Subgroups — C₃²×D₃×#135 #13179144

Order 1728.

1844 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₃×#41	C₂	1	2	yes	maximal, complemented
2	C₃³×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#950	C₂	1	2	yes	maximal, complemented
4	C₃²×#919	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×D₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#916	C₂	1	2	yes	maximal, complemented
7	C₃²×D₃×SD₁₆	C₂	1	2	yes	maximal, complemented
8	C₃²×#912	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×D₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃²×#904	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃²×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃²×#135	—	3	1	no	maximal
13	C₃×D₃×#135	C₃	1	4	yes	maximal, direct product factor
14	C₃³×#41	C₂²	1	2	yes	complemented
15	C₃²×#239	C₂²	1	2	yes	complemented
16	C₃²×D₃×Q₈	C₂²	1	2	yes	—
17	C₃²×#237	—	2	2	no	—
18	C₃²×#237	C₂²	1	2	yes	complemented
19	C₃²×D₃×D₄	—	2	2	no	—
20	C₃²×D₃×D₄	C₂²	1	2	yes	—
21	C₃²×#235	—	2	2	no	—
22	C₃²×#235	C₂²	1	2	yes	complemented
23	C₂×C₃²×C₄×D₃	—	2	2	no	—
24	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
25	C₃³×Q₁₆	C₂²	1	1	yes	characteristic, complemented
26	C₃³×SD₁₆	C₂²	1	2	yes	complemented
27	C₃³×D₈	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic, complemented
29	C₃²×#216	C₂²	1	2	yes	complemented
30	C₃²×#215	C₂²	1	2	yes	complemented
31	C₃²×#214	C₂²	1	2	yes	complemented
32	C₃²×#213	C₂²	1	2	yes	complemented
33	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
34	C₃²×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
35	C₃²×D₂₄	C₂²	1	1	yes	characteristic, complemented
36	C₃²×#204	C₂²	1	2	yes	complemented
37	C₃²×D₃×C₈	C₂²	1	1	yes	characteristic
38	C₃²×D₃×C₈	C₂²	1	1	yes	characteristic
39	C₃²×D₃×C₈	C₂²	1	1	yes	characteristic, complemented
40	C₃²×D₃×C₈	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₃²×#41	—	3	2	no	—
42	C₃×D₃×#41	C₂×C₃	1	8	yes	complemented
43	C₃²×#135	—	2	4	no	—
44	C₃²×#135	—	3	2	no	—
45	C₃²×#135	—	3	1	no	—
46	C₃²×#135	—	3	2	no	—
47	C₃²×#135	—	3	1	no	—
48	C₃²×#135	—	3	1	no	—
49	C₃²×#135	D₃	1	1	yes	characteristic, direct product factor
50	C₃²×#135	C₂×C₃	1	4	yes	complemented
51	C₂×C₃²×Q₁₆	—	3	1	no	—
52	C₂×C₃²×SD₁₆	—	3	2	no	—
53	C₂×C₃²×D₈	—	3	1	no	—
54	C₂²×C₃²×C₈	—	3	1	no	—
55	C₃×#950	C₂×C₃	1	8	yes	complemented
56	C₃×#919	C₂×C₃	1	4	yes	complemented
57	C₃×D₃×Q₁₆	C₂×C₃	1	4	yes	complemented
58	C₃×#916	C₂×C₃	1	8	yes	complemented
59	C₃×D₃×SD₁₆	C₂×C₃	1	8	yes	complemented
60	C₃×#912	C₂×C₃	1	4	yes	complemented
61	C₃×D₃×D₈	C₂×C₃	1	4	yes	complemented
62	C₃×#904	C₂×C₃	1	4	yes	complemented
63	C₂×C₃×D₃×C₈	C₂×C₃	1	4	yes	complemented
64	C₂²×C₃²×D₃	—	2	2	no	—
65	C₂²×C₃²×D₃	D₄	1	1	yes	characteristic
66	C₃³×Q₈	C₂³	1	2	yes	—
67	C₃³×D₄	—	2	2	no	—
68	C₃³×D₄	C₂³	1	2	yes	—
