192	#912	—	3	1	no	—
193	#912	—	9	1	no	—
194	D₃×D₈	—	3	1	no	—
195	D₃×D₈	—	9	1	no	—
196	#904	—	3	1	no	—
197	#904	—	9	1	no	—
198	C₂×D₃×C₈	—	3	1	no	—
199	C₂×D₃×C₈	—	9	1	no	—
200	C₂²×C₃⋊D₃	—	2	2	no	—
201	C₂²×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
202	C₂²×C₃×D₃	—	3	1	no	—
203	C₂²×C₃×D₃	—	6	2	no	—
204	C₃²×Q₈	—	2	2	no	—
205	C₃²×Q₈	C₂²×D₃	1	2	yes	—
206	C₃²×Q₈	C₂³×C₃	1	2	yes	—
207	C₃²×D₄	—	2	2	no	—
208	C₃²×D₄	—	2	2	no	—
209	C₃²×D₄	—	2	2	no	—
210	C₃²×D₄	—	4	2	no	—
211	C₃²×D₄	C₂²×D₃	1	2	yes	—
212	C₃²×D₄	C₂³×C₃	1	2	yes	—
213	C₂×C₃²×C₄	—	2	2	no	—
214	C₂×C₃²×C₄	—	2	2	no	—
215	C₂×C₃²×C₄	—	2	1	no	—
216	C₂×C₃²×C₄	—	4	2	no	—
217	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
218	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
219	#638	—	2	2	no	—
220	#638	—	2	2	no	—
221	#638	—	2	2	no	—
222	C₂×C₃⋊Dic₃	—	2	2	no	—
223	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
224	C₄⋊(C₃⋊D₃)	—	2	2	no	—
225	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	2	yes	—
226	C₄×C₃⋊D₃	—	2	2	no	—
227	C₄×C₃⋊D₃	—	2	2	no	—
228	C₄×C₃⋊D₃	—	2	2	no	—
229	C₄×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
230	C₄×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
231	C₄×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
232	C₄×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
233	#634	—	2	2	no	—
234	#634	C₂³×C₃	1	2	yes	—
235	C₃×#67	—	6	2	no	—
236	C₃×#67	—	6	2	no	—
237	C₃×#67	—	6	2	no	—
238	C₂×C₃×Dic₃	—	3	1	no	—
239	C₂×C₃×Dic₃	—	6	2	no	—
240	C₃×D₁₂	—	3	2	no	—
241	C₃×D₁₂	—	6	2	no	—
242	C₃×C₄×D₃	—	3	1	no	—
243	C₃×C₄×D₃	—	3	1	no	—
244	C₃×C₄×D₃	—	3	1	no	—
245	C₃×C₄×D₃	—	3	1	no	—
246	C₃×C₄×D₃	—	6	2	no	—
247	C₃×C₄×D₃	—	6	2	no	—
248	C₃×C₄×D₃	—	6	2	no	—
249	C₃×Dic₆	—	3	2	no	—
250	C₃×Dic₆	—	6	2	no	—
251	C₃²×C₈	—	2	1	no	—
252	C₃²×C₈	—	2	1	no	—
253	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
254	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
255	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
256	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
257	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	1	yes	characteristic
258	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	1	yes	characteristic
