Subgroups — C₃×#10596200 #13179480

Order 1728.

1508 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64155	C₂	1	2	yes	maximal
3	C₃×#64153	C₂	1	2	yes	maximal, complemented
4	C₃×#64105	C₂	1	2	yes	maximal
5	C₃×#64101	C₂	1	1	yes	maximal, characteristic
6	C₃×C₄⋊C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
7	C₃×#64099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#64098	C₂	1	2	yes	maximal, complemented
9	C₃×#64097	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#64081	C₂	1	1	yes	maximal, characteristic
11	C₃×#64079	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596200	C₃	1	1	yes	maximal, characteristic, direct product factor
13	C₃×#5729	—	3	4	no	maximal
14	C₂×C₃³×Q₈	C₂²	1	2	yes	—
15	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	2	yes	—
16	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×#634	C₂²	1	2	yes	—
18	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃³×C₄²	C₂²	1	1	yes	characteristic
22	C₃×#3752	C₂²	1	1	yes	characteristic
23	C₃×#3752	C₂²	1	1	yes	characteristic
24	C₃×#3752	C₂²	1	2	yes	—
25	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
28	C₃×#3750	C₂²	1	2	yes	—
29	C₃×#3750	C₂²	1	2	yes	—
30	C₃×#3750	C₂²	1	1	yes	characteristic
31	C₃×#3750	C₂²	1	1	yes	characteristic
32	C₃×#3750	C₂²	1	2	yes	—
33	C₃×#3750	C₂²	1	2	yes	—
34	C₃×#3750	C₂²	1	2	yes	—
35	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
36	C₃×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
37	C₃²×C₄⋊Q₈	—	2	4	no	—
38	C₃²×C₄⋊Q₈	D₃	1	4	yes	complemented
39	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
40	#64155	C₂×C₃	1	2	yes	—
41	#64153	C₂×C₃	1	2	yes	complemented
42	#64105	C₂×C₃	1	2	yes	—
43	#64101	C₂×C₃	1	1	yes	characteristic
44	C₄⋊C₄×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
45	#64099	C₂×C₃	1	1	yes	characteristic, complemented
46	#64098	C₂×C₃	1	2	yes	complemented
47	#64097	C₂×C₃	1	1	yes	characteristic, complemented
48	#64081	C₂×C₃	1	1	yes	characteristic
49	#64079	C₂×C₃	1	1	yes	characteristic, complemented
50	C₃×#946	—	3	8	no	—
51	C₃×#944	—	3	8	no	—
52	C₃×#896	—	3	8	no	—
53	C₃×#892	—	3	4	no	—
54	C₃×D₃×C₄⋊C₄	—	3	4	no	—
55	C₃×#890	—	3	4	no	—
56	C₃×#889	—	3	8	no	—
57	C₃×#888	—	3	4	no	—
58	C₃×#872	—	3	4	no	—
59	C₃×#870	—	3	4	no	—
60	C₂²×C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
61	C₃³×Q₈	—	4	2	no	—
62	C₂×C₃³×C₄	C₂³	1	2	yes	—
63	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
64	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
65	C₂×C₃³×C₄	C₂³	1	2	yes	—
66	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
69	C₂×C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
71	C₂×C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