69	C₂×C₃³×C₄	—	2	2	no	—
70	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
71	C₃²×#67	—	2	2	no	—
72	C₃²×#67	—	2	2	no	—
73	C₃²×#67	—	2	2	no	—
74	C₂×C₃²×Dic₃	—	2	2	no	—
75	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
76	C₃²×D₁₂	—	2	2	no	—
77	C₃²×D₁₂	C₂³	1	2	yes	—
78	C₃²×C₄×D₃	—	2	2	no	—
79	C₃²×C₄×D₃	—	2	2	no	—
80	C₃²×C₄×D₃	—	2	2	no	—
81	C₃²×C₄×D₃	D₄	1	1	yes	characteristic
82	C₃²×C₄×D₃	D₄	1	1	yes	characteristic
83	C₃²×C₄×D₃	C₂³	1	1	yes	characteristic
84	C₃²×C₄×D₃	C₂³	1	1	yes	characteristic
85	C₃²×Dic₆	—	2	2	no	—
86	C₃²×Dic₆	C₂³	1	2	yes	—
87	C₃³×C₈	C₂³	1	1	yes	characteristic
88	C₃³×C₈	C₂³	1	1	yes	characteristic
89	C₃²×C₃⋊C₈	C₂³	1	1	yes	characteristic
90	C₃²×C₃⋊C₈	C₂³	1	1	yes	characteristic
91	C₂×C₃×#135	—	3	4	no	—
92	D₃×#135	C₃²	1	1	yes	characteristic, direct product factor
93	C₃²×#41	—	2	8	no	—
94	C₃²×#41	—	3	2	no	—
95	C₃²×#41	—	3	2	no	—
96	C₃²×#41	—	3	2	no	—
97	C₃²×#41	—	6	2	no	—
98	C₃²×#41	—	6	2	no	—
99	C₃²×#41	C₂×D₃	1	2	yes	complemented
100	C₃²×#41	C₂²×C₃	1	8	yes	complemented
101	C₂×C₃²×Q₈	—	3	2	no	—
102	C₂×C₃²×D₄	—	3	2	no	—
103	C₂×C₃²×D₄	—	6	2	no	—
104	C₂²×C₃²×C₄	—	3	1	no	—
105	C₂²×C₃²×C₄	—	6	2	no	—
106	C₃×#239	C₂²×C₃	1	8	yes	complemented
107	C₃×D₃×Q₈	C₂²×C₃	1	8	yes	—
108	C₃×#237	—	2	8	no	—
109	C₃×#237	C₂²×C₃	1	8	yes	complemented
110	C₃×D₃×D₄	—	2	8	no	—
111	C₃×D₃×D₄	C₂²×C₃	1	8	yes	—
112	C₃×#235	—	2	8	no	—
113	C₃×#235	C₂²×C₃	1	8	yes	complemented
114	C₂×C₃×C₄×D₃	—	2	8	no	—
115	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
116	C₃²×Q₁₆	—	2	4	no	—
117	C₃²×Q₁₆	—	3	2	no	—
118	C₃²×Q₁₆	—	3	1	no	—
119	C₃²×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
120	C₃²×Q₁₆	C₂²×C₃	1	4	yes	complemented
121	C₃²×SD₁₆	—	2	8	no	—
122	C₃²×SD₁₆	—	3	2	no	—
123	C₃²×SD₁₆	—	3	2	no	—
124	C₃²×SD₁₆	—	3	2	no	—
125	C₃²×SD₁₆	C₂×D₃	1	2	yes	complemented
126	C₃²×SD₁₆	C₂²×C₃	1	8	yes	complemented
127	C₃²×D₈	—	2	4	no	—
128	C₃²×D₈	—	3	2	no	—
129	C₃²×D₈	—	3	1	no	—
130	C₃²×D₈	C₂×D₃	1	1	yes	characteristic, complemented
131	C₃²×D₈	C₂²×C₃	1	4	yes	complemented
132	C₂×C₃²×C₈	—	2	4	no	—
133	C₂×C₃²×C₈	—	3	1	no	—
134	C₂×C₃²×C₈	—	3	1	no	—
135	C₂×C₃²×C₈	—	3	1	no	—
136	C₂×C₃²×C₈	—	3	1	no	—
137	C₂×C₃²×C₈	—	3	1	no	—
138	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
139	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	complemented
140	C₃×#216	C₂²×C₃	1	8	yes	complemented
141	C₃×#215	C₂²×C₃	1	8	yes	complemented
142	C₃×#214	C₂²×C₃	1	8	yes	complemented
143	C₃×#213	C₂²×C₃	1	8	yes	complemented
144	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	complemented
145	C₃×Dic₁₂	C₂²×C₃	1	4	yes	complemented
146	C₃×D₂₄	C₂²×C₃	1	4	yes	complemented
147	C₃×#204	C₂²×C₃	1	8	yes	complemented
148	C₃×D₃×C₈	C₂²×C₃	1	4	yes	—
149	C₃×D₃×C₈	C₂²×C₃	1	4	yes	—
150	C₃×D₃×C₈	C₂²×C₃	1	4	yes	complemented