259	C₃×C₃⋊C₈	—	3	1	no	—
260	C₃×C₃⋊C₈	—	3	1	no	—
261	C₂×#135	—	9	1	no	Sylow
262	C₂×H(3,𝔽₃)	—	2	1	no	—
263	C₂×H(3,𝔽₃)	—	4	2	no	—
264	C₂×H(3,𝔽₃)	C₂²×D₄	1	1	yes	characteristic
265	(C₃⋊D₃)⋊C₃	—	2	1	no	—
266	(C₃⋊D₃)⋊C₃	—	4	2	no	—
267	(C₃⋊D₃)⋊C₃	#135	1	2	yes	direct product factor
268	C₃×#41	—	3	2	no	—
269	C₃×#41	—	3	2	no	—
270	C₃×#41	—	6	2	no	—
271	C₃×#41	—	9	2	no	—
272	C₃×#41	—	9	2	no	—
273	C₃×#41	—	9	2	no	—
274	C₃×#41	—	18	2	no	—
275	C₃×#41	—	18	2	no	—
276	C₃×#41	C₂×C₃×D₃	1	2	yes	—
277	C₂×C₃×Q₈	—	9	2	no	—
278	C₂×C₃×D₄	—	9	2	no	—
279	C₂×C₃×D₄	—	18	2	no	—
280	C₂²×C₃×C₄	—	9	1	no	—
281	C₂²×C₃×C₄	—	18	2	no	—
282	#239	—	3	2	no	—
283	#239	—	9	2	no	—
284	D₃×Q₈	—	3	2	no	—
285	D₃×Q₈	—	9	2	no	—
286	#237	—	3	2	no	—
287	#237	—	6	2	no	—
288	#237	—	9	2	no	—
289	#237	—	18	2	no	—
290	D₃×D₄	—	3	2	no	—
291	D₃×D₄	—	6	2	no	—
292	D₃×D₄	—	9	2	no	—
293	D₃×D₄	—	18	2	no	—
294	#235	—	3	2	no	—
295	#235	—	6	2	no	—
296	#235	—	9	2	no	—
297	#235	—	18	2	no	—
298	C₂×C₄×D₃	—	3	1	no	—
299	C₂×C₄×D₃	—	6	2	no	—
300	C₂×C₄×D₃	—	9	1	no	—
301	C₂×C₄×D₃	—	18	2	no	—
302	C₃×Q₁₆	—	3	1	no	—
303	C₃×Q₁₆	—	3	1	no	—
304	C₃×Q₁₆	—	6	1	no	—
305	C₃×Q₁₆	—	9	1	no	—
306	C₃×Q₁₆	—	9	2	no	—
307	C₃×Q₁₆	C₂×C₃×D₃	1	1	yes	characteristic
308	C₃×SD₁₆	—	3	2	no	—
309	C₃×SD₁₆	—	3	2	no	—
310	C₃×SD₁₆	—	6	2	no	—
311	C₃×SD₁₆	—	9	2	no	—
312	C₃×SD₁₆	—	9	2	no	—
313	C₃×SD₁₆	—	9	2	no	—
314	C₃×SD₁₆	C₂×C₃×D₃	1	2	yes	—
315	C₃×D₈	—	3	1	no	—
316	C₃×D₈	—	3	1	no	—
317	C₃×D₈	—	6	1	no	—
318	C₃×D₈	—	9	1	no	—
319	C₃×D₈	—	9	2	no	—
320	C₃×D₈	C₂×C₃×D₃	1	1	yes	characteristic
321	C₂×C₃×C₈	—	3	1	no	—
322	C₂×C₃×C₈	—	3	1	no	—
323	C₂×C₃×C₈	—	6	1	no	—
324	C₂×C₃×C₈	—	9	1	no	—
325	C₂×C₃×C₈	—	9	1	no	—
326	C₂×C₃×C₈	—	9	1	no	—
327	C₂×C₃×C₈	—	9	1	no	—
328	C₂×C₃×C₈	—	9	1	no	—
329	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
330	#216	—	3	2	no	—
331	#216	—	9	2	no	—
332	#215	—	3	2	no	—
333	#215	—	9	2	no	—
334	#214	—	3	2	no	—
335	#214	—	9	2	no	—
336	#213	—	3	2	no	—
337	#213	—	9	2	no	—
338	C₂×C₃⋊C₈	—	3	1	no	—
339	C₂×C₃⋊C₈	—	9	1	no	—
340	Dic₁₂	—	3	1	no	—
341	Dic₁₂	—	9	1	no	—
342	D₂₄	—	3	1	no	—
343	D₂₄	—	9	1	no	—
344	#204	—	3	2	no	—
345	#204	—	9	2	no	—