72	C₃×C₄×C₃⋊D₃	—	4	2	no	—
73	C₃×C₄×C₃⋊D₃	Q₈	1	4	yes	complemented
74	C₃×#634	—	4	2	no	—
75	C₃×#544	—	9	1	no	—
76	#5729	—	3	4	no	—
77	C₂×C₃²×Q₈	—	2	8	no	—
78	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
79	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
80	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	2	yes	—
81	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
82	C₂×#634	C₂²×C₃	1	2	yes	—
83	C₂×C₃×C₄×D₃	—	3	4	no	—
84	C₂×C₃×C₄×D₃	—	3	8	no	—
85	C₂×C₃×Dic₆	—	3	8	no	—
86	C₃²×C₄⋊C₄	—	2	4	no	—
87	C₃²×C₄⋊C₄	—	2	8	no	—
88	C₃²×C₄⋊C₄	—	2	4	no	—
89	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
90	C₃²×C₄⋊C₄	C₂×D₃	1	8	yes	—
91	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
92	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
94	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₃²×C₄²	—	2	4	no	—
96	C₃²×C₄²	C₂×D₃	1	4	yes	—
97	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
98	#3752	C₂²×C₃	1	1	yes	characteristic
99	#3752	C₂²×C₃	1	1	yes	characteristic
100	#3752	C₂²×C₃	1	2	yes	—
101	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
102	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
103	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
104	#3750	C₂²×C₃	1	2	yes	—
105	#3750	C₂²×C₃	1	2	yes	—
106	#3750	C₂²×C₃	1	1	yes	characteristic
107	#3750	C₂²×C₃	1	1	yes	characteristic
108	#3750	C₂²×C₃	1	2	yes	—
109	#3750	C₂²×C₃	1	2	yes	—
110	#3750	C₂²×C₃	1	2	yes	—
111	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
112	C₄×C₃⋊Dic₃	C₂²×C₃	1	2	yes	—
113	C₃×(C₂×D₃)⋊C₄	—	3	8	no	—
114	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
115	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
116	C₃×C₄⋊Dic₃	—	3	4	no	—
117	C₃×C₄⋊Dic₃	—	3	8	no	—
118	C₃×C₄⋊Dic₃	—	3	4	no	—
119	C₃×Dic₃⋊C₄	—	3	8	no	—
120	C₃×Dic₃⋊C₄	—	3	4	no	—
121	C₃×Dic₃⋊C₄	—	3	8	no	—
122	C₃×Dic₃⋊C₄	—	3	8	no	—
123	C₃×Dic₃⋊C₄	—	3	8	no	—
124	C₃×Dic₃⋊C₄	—	3	4	no	—
125	C₃×Dic₃⋊C₄	—	3	8	no	—
126	C₃×C₄×Dic₃	—	3	4	no	—
127	C₃×C₄×Dic₃	—	3	8	no	—
128	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
129	C₂×C₃×C₃⋊D₃	—	2	1	no	—
130	C₂×C₃×C₃⋊D₃	—	2	1	no	—
131	C₂×C₃×C₃⋊D₃	C₂×Q₈	1	2	yes	—
132	C₃³×C₄	—	2	1	no	—
133	C₃³×C₄	—	2	2	no	—
134	C₃³×C₄	—	2	2	no	—
135	C₃³×C₄	—	2	1	no	—
136	C₃³×C₄	C₂×Q₈	1	2	yes	—
137	C₃×C₃⋊Dic₃	—	2	1	no	—
138	C₃×C₃⋊Dic₃	—	2	2	no	—
139	C₃×C₃⋊Dic₃	—	2	1	no	—
140	C₃×C₃⋊Dic₃	—	2	2	no	—
141	C₃×C₃⋊Dic₃	C₂×Q₈	1	2	yes	—
142	C₃×C₄⋊Q₈	—	2	4	no	—
143	C₃×C₄⋊Q₈	—	9	2	no	—
144	C₃×C₄⋊Q₈	—	9	1	no	—
145	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
146	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
147	C₃×#125	—	9	2	no	—
148	C₃×#125	—	9	1	no	—