151	C₃×D₃×C₈	C₂²×C₃	1	4	yes	complemented
152	C₂²×C₃³	—	2	2	no	—
153	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
154	C₂×C₃²×D₃	—	2	2	no	—
155	C₂×C₃²×D₃	—	2	2	no	—
156	C₂×C₃²×D₃	—	4	4	no	—
157	C₂×C₃²×D₃	C₂×D₄	1	1	yes	characteristic
158	C₂×C₃²×D₃	C₂×D₄	1	1	yes	characteristic
159	C₃³×C₄	—	2	2	no	—
160	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
161	C₃³×C₄	C₂⁴	1	1	yes	characteristic
162	C₃²×Dic₃	—	2	2	no	—
163	C₃²×Dic₃	C₂×D₄	1	1	yes	characteristic
164	C₃²×Dic₃	C₂×D₄	1	1	yes	characteristic
165	C₂×C₃×#41	—	3	8	no	—
166	D₃×#41	C₂×C₃²	1	2	yes	complemented
167	C₃×#135	—	2	4	no	—
168	C₃×#135	—	3	8	no	—
169	C₃×#135	—	3	4	no	—
170	C₃×#135	—	3	8	no	—
171	C₃×#135	—	3	4	no	—
172	C₃×#135	—	3	4	no	—
173	C₃×#135	C₃×D₃	1	4	yes	direct product factor
174	C₃×#135	C₂×C₃²	1	1	yes	characteristic, complemented
175	C₂×C₃×Q₁₆	—	3	4	no	—
176	C₂×C₃×SD₁₆	—	3	8	no	—
177	C₂×C₃×D₈	—	3	4	no	—
178	C₂²×C₃×C₈	—	3	4	no	—
179	#950	C₂×C₃²	1	2	yes	complemented
180	#919	C₂×C₃²	1	1	yes	characteristic, complemented
181	D₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
182	#916	C₂×C₃²	1	2	yes	complemented
183	D₃×SD₁₆	C₂×C₃²	1	2	yes	complemented
184	#912	C₂×C₃²	1	1	yes	characteristic, complemented
185	D₃×D₈	C₂×C₃²	1	1	yes	characteristic, complemented
186	#904	C₂×C₃²	1	1	yes	characteristic, complemented
187	C₂×D₃×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
188	C₂³×C₃²	—	3	1	no	—
189	C₂³×C₃²	—	6	2	no	—
190	C₂²×C₃×D₃	—	2	8	no	—
191	C₂²×C₃×D₃	C₃×D₄	1	4	yes	—
192	C₃²×Q₈	—	2	8	no	—
193	C₃²×Q₈	—	3	2	no	—
194	C₃²×Q₈	—	6	2	no	—
195	C₃²×Q₈	C₂²×D₃	1	2	yes	—
196	C₃²×Q₈	C₂³×C₃	1	8	yes	—
197	C₃²×D₄	—	2	2	no	—
198	C₃²×D₄	—	2	8	no	—
199	C₃²×D₄	—	2	8	no	—
200	C₃²×D₄	—	3	2	no	—
201	C₃²×D₄	—	4	8	no	—
202	C₃²×D₄	—	6	2	no	—
203	C₃²×D₄	—	6	2	no	—
204	C₃²×D₄	—	6	2	no	—
205	C₃²×D₄	—	6	2	no	—
206	C₃²×D₄	C₂²×D₃	1	2	yes	—
207	C₃²×D₄	C₂³×C₃	1	8	yes	—
208	C₂×C₃²×C₄	—	2	2	no	—
209	C₂×C₃²×C₄	—	2	4	no	—
210	C₂×C₃²×C₄	—	2	8	no	—
211	C₂×C₃²×C₄	—	3	1	no	—
212	C₂×C₃²×C₄	—	3	1	no	—
213	C₂×C₃²×C₄	—	3	1	no	—
214	C₂×C₃²×C₄	—	3	1	no	—
215	C₂×C₃²×C₄	—	3	1	no	—
216	C₂×C₃²×C₄	—	4	8	no	—
217	C₂×C₃²×C₄	—	6	2	no	—
218	C₂×C₃²×C₄	—	6	2	no	—
219	C₂×C₃²×C₄	—	6	2	no	—
220	C₂×C₃²×C₄	—	6	2	no	—
221	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
222	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
223	C₃×#67	—	2	8	no	—
224	C₃×#67	—	2	8	no	—
225	C₃×#67	—	2	8	no	—
226	C₂×C₃×Dic₃	—	2	8	no	—
227	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
228	C₃×D₁₂	—	2	8	no	—
229	C₃×D₁₂	C₂³×C₃	1	8	yes	—
230	C₃×C₄×D₃	—	2	8	no	—
231	C₃×C₄×D₃	—	2	8	no	—
232	C₃×C₄×D₃	—	2	8	no	—
233	C₃×C₄×D₃	C₃×D₄	1	4	yes	—
234	C₃×C₄×D₃	C₃×D₄	1	4	yes	—
235	C₃×C₄×D₃	C₂³×C₃	1	4	yes	—
236	C₃×C₄×D₃	C₂³×C₃	1	4	yes	—
237	C₃×Dic₆	—	2	8	no	—
238	C₃×Dic₆	C₂³×C₃	1	8	yes	—