346	D₃×C₈	—	3	1	no	—
347	D₃×C₈	—	3	1	no	—
348	D₃×C₈	—	3	1	no	—
349	D₃×C₈	—	3	1	no	—
350	D₃×C₈	—	9	1	no	—
351	D₃×C₈	—	9	1	no	—
352	D₃×C₈	—	9	1	no	—
353	D₃×C₈	—	9	1	no	—
354	C₂²×C₃²	—	2	2	no	—
355	C₂²×C₃²	—	2	2	no	—
356	C₂²×C₃²	—	2	1	no	—
357	C₂²×C₃²	—	4	2	no	—
358	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
359	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
360	C₂×C₃⋊D₃	—	2	2	no	—
361	C₂×C₃⋊D₃	—	2	2	no	—
362	C₂×C₃⋊D₃	—	4	4	no	—
363	C₂×C₃⋊D₃	C₂×C₃×D₄	1	1	yes	characteristic
364	C₂×C₃⋊D₃	C₂×C₃×D₄	1	1	yes	characteristic
365	C₂×C₃×D₃	—	3	1	no	—
366	C₂×C₃×D₃	—	3	1	no	—
367	C₂×C₃×D₃	—	6	2	no	—
368	C₂×C₃×D₃	—	6	2	no	—
369	C₂×C₃×D₃	—	12	4	no	—
370	C₃²×C₄	—	2	2	no	—
371	C₃²×C₄	—	2	2	no	—
372	C₃²×C₄	—	2	1	no	—
373	C₃²×C₄	—	2	1	no	—
374	C₃²×C₄	—	4	2	no	—
375	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
376	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
377	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
378	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
379	C₃⋊Dic₃	—	2	2	no	—
380	C₃⋊Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
381	C₃⋊Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
382	C₃×Dic₃	—	3	1	no	—
383	C₃×Dic₃	—	3	1	no	—
384	C₃×Dic₃	—	6	2	no	—
385	C₂×#41	—	9	2	no	—
386	#135	—	9	2	no	—
387	#135	—	9	1	no	—
388	#135	—	9	2	no	—
389	#135	—	9	1	no	—
390	#135	—	9	1	no	—
391	#135	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, direct product factor
392	C₂×Q₁₆	—	9	1	no	—
393	C₂×SD₁₆	—	9	2	no	—
394	C₂×D₈	—	9	1	no	—
395	C₂²×C₈	—	9	1	no	—
396	H(3,𝔽₃)	C₂×#135	1	1	yes	characteristic, complemented, Sylow
397	C₂³×C₃	—	9	1	no	—
398	C₂³×C₃	—	18	2	no	—
399	C₂²×D₃	—	3	1	no	—
400	C₂²×D₃	—	6	2	no	—
401	C₂²×D₃	—	9	1	no	—
402	C₂²×D₃	—	18	2	no	—
403	C₃×Q₈	—	3	2	no	—
404	C₃×Q₈	—	3	2	no	—
405	C₃×Q₈	—	6	2	no	—
406	C₃×Q₈	—	9	2	no	—
407	C₃×Q₈	—	18	2	no	—
408	C₃×Q₈	C₂²×C₃×D₃	1	2	yes	—
409	C₃×D₄	—	2	2	no	—
410	C₃×D₄	—	3	2	no	—
411	C₃×D₄	—	3	2	no	—
412	C₃×D₄	—	6	2	no	—
413	C₃×D₄	—	6	2	no	—
414	C₃×D₄	—	6	2	no	—
415	C₃×D₄	—	9	2	no	—
416	C₃×D₄	—	12	2	no	—
417	C₃×D₄	—	18	2	no	—
418	C₃×D₄	—	18	2	no	—
419	C₃×D₄	—	18	2	no	—
420	C₃×D₄	—	18	2	no	—
421	C₃×D₄	C₂²×C₃×D₃	1	2	yes	—