149	C₃×#125	—	9	1	no	—
150	C₃×C₂²⋊Q₈	—	9	2	no	—
151	C₃×C₂²⋊Q₈	—	9	2	no	—
152	C₃×#117	—	9	1	no	—
153	C₃×#117	—	9	1	no	—
154	C₂×C₃×C₄⋊C₄	—	9	1	no	—
155	#946	—	3	8	no	—
156	#944	—	3	8	no	—
157	#896	—	3	8	no	—
158	#892	—	3	4	no	—
159	D₃×C₄⋊C₄	—	3	4	no	—
160	#890	—	3	4	no	—
161	#889	—	3	8	no	—
162	#888	—	3	4	no	—
163	#872	—	3	4	no	—
164	#870	—	3	4	no	—
165	C₂²×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
166	C₂²×C₃×D₃	—	3	4	no	—
167	C₃²×Q₈	—	4	8	no	—
168	C₃²×Q₈	—	4	2	no	—
169	C₃²×Q₈	—	4	16	no	—
170	C₂×C₃²×C₄	—	2	4	no	—
171	C₂×C₃²×C₄	—	2	4	no	—
172	C₂×C₃²×C₄	—	2	4	no	—
173	C₂×C₃²×C₄	—	2	8	no	—
174	C₂×C₃²×C₄	—	2	8	no	—
175	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
176	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
177	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
178	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
179	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
180	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
181	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
182	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
183	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
184	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
185	C₂×C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
186	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
187	C₂×C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
188	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
189	C₂×C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
190	C₄×C₃⋊D₃	—	4	2	no	—
191	C₄×C₃⋊D₃	C₃×Q₈	1	4	yes	complemented
192	#634	—	4	2	no	—
193	C₂×C₃×Dic₃	—	3	4	no	—
194	C₂×C₃×Dic₃	—	3	4	no	—
195	C₂×C₃×Dic₃	—	3	8	no	—
196	C₂×C₃×Dic₃	—	3	4	no	—
197	C₂×C₃×Dic₃	—	3	8	no	—
198	C₃×C₄×D₃	—	3	16	no	—
199	C₃×C₄×D₃	—	12	8	no	—
200	C₃×Dic₆	—	12	8	no	—
201	#544	—	9	1	no	Sylow
202	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
203	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
205	C₃×C₃⋊D₃	—	2	2	no	—
206	C₂×C₃×Q₈	—	2	8	no	—
207	C₂×C₃×Q₈	—	9	2	no	—
208	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
209	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
210	C₂²×C₃×C₄	—	9	1	no	—
211	C₂²×C₃×C₄	—	9	2	no	—
212	C₂×C₄×D₃	—	3	4	no	—
213	C₂×C₄×D₃	—	3	8	no	—
214	C₂×Dic₆	—	3	8	no	—
215	C₃×C₄⋊C₄	—	2	4	no	—
216	C₃×C₄⋊C₄	—	2	4	no	—
217	C₃×C₄⋊C₄	—	2	8	no	—
218	C₃×C₄⋊C₄	—	9	2	no	—
219	C₃×C₄⋊C₄	—	9	1	no	—
220	C₃×C₄⋊C₄	—	9	1	no	—
221	C₃×C₄⋊C₄	—	9	2	no	—
222	C₃×C₄⋊C₄	—	9	2	no	—
223	C₃×C₄⋊C₄	—	9	2	no	—
224	C₃×C₄⋊C₄	—	9	2	no	—
225	C₃×C₄⋊C₄	—	9	1	no	—
226	C₃×C₄⋊C₄	—	9	1	no	—
227	C₃×C₄⋊C₄	—	9	2	no	—
228	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