239	C₃²×C₈	—	2	4	no	—
240	C₃²×C₈	—	2	4	no	—
241	C₃²×C₈	—	3	1	no	—
242	C₃²×C₈	—	3	1	no	—
243	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
244	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
245	C₃²×C₈	C₂³×C₃	1	4	yes	—
246	C₃²×C₈	C₂³×C₃	1	4	yes	—
247	C₃×C₃⋊C₈	C₂³×C₃	1	4	yes	—
248	C₃×C₃⋊C₈	C₂³×C₃	1	4	yes	—
249	C₂×#135	—	3	1	no	Sylow
250	C₂×C₃³	—	2	1	no	—
251	C₂×C₃³	—	4	2	no	—
252	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
253	C₃²×D₃	—	2	1	no	—
254	C₃²×D₃	—	4	2	no	—
255	C₃²×D₃	#135	1	2	yes	direct product factor
256	C₃×#41	—	2	8	no	—
257	C₃×#41	—	3	8	no	—
258	C₃×#41	—	3	8	no	—
259	C₃×#41	—	3	8	no	—
260	C₃×#41	—	6	8	no	—
261	C₃×#41	—	6	8	no	—
262	C₃×#41	C₂×C₃×D₃	1	8	yes	complemented
263	C₃×#41	C₂²×C₃²	1	2	yes	complemented
264	C₂×C₃×Q₈	—	3	8	no	—
265	C₂×C₃×D₄	—	3	8	no	—
266	C₂×C₃×D₄	—	6	8	no	—
267	C₂²×C₃×C₄	—	3	4	no	—
268	C₂²×C₃×C₄	—	6	8	no	—
269	#239	C₂²×C₃²	1	2	yes	complemented
270	D₃×Q₈	C₂²×C₃²	1	2	yes	—
271	#237	—	2	2	no	—
272	#237	C₂²×C₃²	1	2	yes	complemented
273	D₃×D₄	—	2	2	no	—
274	D₃×D₄	C₂²×C₃²	1	2	yes	—
275	#235	—	2	2	no	—
276	#235	C₂²×C₃²	1	2	yes	complemented
277	C₂×C₄×D₃	—	2	2	no	—
278	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
279	C₃×Q₁₆	—	2	4	no	—
280	C₃×Q₁₆	—	3	8	no	—
281	C₃×Q₁₆	—	3	4	no	—
282	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	complemented
283	C₃×Q₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
284	C₃×SD₁₆	—	2	8	no	—
285	C₃×SD₁₆	—	3	8	no	—
286	C₃×SD₁₆	—	3	8	no	—
287	C₃×SD₁₆	—	3	8	no	—
288	C₃×SD₁₆	C₂×C₃×D₃	1	8	yes	complemented
289	C₃×SD₁₆	C₂²×C₃²	1	2	yes	complemented
290	C₃×D₈	—	2	4	no	—
291	C₃×D₈	—	3	8	no	—
292	C₃×D₈	—	3	4	no	—
293	C₃×D₈	C₂×C₃×D₃	1	4	yes	complemented
294	C₃×D₈	C₂²×C₃²	1	1	yes	characteristic, complemented
295	C₂×C₃×C₈	—	2	4	no	—
296	C₂×C₃×C₈	—	3	4	no	—
297	C₂×C₃×C₈	—	3	4	no	—
298	C₂×C₃×C₈	—	3	4	no	—
299	C₂×C₃×C₈	—	3	4	no	—
300	C₂×C₃×C₈	—	3	4	no	—
301	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	complemented
302	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
303	#216	C₂²×C₃²	1	2	yes	complemented
304	#215	C₂²×C₃²	1	2	yes	complemented
305	#214	C₂²×C₃²	1	2	yes	complemented
306	#213	C₂²×C₃²	1	2	yes	complemented
307	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
308	Dic₁₂	C₂²×C₃²	1	1	yes	characteristic, complemented
309	D₂₄	C₂²×C₃²	1	1	yes	characteristic, complemented
310	#204	C₂²×C₃²	1	2	yes	complemented
311	D₃×C₈	C₂²×C₃²	1	1	yes	characteristic
312	D₃×C₈	C₂²×C₃²	1	1	yes	characteristic
313	D₃×C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
314	D₃×C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
315	C₂²×C₃²	—	2	2	no	—
316	C₂²×C₃²	—	2	4	no	—
317	C₂²×C₃²	—	2	8	no	—
318	C₂²×C₃²	—	3	1	no	—
319	C₂²×C₃²	—	3	1	no	—
320	C₂²×C₃²	—	4	8	no	—
321	C₂²×C₃²	—	6	2	no	—