422	C₂×C₃×C₄	—	2	2	no	—
423	C₂×C₃×C₄	—	3	1	no	—
424	C₂×C₃×C₄	—	3	1	no	—
425	C₂×C₃×C₄	—	6	2	no	—
426	C₂×C₃×C₄	—	6	2	no	—
427	C₂×C₃×C₄	—	6	1	no	—
428	C₂×C₃×C₄	—	9	1	no	—
429	C₂×C₃×C₄	—	9	1	no	—
430	C₂×C₃×C₄	—	9	1	no	—
431	C₂×C₃×C₄	—	9	1	no	—
432	C₂×C₃×C₄	—	9	1	no	—
433	C₂×C₃×C₄	—	12	2	no	—
434	C₂×C₃×C₄	—	18	2	no	—
435	C₂×C₃×C₄	—	18	2	no	—
436	C₂×C₃×C₄	—	18	2	no	—
437	C₂×C₃×C₄	—	18	2	no	—
438	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
439	#67	—	6	2	no	—
440	#67	—	6	2	no	—
441	#67	—	6	2	no	—
442	#67	—	18	2	no	—
443	#67	—	18	2	no	—
444	#67	—	18	2	no	—
445	C₂×Dic₃	—	3	1	no	—
446	C₂×Dic₃	—	6	2	no	—
447	C₂×Dic₃	—	9	1	no	—
448	C₂×Dic₃	—	18	2	no	—
449	D₁₂	—	3	2	no	—
450	D₁₂	—	6	2	no	—
451	D₁₂	—	9	2	no	—
452	D₁₂	—	18	2	no	—
453	C₄×D₃	—	3	1	no	—
454	C₄×D₃	—	3	1	no	—
455	C₄×D₃	—	3	1	no	—
456	C₄×D₃	—	3	1	no	—
457	C₄×D₃	—	6	2	no	—
458	C₄×D₃	—	6	2	no	—
459	C₄×D₃	—	6	2	no	—
460	C₄×D₃	—	9	1	no	—
461	C₄×D₃	—	9	1	no	—
462	C₄×D₃	—	9	1	no	—
463	C₄×D₃	—	9	1	no	—
464	C₄×D₃	—	18	2	no	—
465	C₄×D₃	—	18	2	no	—
466	C₄×D₃	—	18	2	no	—
467	Dic₆	—	3	2	no	—
468	Dic₆	—	6	2	no	—
469	Dic₆	—	9	2	no	—
470	Dic₆	—	18	2	no	—
471	C₃×C₈	—	3	1	no	—
472	C₃×C₈	—	3	1	no	—
473	C₃×C₈	—	3	1	no	—
474	C₃×C₈	—	3	1	no	—
475	C₃×C₈	—	6	1	no	—
476	C₃×C₈	—	6	1	no	—
477	C₃×C₈	—	9	1	no	—
478	C₃×C₈	—	9	1	no	—
479	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
480	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
481	C₃⋊C₈	—	3	1	no	—
482	C₃⋊C₈	—	3	1	no	—
483	C₃⋊C₈	—	9	1	no	—
484	C₃⋊C₈	—	9	1	no	—
485	C₂×C₃²	—	2	1	no	—
486	C₂×C₃²	—	2	1	no	—
487	C₂×C₃²	—	2	1	no	—
488	C₂×C₃²	—	4	1	no	—
489	C₂×C₃²	—	4	2	no	—
490	C₂×C₃²	—	4	2	no	—
491	C₂×C₃²	—	8	2	no	—
492	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
493	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
494	C₃⋊D₃	—	2	1	no	—
495	C₃⋊D₃	—	4	2	no	—
496	C₃⋊D₃	C₃×#135	1	2	yes	complemented
497	C₃×D₃	—	3	2	no	—
498	C₃×D₃	—	6	1	no	—
499	C₃×D₃	—	12	2	no	—
500	#41	—	9	2	no	—
501	#41	—	9	2	no	—
502	#41	—	9	2	no	—
503	#41	—	18	2	no	—
504	#41	—	18	2	no	—
505	#41	C₂×(C₃⋊D₃)⋊C₃	1	2	yes	complemented
506	C₂×Q₈	—	9	2	no	—
507	C₂×D₄	—	9	2	no	—
508	C₂×D₄	—	18	2	no	—
509	C₂²×C₄	—	9	1	no	—