229	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
230	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
231	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
232	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
233	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
234	C₃×C₂²⋊C₄	—	9	2	no	—
235	C₃×C₂²⋊C₄	—	9	1	no	—
236	C₃×C₂²⋊C₄	—	9	1	no	—
237	C₃×C₄²	—	2	4	no	—
238	C₃×C₄²	—	9	2	no	—
239	C₃×C₄²	—	9	1	no	—
240	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
241	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
242	(C₂×D₃)⋊C₄	—	3	4	no	—
243	(C₂×D₃)⋊C₄	—	3	4	no	—
244	(C₂×D₃)⋊C₄	—	3	8	no	—
245	C₄⋊Dic₃	—	3	4	no	—
246	C₄⋊Dic₃	—	3	4	no	—
247	C₄⋊Dic₃	—	3	8	no	—
248	Dic₃⋊C₄	—	3	8	no	—
249	Dic₃⋊C₄	—	3	8	no	—
250	Dic₃⋊C₄	—	3	4	no	—
251	Dic₃⋊C₄	—	3	4	no	—
252	Dic₃⋊C₄	—	3	8	no	—
253	Dic₃⋊C₄	—	3	8	no	—
254	Dic₃⋊C₄	—	3	8	no	—
255	C₄×Dic₃	—	3	4	no	—
256	C₄×Dic₃	—	3	8	no	—
257	C₂²×C₃²	—	2	4	no	—
258	C₂²×C₃²	C₂³×D₃	1	4	yes	—
259	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
260	C₂×C₃⋊D₃	—	2	1	no	—
261	C₂×C₃⋊D₃	—	2	1	no	—
262	C₂×C₃⋊D₃	C₂×C₃×Q₈	1	2	yes	—
263	C₂×C₃×D₃	—	3	8	no	—
264	C₂×C₃×D₃	—	6	4	no	—
265	C₂×C₃×D₃	—	6	4	no	—
266	C₃²×C₄	—	2	8	no	—
267	C₃²×C₄	—	2	8	no	—
268	C₃²×C₄	—	2	4	no	—
269	C₃²×C₄	—	2	4	no	—
270	C₃²×C₄	—	2	2	no	—
271	C₃²×C₄	—	2	2	no	—
272	C₃²×C₄	—	2	1	no	—
273	C₃²×C₄	—	2	1	no	—
274	C₃²×C₄	—	2	8	no	—
275	C₃²×C₄	—	2	8	no	—
276	C₃²×C₄	—	2	8	no	—
277	C₃²×C₄	—	4	8	no	—
278	C₃²×C₄	—	4	8	no	—
279	C₃²×C₄	D₃×Q₈	1	8	yes	—
280	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
281	C₃⋊Dic₃	—	2	1	no	—
282	C₃⋊Dic₃	—	2	2	no	—
283	C₃⋊Dic₃	—	2	1	no	—
284	C₃⋊Dic₃	—	2	2	no	—
285	C₃⋊Dic₃	C₂×C₃×Q₈	1	2	yes	—
286	C₃×Dic₃	—	3	8	no	—
287	C₃×Dic₃	—	6	4	no	—
288	C₃×Dic₃	—	6	8	no	—
289	C₃×Dic₃	—	6	4	no	—
290	C₃×Dic₃	—	6	8	no	—
291	C₄⋊Q₈	—	9	2	no	—
292	C₄⋊Q₈	—	9	1	no	—
293	C₄⋊Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
294	#125	—	9	2	no	—
295	#125	—	9	1	no	—
296	#125	—	9	1	no	—
297	C₂²⋊Q₈	—	9	2	no	—
298	C₂²⋊Q₈	—	9	2	no	—
299	#117	—	9	1	no	—
300	#117	—	9	1	no	—
301	C₂×C₄⋊C₄	—	9	1	no	—
302	C₃³	#544	1	1	yes	characteristic, complemented, Sylow
303	C₂³×C₃	—	9	1	no	—
304	C₂²×D₃	—	3	4	no	—
305	C₃×Q₈	—	4	8	no	—
306	C₃×Q₈	—	4	2	no	—
307	C₃×Q₈	—	4	16	no	—
308	C₃×Q₈	—	36	2	no	—
309	C₂×C₃×C₄	—	2	4	no	—
310	C₂×C₃×C₄	—	2	4	no	—
311	C₂×C₃×C₄	—	2	4	no	—
312	C₂×C₃×C₄	—	2	8	no	—
313	C₂×C₃×C₄	—	2	8	no	—
314	C₂×C₃×C₄	—	9	1	no	—
315	C₂×C₃×C₄	—	9	4	no	—
316	C₂×C₃×C₄	—	9	1	no	—
317	C₂×C₃×C₄	—	9	2	no	—
318	C₂×C₃×C₄	—	9	1	no	—
319	C₂×C₃×C₄	—	9	2	no	—
320	C₂×C₃×C₄	—	36	2	no	—
321	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