322	C₂²×C₃²	—	6	2	no	—
323	C₂²×C₃²	—	12	4	no	—
324	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
325	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
326	C₂×C₃×D₃	—	2	8	no	—
327	C₂×C₃×D₃	—	2	8	no	—
328	C₂×C₃×D₃	—	4	16	no	—
329	C₂×C₃×D₃	C₂×C₃×D₄	1	4	yes	—
330	C₂×C₃×D₃	C₂×C₃×D₄	1	4	yes	—
331	C₃²×C₄	—	2	2	no	—
332	C₃²×C₄	—	2	4	no	—
333	C₃²×C₄	—	2	4	no	—
334	C₃²×C₄	—	2	8	no	—
335	C₃²×C₄	—	3	1	no	—
336	C₃²×C₄	—	3	1	no	—
337	C₃²×C₄	—	4	8	no	—
338	C₃²×C₄	—	6	2	no	—
339	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
340	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
341	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
342	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
343	C₃×Dic₃	—	2	8	no	—
344	C₃×Dic₃	C₂×C₃×D₄	1	4	yes	—
345	C₃×Dic₃	C₂×C₃×D₄	1	4	yes	—
346	C₂×#41	—	3	2	no	—
347	#135	—	3	2	no	—
348	#135	—	3	1	no	—
349	#135	—	3	2	no	—
350	#135	—	3	1	no	—
351	#135	—	3	1	no	—
352	#135	C₃²×D₃	1	1	yes	characteristic, direct product factor
353	C₂×Q₁₆	—	3	1	no	—
354	C₂×SD₁₆	—	3	2	no	—
355	C₂×D₈	—	3	1	no	—
356	C₂²×C₈	—	3	1	no	—
357	C₃³	C₂×#135	1	1	yes	characteristic, complemented, Sylow
358	C₂³×C₃	—	3	4	no	—
359	C₂³×C₃	—	6	8	no	—
360	C₂²×D₃	—	2	2	no	—
361	C₂²×D₃	C₃²×D₄	1	1	yes	characteristic
362	C₃×Q₈	—	2	8	no	—
363	C₃×Q₈	—	3	8	no	—
364	C₃×Q₈	—	6	8	no	—
365	C₃×Q₈	C₂²×C₃×D₃	1	8	yes	—
366	C₃×Q₈	C₂³×C₃²	1	2	yes	—
367	C₃×D₄	—	2	8	no	—
368	C₃×D₄	—	2	2	no	—
369	C₃×D₄	—	2	8	no	—
370	C₃×D₄	—	3	8	no	—
371	C₃×D₄	—	4	8	no	—
372	C₃×D₄	—	6	8	no	—
373	C₃×D₄	—	6	8	no	—
374	C₃×D₄	—	6	8	no	—
375	C₃×D₄	—	6	8	no	—
376	C₃×D₄	C₂²×C₃×D₃	1	8	yes	—
377	C₃×D₄	C₂³×C₃²	1	2	yes	—
378	C₂×C₃×C₄	—	2	8	no	—
379	C₂×C₃×C₄	—	2	2	no	—
380	C₂×C₃×C₄	—	2	4	no	—
381	C₂×C₃×C₄	—	3	4	no	—
382	C₂×C₃×C₄	—	3	4	no	—
383	C₂×C₃×C₄	—	3	4	no	—
384	C₂×C₃×C₄	—	3	4	no	—
385	C₂×C₃×C₄	—	3	4	no	—
386	C₂×C₃×C₄	—	4	8	no	—
387	C₂×C₃×C₄	—	6	8	no	—
388	C₂×C₃×C₄	—	6	8	no	—
389	C₂×C₃×C₄	—	6	8	no	—
390	C₂×C₃×C₄	—	6	8	no	—
391	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
392	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
393	#67	—	2	2	no	—
394	#67	—	2	2	no	—
395	#67	—	2	2	no	—
396	C₂×Dic₃	—	2	2	no	—
397	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
398	D₁₂	—	2	2	no	—
399	D₁₂	C₂³×C₃²	1	2	yes	—
400	C₄×D₃	—	2	2	no	—
401	C₄×D₃	—	2	2	no	—
402	C₄×D₃	—	2	2	no	—
403	C₄×D₃	C₃²×D₄	1	1	yes	characteristic
404	C₄×D₃	C₃²×D₄	1	1	yes	characteristic
405	C₄×D₃	C₂³×C₃²	1	1	yes	characteristic
406	C₄×D₃	C₂³×C₃²	1	1	yes	characteristic
407	Dic₆	—	2	2	no	—
408	Dic₆	C₂³×C₃²	1	2	yes	—
409	C₃×C₈	—	2	4	no	—
410	C₃×C₈	—	2	4	no	—
411	C₃×C₈	—	3	4	no	—
412	C₃×C₈	—	3	4	no	—
413	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
414	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
415	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