510	C₂²×C₄	—	18	2	no	—
511	Q₁₆	—	9	2	no	—
512	Q₁₆	—	9	1	no	—
513	Q₁₆	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
514	SD₁₆	—	9	2	no	—
515	SD₁₆	—	9	2	no	—
516	SD₁₆	—	9	2	no	—
517	SD₁₆	C₂×(C₃⋊D₃)⋊C₃	1	2	yes	complemented
518	D₈	—	9	2	no	—
519	D₈	—	9	1	no	—
520	D₈	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
521	C₂×C₈	—	9	1	no	—
522	C₂×C₈	—	9	1	no	—
523	C₂×C₈	—	9	1	no	—
524	C₂×C₈	—	9	1	no	—
525	C₂×C₈	—	9	1	no	—
526	C₂×C₈	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
527	C₂²×C₃	—	2	2	no	—
528	C₂²×C₃	—	3	1	no	—
529	C₂²×C₃	—	3	1	no	—
530	C₂²×C₃	—	6	2	no	—
531	C₂²×C₃	—	6	2	no	—
532	C₂²×C₃	—	6	1	no	—
533	C₂²×C₃	—	9	1	no	—
534	C₂²×C₃	—	9	1	no	—
535	C₂²×C₃	—	12	2	no	—
536	C₂²×C₃	—	18	2	no	—
537	C₂²×C₃	—	18	2	no	—
538	C₂²×C₃	—	36	4	no	—
539	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
540	C₂×D₃	—	3	1	no	—
541	C₂×D₃	—	3	1	no	—
542	C₂×D₃	—	6	2	no	—
543	C₂×D₃	—	6	2	no	—
544	C₂×D₃	—	9	1	no	—
545	C₂×D₃	—	9	1	no	—
546	C₂×D₃	—	12	4	no	—
547	C₂×D₃	—	18	2	no	—
548	C₂×D₃	—	18	2	no	—
549	C₂×D₃	—	36	4	no	—
550	C₃×C₄	—	2	2	no	—
551	C₃×C₄	—	3	1	no	—
552	C₃×C₄	—	3	1	no	—
553	C₃×C₄	—	3	1	no	—
554	C₃×C₄	—	3	1	no	—
555	C₃×C₄	—	6	2	no	—
556	C₃×C₄	—	6	2	no	—
557	C₃×C₄	—	6	1	no	—
558	C₃×C₄	—	6	1	no	—
559	C₃×C₄	—	9	1	no	—
560	C₃×C₄	—	9	1	no	—
561	C₃×C₄	—	12	2	no	—
562	C₃×C₄	—	18	2	no	—
563	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
564	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
565	Dic₃	—	3	1	no	—
566	Dic₃	—	3	1	no	—
567	Dic₃	—	6	2	no	—
568	Dic₃	—	9	1	no	—
569	Dic₃	—	9	1	no	—
570	Dic₃	—	18	2	no	—
571	C₃²	—	2	1	no	—
572	C₃²	D₃×#135	1	1	yes	characteristic, complemented
573	C₃²	C₂×C₃×#135	1	1	yes	characteristic, complemented
574	C₂³	—	9	1	no	—
575	C₂³	—	18	2	no	—
576	Q₈	—	9	2	no	—
577	Q₈	—	18	2	no	—
578	Q₈	C₂²×(C₃⋊D₃)⋊C₃	1	2	yes	—
579	D₄	—	2	2	no	—
580	D₄	—	9	2	no	—
581	D₄	—	18	2	no	—
582	D₄	—	18	2	no	—
583	D₄	—	18	2	no	—
584	D₄	—	18	2	no	—
585	D₄	C₂²×(C₃⋊D₃)⋊C₃	1	2	yes	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	9	1	no	—
588	C₂×C₄	—	9	1	no	—
589	C₂×C₄	—	9	1	no	—
590	C₂×C₄	—	9	1	no	—
591	C₂×C₄	—	9	1	no	—
592	C₂×C₄	—	18	2	no	—
593	C₂×C₄	—	18	2	no	—