322	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
323	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
324	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
325	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
326	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
327	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
328	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
329	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
330	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
331	C₂×Dic₃	—	3	4	no	—
332	C₂×Dic₃	—	3	4	no	—
333	C₂×Dic₃	—	3	8	no	—
334	C₂×Dic₃	—	3	4	no	—
335	C₂×Dic₃	—	3	8	no	—
336	C₄×D₃	—	3	16	no	—
337	C₄×D₃	—	12	8	no	—
338	Dic₆	—	12	8	no	—
339	C₂×C₃²	—	2	4	no	—
340	C₂×C₃²	—	2	4	no	—
341	C₂×C₃²	—	2	4	no	—
342	C₂×C₃²	#1004	1	4	yes	—
343	C₂×C₃²	C₂×D₃×Q₈	1	4	yes	—
344	C₂×C₃²	#1007	1	4	yes	—
345	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
346	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
347	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
348	C₃⋊D₃	—	2	2	no	—
349	C₃×D₃	—	6	8	no	—
350	C₂×Q₈	—	9	2	no	—
351	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
352	C₂²×C₄	—	9	1	no	—
353	C₂²×C₄	—	9	2	no	—
354	C₄⋊C₄	—	9	2	no	—
355	C₄⋊C₄	—	9	1	no	—
356	C₄⋊C₄	—	9	1	no	—
357	C₄⋊C₄	—	9	1	no	—
358	C₄⋊C₄	—	9	1	no	—
359	C₄⋊C₄	—	9	2	no	—
360	C₄⋊C₄	—	9	2	no	—
361	C₄⋊C₄	—	9	2	no	—
362	C₄⋊C₄	—	9	2	no	—
363	C₄⋊C₄	—	9	2	no	—
364	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
365	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
366	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
367	C₂²⋊C₄	—	9	2	no	—
368	C₂²⋊C₄	—	9	1	no	—
369	C₂²⋊C₄	—	9	1	no	—
370	C₄²	—	9	2	no	—
371	C₄²	—	9	1	no	—
372	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
373	C₂²×C₃	—	2	4	no	—
374	C₂²×C₃	—	9	2	no	—
375	C₂²×C₃	—	18	1	no	—
376	C₂²×C₃	—	18	1	no	—
377	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
378	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
379	C₂×D₃	—	3	8	no	—
380	C₂×D₃	—	6	4	no	—
381	C₂×D₃	—	6	4	no	—
382	C₃×C₄	—	2	8	no	—
383	C₃×C₄	—	2	8	no	—
384	C₃×C₄	—	2	8	no	—
385	C₃×C₄	—	2	8	no	—
386	C₃×C₄	—	2	4	no	—
387	C₃×C₄	—	2	4	no	—
388	C₃×C₄	—	2	2	no	—
389	C₃×C₄	—	2	1	no	—
390	C₃×C₄	—	2	2	no	—
391	C₃×C₄	—	2	1	no	—
392	C₃×C₄	—	2	8	no	—
393	C₃×C₄	—	4	8	no	—
394	C₃×C₄	—	4	8	no	—
395	C₃×C₄	—	9	2	no	—
396	C₃×C₄	—	18	2	no	—
397	C₃×C₄	—	18	1	no	—
398	C₃×C₄	—	18	1	no	—
399	C₃×C₄	—	18	2	no	—
400	C₃×C₄	C₃×D₃×Q₈	1	8	yes	—
401	C₃×C₄	Q₈×C₃⋊D₃	1	2	yes	—
402	Dic₃	—	3	8	no	—
403	Dic₃	—	6	4	no	—
404	Dic₃	—	6	8	no	—
405	Dic₃	—	6	4	no	—
406	Dic₃	—	6	8	no	—
407	C₃²	—	2	4	no	—
408	C₃²	#5729	1	4	yes	complemented