416	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
417	C₃⋊C₈	C₂³×C₃²	1	1	yes	characteristic
418	C₃⋊C₈	C₂³×C₃²	1	1	yes	characteristic
419	C₂×C₃²	—	2	1	no	—
420	C₂×C₃²	—	2	4	no	—
421	C₂×C₃²	—	2	4	no	—
422	C₂×C₃²	—	3	2	no	—
423	C₂×C₃²	—	4	4	no	—
424	C₂×C₃²	—	4	2	no	—
425	C₂×C₃²	—	4	8	no	—
426	C₂×C₃²	—	6	1	no	—
427	C₂×C₃²	—	8	8	no	—
428	C₂×C₃²	—	12	2	no	—
429	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
430	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
431	C₃×D₃	—	2	4	no	—
432	C₃×D₃	—	4	8	no	—
433	C₃×D₃	C₃×#135	1	8	yes	direct product factor
434	#41	—	3	2	no	—
435	#41	—	3	2	no	—
436	#41	—	3	2	no	—
437	#41	—	6	2	no	—
438	#41	—	6	2	no	—
439	#41	C₂×C₃²×D₃	1	2	yes	complemented
440	C₂×Q₈	—	3	2	no	—
441	C₂×D₄	—	3	2	no	—
442	C₂×D₄	—	6	2	no	—
443	C₂²×C₄	—	3	1	no	—
444	C₂²×C₄	—	6	2	no	—
445	Q₁₆	—	3	2	no	—
446	Q₁₆	—	3	1	no	—
447	Q₁₆	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
448	SD₁₆	—	3	2	no	—
449	SD₁₆	—	3	2	no	—
450	SD₁₆	—	3	2	no	—
451	SD₁₆	C₂×C₃²×D₃	1	2	yes	complemented
452	D₈	—	3	2	no	—
453	D₈	—	3	1	no	—
454	D₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
455	C₂×C₈	—	3	1	no	—
456	C₂×C₈	—	3	1	no	—
457	C₂×C₈	—	3	1	no	—
458	C₂×C₈	—	3	1	no	—
459	C₂×C₈	—	3	1	no	—
460	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
461	C₂²×C₃	—	2	8	no	—
462	C₂²×C₃	—	2	2	no	—
463	C₂²×C₃	—	2	4	no	—
464	C₂²×C₃	—	3	4	no	—
465	C₂²×C₃	—	3	4	no	—
466	C₂²×C₃	—	4	8	no	—
467	C₂²×C₃	—	6	8	no	—
468	C₂²×C₃	—	6	8	no	—
469	C₂²×C₃	—	12	16	no	—
470	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
471	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
472	C₂×D₃	—	2	2	no	—
473	C₂×D₃	—	2	2	no	—
474	C₂×D₃	—	4	4	no	—
475	C₂×D₃	C₂×C₃²×D₄	1	1	yes	characteristic
476	C₂×D₃	C₂×C₃²×D₄	1	1	yes	characteristic
477	C₃×C₄	—	2	8	no	—
478	C₃×C₄	—	2	2	no	—
479	C₃×C₄	—	2	4	no	—
480	C₃×C₄	—	2	4	no	—
481	C₃×C₄	—	3	4	no	—
482	C₃×C₄	—	3	4	no	—
483	C₃×C₄	—	4	8	no	—
484	C₃×C₄	—	6	8	no	—
485	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
486	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
487	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
488	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
489	Dic₃	—	2	2	no	—
490	Dic₃	C₂×C₃²×D₄	1	1	yes	characteristic
491	Dic₃	C₂×C₃²×D₄	1	1	yes	characteristic
492	C₃²	—	2	4	no	—
493	C₃²	D₃×#135	1	1	yes	characteristic, direct product factor
494	C₃²	C₂×C₃×#135	1	4	yes	complemented
495	C₂³	—	3	1	no	—
496	C₂³	—	6	2	no	—
497	Q₈	—	3	2	no	—
498	Q₈	—	6	2	no	—
499	Q₈	C₂²×C₃²×D₃	1	2	yes	—
500	D₄	—	2	2	no	—
501	D₄	—	3	2	no	—
502	D₄	—	6	2	no	—
503	D₄	—	6	2	no	—
504	D₄	—	6	2	no	—
505	D₄	—	6	2	no	—
506	D₄	C₂²×C₃²×D₃	1	2	yes	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	3	1	no	—
509	C₂×C₄	—	3	1	no	—
510	C₂×C₄	—	3	1	no	—
511	C₂×C₄	—	3	1	no	—