594	C₂×C₄	—	18	2	no	—
595	C₂×C₄	—	18	2	no	—
596	C₂×C₄	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
597	C₈	—	9	1	no	—
598	C₈	—	9	1	no	—
599	C₈	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
600	C₈	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
601	C₂×C₃	—	2	1	no	—
602	C₂×C₃	—	3	1	no	—
603	C₂×C₃	—	3	1	no	—
604	C₂×C₃	—	4	2	no	—
605	C₂×C₃	—	6	1	no	—
606	C₂×C₃	—	6	1	no	—
607	C₂×C₃	—	6	1	no	—
608	C₂×C₃	—	9	2	no	—
609	C₂×C₃	—	12	1	no	—
610	C₂×C₃	—	12	2	no	—
611	C₂×C₃	—	12	2	no	—
612	C₂×C₃	—	18	1	no	—
613	C₂×C₃	—	24	2	no	—
614	C₂×C₃	—	36	2	no	—
615	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
616	D₃	—	3	2	no	—
617	D₃	—	6	1	no	—
618	D₃	—	9	2	no	—
619	D₃	—	12	2	no	—
620	D₃	—	18	1	no	—
621	D₃	—	36	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	9	1	no	—
624	C₂²	—	9	1	no	—
625	C₂²	—	18	2	no	—
626	C₂²	—	18	2	no	—
627	C₂²	—	36	4	no	—
628	C₂²	D₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
629	C₄	—	2	2	no	—
630	C₄	—	9	1	no	—
631	C₄	—	9	1	no	—
632	C₄	—	18	2	no	—
633	C₄	D₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
634	C₄	C₂³×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
635	C₃	—	3	1	no	—
636	C₃	—	3	1	no	—
637	C₃	—	6	1	no	—
638	C₃	C₃×D₃×#135	1	1	yes	characteristic
639	C₂	—	2	1	no	—
640	C₂	—	4	2	no	—
641	C₂	—	9	2	no	—
642	C₂	—	18	1	no	—
643	C₂	—	36	2	no	—
644	C₂	C₂×D₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	H(3,𝔽₃)×#135	C₂
socle	C₂×C₃	C₂×C₃×D₃×D₄
hypercenter	#135	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×C₃×#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #41×(C₃⋊D₃)⋊C₃	C₂	semidirect
2. H(3,𝔽₃)×#135	C₂	semidirect
3. #64159⋊C₃	C₂	semidirect
4. #64128⋊C₃	C₂	semidirect
5. Q₁₆×(C₃⋊D₃)⋊C₃	C₂	semidirect
6. #64125⋊C₃	C₂	semidirect
7. SD₁₆×(C₃⋊D₃)⋊C₃	C₂	semidirect
8. #64121⋊C₃	C₂	semidirect
9. D₈×(C₃⋊D₃)⋊C₃	C₂	semidirect
10. #64113⋊C₃	C₂	semidirect
11. C₂×C₈×(C₃⋊D₃)⋊C₃	C₂	semidirect
12. C₃⋊D₃×#135	C₃	semidirect
14. #41×H(3,𝔽₃)	C₂²	semidirect
15. #88722	C₂²	semidirect
18. #3830⋊C₃	C₂²	semidirect
22. #3828⋊C₃	C₂²	semidirect
25. Q₁₆×H(3,𝔽₃)	C₂²	semidirect
26. SD₁₆×H(3,𝔽₃)	C₂²	semidirect
27. D₈×H(3,𝔽₃)	C₂²	semidirect