512	C₂×C₄	—	3	1	no	—
513	C₂×C₄	—	6	2	no	—
514	C₂×C₄	—	6	2	no	—
515	C₂×C₄	—	6	2	no	—
516	C₂×C₄	—	6	2	no	—
517	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
518	C₈	—	3	1	no	—
519	C₈	—	3	1	no	—
520	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
521	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
522	C₂×C₃	—	2	1	no	—
523	C₂×C₃	—	2	4	no	—
524	C₂×C₃	—	2	4	no	—
525	C₂×C₃	—	3	8	no	—
526	C₂×C₃	—	4	4	no	—
527	C₂×C₃	—	4	2	no	—
528	C₂×C₃	—	4	8	no	—
529	C₂×C₃	—	6	4	no	—
530	C₂×C₃	—	8	8	no	—
531	C₂×C₃	—	12	8	no	—
532	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
533	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
534	D₃	—	2	1	no	—
535	D₃	—	4	2	no	—
536	D₃	C₃²×#135	1	2	yes	direct product factor
537	C₂²	—	2	2	no	—
538	C₂²	—	3	1	no	—
539	C₂²	—	3	1	no	—
540	C₂²	—	6	2	no	—
541	C₂²	—	6	2	no	—
542	C₂²	—	12	4	no	—
543	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	3	1	no	—
546	C₄	—	3	1	no	—
547	C₄	—	6	2	no	—
548	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
549	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
550	C₃	—	2	4	no	—
551	C₃	C₃×D₃×#135	1	4	yes	direct product factor
552	C₃	C₂×C₃²×#135	1	1	yes	characteristic, complemented
553	C₂	—	2	1	no	—
554	C₂	—	3	2	no	—
555	C₂	—	4	2	no	—
556	C₂	—	6	1	no	—
557	C₂	—	12	2	no	—
558	C₂	C₂×C₃²×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₃³×#135	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₃²×#135	D₃
hypocenter	C₃	C₂×C₃²×#135
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ C₃²×#135

Complements

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

N	H ≅ G/N	type
1. C₃²×D₃×#41	C₂	semidirect
2. C₃³×#135	C₂	semidirect
3. C₃²×#950	C₂	semidirect
4. C₃²×#919	C₂	semidirect
5. C₃²×D₃×Q₁₆	C₂	semidirect
6. C₃²×#916	C₂	semidirect
7. C₃²×D₃×SD₁₆	C₂	semidirect
8. C₃²×#912	C₂	semidirect
9. C₃²×D₃×D₈	C₂	semidirect
10. C₃²×#904	C₂	semidirect
11. C₂×C₃²×D₃×C₈	C₂	semidirect
13. C₃×D₃×#135	C₃	direct (N × H)
14. C₃³×#41	C₂²	semidirect
15. C₃²×#239	C₂²	semidirect
18. C₃²×#237	C₂²	semidirect
22. C₃²×#235	C₂²	semidirect
25. C₃³×Q₁₆	C₂²	semidirect
26. C₃³×SD₁₆	C₂²	semidirect
27. C₃³×D₈	C₂²	semidirect
28. C₂×C₃³×C₈	C₂²	semidirect
29. C₃²×#216	C₂²	semidirect
30. C₃²×#215	C₂²	semidirect
31. C₃²×#214	C₂²	semidirect
32. C₃²×#213	C₂²	semidirect
33. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
34. C₃²×Dic₁₂	C₂²	semidirect
35. C₃²×D₂₄	C₂²	semidirect
36. C₃²×#204	C₂²	semidirect
39. C₃²×D₃×C₈	C₂²	semidirect
40. C₃²×D₃×C₈	C₂²	semidirect
42. C₃×D₃×#41	C₂×C₃	semidirect
49. C₃²×#135	D₃	direct (N × H)
50. C₃²×#135	C₂×C₃	semidirect
55. C₃×#950	C₂×C₃	semidirect
56. C₃×#919	C₂×C₃	semidirect
57. C₃×D₃×Q₁₆	C₂×C₃	semidirect
58. C₃×#916	C₂×C₃	semidirect
59. C₃×D₃×SD₁₆	C₂×C₃	semidirect
60. C₃×#912	C₂×C₃	semidirect
61. C₃×D₃×D₈	C₂×C₃	semidirect
62. C₃×#904	C₂×C₃	semidirect
63. C₂×C₃×D₃×C₈	C₂×C₃	semidirect
92. D₃×#135	C₃²	direct (N × H)
99. C₃²×#41	C₂×D₃	semidirect
100. C₃²×#41	C₂²×C₃	semidirect
106. C₃×#239	C₂²×C₃	semidirect
109. C₃×#237	C₂²×C₃	semidirect
113. C₃×#235	C₂²×C₃	semidirect
119. C₃²×Q₁₆	C₂×D₃	semidirect
120. C₃²×Q₁₆	C₂²×C₃	semidirect
125. C₃²×SD₁₆	C₂×D₃	semidirect
126. C₃²×SD₁₆	C₂²×C₃	semidirect
130. C₃²×D₈	C₂×D₃	semidirect
131. C₃²×D₈	C₂²×C₃	semidirect
138. C₂×C₃²×C₈	C₂×D₃	semidirect
139. C₂×C₃²×C₈	C₂²×C₃	semidirect
140. C₃×#216	C₂²×C₃	semidirect
141. C₃×#215	C₂²×C₃	semidirect
142. C₃×#214	C₂²×C₃	semidirect
143. C₃×#213	C₂²×C₃	semidirect
144. C₂×C₃×C₃⋊C₈	C₂²×C₃	semidirect
145. C₃×Dic₁₂	C₂²×C₃	semidirect
146. C₃×D₂₄	C₂²×C₃	semidirect
147. C₃×#204	C₂²×C₃	semidirect
150. C₃×D₃×C₈	C₂²×C₃	semidirect
151. C₃×D₃×C₈	C₂²×C₃	semidirect
166. D₃×#41	C₂×C₃²	semidirect
173. C₃×#135	C₃×D₃	direct (N × H)
174. C₃×#135	C₂×C₃²	semidirect
179. #950	C₂×C₃²	semidirect
180. #919	C₂×C₃²	semidirect
181. D₃×Q₁₆	C₂×C₃²	semidirect
182. #916	C₂×C₃²	semidirect
183. D₃×SD₁₆	C₂×C₃²	semidirect
184. #912	C₂×C₃²	semidirect
185. D₃×D₈	C₂×C₃²	semidirect
186. #904	C₂×C₃²	semidirect
187. C₂×D₃×C₈	C₂×C₃²	semidirect
255. C₃²×D₃	#135	direct (N × H)
262. C₃×#41	C₂×C₃×D₃	semidirect
263. C₃×#41	C₂²×C₃²	semidirect
269. #239	C₂²×C₃²	semidirect
272. #237	C₂²×C₃²	semidirect
276. #235	C₂²×C₃²	semidirect
282. C₃×Q₁₆	C₂×C₃×D₃	semidirect
283. C₃×Q₁₆	C₂²×C₃²	semidirect
288. C₃×SD₁₆	C₂×C₃×D₃	semidirect
289. C₃×SD₁₆	C₂²×C₃²	semidirect
293. C₃×D₈	C₂×C₃×D₃	semidirect
294. C₃×D₈	C₂²×C₃²	semidirect
301. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
302. C₂×C₃×C₈	C₂²×C₃²	semidirect
303. #216	C₂²×C₃²	semidirect
304. #215	C₂²×C₃²	semidirect
305. #214	C₂²×C₃²	semidirect
306. #213	C₂²×C₃²	semidirect
307. C₂×C₃⋊C₈	C₂²×C₃²	semidirect
308. Dic₁₂	C₂²×C₃²	semidirect
309. D₂₄	C₂²×C₃²	semidirect
310. #204	C₂²×C₃²	semidirect
313. D₃×C₈	C₂²×C₃²	semidirect
314. D₃×C₈	C₂²×C₃²	semidirect
352. #135	C₃²×D₃	direct (N × H)
357. C₃³	C₂×#135	semidirect
433. C₃×D₃	C₃×#135	direct (N × H)
439. #41	C₂×C₃²×D₃	semidirect
447. Q₁₆	C₂×C₃²×D₃	semidirect
451. SD₁₆	C₂×C₃²×D₃	semidirect
454. D₈	C₂×C₃²×D₃	semidirect
460. C₂×C₈	C₂×C₃²×D₃	semidirect
493. C₃²	D₃×#135	direct (N × H)
494. C₃²	C₂×C₃×#135	semidirect
536. D₃	C₃²×#135	direct (N × H)
551. C₃	C₃×D₃×#135	direct (N × H)
552. C₃	C₂×C₃²×#135	semidirect

Maximal subgroups

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

subgroup	index
C₃²×D₃×#41	2
C₃³×#135	2
C₃²×#950	2
C₃²×#919	2
C₃²×D₃×Q₁₆	2
C₃²×#916	2
C₃²×D₃×SD₁₆	2
C₃²×#912	2
C₃²×D₃×D₈	2
C₃²×#904	2
C₂×C₃²×D₃×C₈	2
C₂×C₃²×#135	3
C₃×D₃×#135	3

Subgroups — C32×D3×#135 #13179144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₃×#135 #13179144