28. C₂×C₈×H(3,𝔽₃)	C₂²	semidirect
29. #3755⋊C₃	C₂²	semidirect
30. #3756⋊C₃	C₂²	semidirect
31. #3753⋊C₃	C₂²	semidirect
32. #3754⋊C₃	C₂²	semidirect
33. C₂×(C₃⋊(C₃⋊C₈))⋊C₃	C₂²	semidirect
34. #3745⋊C₃	C₂²	semidirect
35. #3744⋊C₃	C₂²	semidirect
38. C₈×(C₃⋊D₃)⋊C₃	C₂²	semidirect
39. C₈×(C₃⋊D₃)⋊C₃	C₂²	semidirect
40. #3746⋊C₃	C₂²	semidirect
41. #41×C₃⋊D₃	C₂×C₃	semidirect
44. C₃²×#135	D₃	semidirect
45. C₃²×#135	C₂×C₃	semidirect
46. #64159	C₂×C₃	semidirect
47. #64128	C₂×C₃	semidirect
48. Q₁₆×C₃⋊D₃	C₂×C₃	semidirect
49. #64125	C₂×C₃	semidirect
50. SD₁₆×C₃⋊D₃	C₂×C₃	semidirect
51. #64121	C₂×C₃	semidirect
52. D₈×C₃⋊D₃	C₂×C₃	semidirect
53. #64113	C₂×C₃	semidirect
54. C₂×C₈×C₃⋊D₃	C₂×C₃	semidirect
95. C₃²×#41	C₂×D₃	semidirect
96. C₃²×#41	C₂²×C₃	semidirect
97. #3832	C₂²×C₃	semidirect
100. #3830	C₂²×C₃	semidirect
104. #3828	C₂²×C₃	semidirect
118. C₃²×Q₁₆	C₂×D₃	semidirect
119. C₃²×Q₁₆	C₂²×C₃	semidirect
121. C₃²×SD₁₆	C₂×D₃	semidirect
122. C₃²×SD₁₆	C₂²×C₃	semidirect
124. C₃²×D₈	C₂×D₃	semidirect
125. C₃²×D₈	C₂²×C₃	semidirect
127. C₂×C₃²×C₈	C₂×D₃	semidirect
128. C₂×C₃²×C₈	C₂²×C₃	semidirect
129. #3756	C₂²×C₃	semidirect
130. #3755	C₂²×C₃	semidirect
131. #3754	C₂²×C₃	semidirect
132. #3753	C₂²×C₃	semidirect
133. C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	semidirect
134. #3746	C₂²×C₃	semidirect
135. #3745	C₂²×C₃	semidirect
136. #3744	C₂²×C₃	semidirect
139. C₈×C₃⋊D₃	C₂²×C₃	semidirect
140. C₈×C₃⋊D₃	C₂²×C₃	semidirect
267. (C₃⋊D₃)⋊C₃	#135	direct (N × H)
391. #135	(C₃⋊D₃)⋊C₃	direct (N × H)
396. H(3,𝔽₃)	C₂×#135	semidirect
496. C₃⋊D₃	C₃×#135	semidirect
505. #41	C₂×(C₃⋊D₃)⋊C₃	semidirect
513. Q₁₆	C₂×(C₃⋊D₃)⋊C₃	semidirect
517. SD₁₆	C₂×(C₃⋊D₃)⋊C₃	semidirect
520. D₈	C₂×(C₃⋊D₃)⋊C₃	semidirect
526. C₂×C₈	C₂×(C₃⋊D₃)⋊C₃	semidirect
572. C₃²	D₃×#135	semidirect
573. C₃²	C₂×C₃×#135	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#41×(C₃⋊D₃)⋊C₃	2
H(3,𝔽₃)×#135	2
#64159⋊C₃	2
#64128⋊C₃	2
Q₁₆×(C₃⋊D₃)⋊C₃	2
#64125⋊C₃	2
SD₁₆×(C₃⋊D₃)⋊C₃	2
#64121⋊C₃	2
D₈×(C₃⋊D₃)⋊C₃	2
#64113⋊C₃	2
C₂×C₈×(C₃⋊D₃)⋊C₃	2
C₃⋊D₃×#135	3
C₃×D₃×#135	3

Subgroups — #135×(C3⋊D3)⋊C3 #13153481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #135×(C₃⋊D₃)⋊C₃ #13153481