409	C₃²	C₃×#544	1	1	yes	characteristic, complemented
410	C₂³	—	9	1	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	36	2	no	—
413	C₂×C₄	—	9	1	no	—
414	C₂×C₄	—	9	4	no	—
415	C₂×C₄	—	9	1	no	—
416	C₂×C₄	—	9	2	no	—
417	C₂×C₄	—	9	1	no	—
418	C₂×C₄	—	9	2	no	—
419	C₂×C₄	—	36	2	no	—
420	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
421	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
422	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
423	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
424	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
425	C₂×C₃	—	2	4	no	—
426	C₂×C₃	—	2	4	no	—
427	C₂×C₃	—	2	4	no	—
428	C₂×C₃	—	18	2	no	—
429	C₂×C₃	C₃×#1004	1	4	yes	—
430	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
431	C₂×C₃	C₃×#1007	1	4	yes	—
432	C₂×C₃	#64361	1	1	yes	characteristic
433	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
434	C₂×C₃	#64364	1	1	yes	characteristic
435	D₃	—	6	8	no	—
436	C₂²	—	9	2	no	—
437	C₂²	—	18	1	no	—
438	C₂²	—	18	1	no	—
439	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
440	C₄	—	2	2	no	—
441	C₄	—	2	1	no	—
442	C₄	—	2	2	no	—
443	C₄	—	2	1	no	—
444	C₄	—	9	2	no	—
445	C₄	—	18	2	no	—
446	C₄	—	18	1	no	—
447	C₄	—	18	1	no	—
448	C₄	—	18	2	no	—
449	C₄	C₃×Q₈×C₃⋊D₃	1	2	yes	—
450	C₃	—	2	4	no	—
451	C₃	C₃×#5729	1	4	yes	complemented
452	C₃	#10596200	1	1	yes	characteristic, direct product factor
453	C₂	—	18	2	no	—
454	C₂	C₃×#64361	1	1	yes	characteristic
455	C₂	C₂×C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
456	C₂	C₃×#64364	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle	C₂²×C₃³	C₂⁴
prime radical	C₂²×C₃×C₃⋊D₃	C₂³
hypercenter	C₃×C₄⋊Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×#544
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄⋊Q₈	C₂	semidirect
3. C₃×#64153	C₂	semidirect
7. C₃×#64099	C₂	semidirect
8. C₃×#64098	C₂	semidirect
9. C₃×#64097	C₂	semidirect
11. C₃×#64079	C₂	semidirect
12. #10596200	C₃	direct (N × H)
38. C₃²×C₄⋊Q₈	D₃	semidirect
39. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
41. #64153	C₂×C₃	semidirect
45. #64099	C₂×C₃	semidirect
46. #64098	C₂×C₃	semidirect
47. #64097	C₂×C₃	semidirect
49. #64079	C₂×C₃	semidirect
73. C₃×C₄×C₃⋊D₃	Q₈	semidirect
145. C₃×C₄⋊Q₈	C₃×D₃	semidirect
146. C₃×C₄⋊Q₈	C₃⋊D₃	semidirect
191. C₄×C₃⋊D₃	C₃×Q₈	semidirect
293. C₄⋊Q₈	C₃×C₃⋊D₃	semidirect
302. C₃³	#544	semidirect
408. C₃²	#5729	semidirect
409. C₃²	C₃×#544	semidirect
451. C₃	C₃×#5729	semidirect
452. C₃	#10596200	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₃×#64155	2
C₃×#64153	2
C₃×#64105	2
C₃×#64101	2
C₃×C₄⋊C₄×C₃⋊D₃	2
C₃×#64099	2
C₃×#64098	2
C₃×#64097	2
C₃×#64081	2
C₃×#64079	2
#10596200	3
C₃×#5729	3

Subgroups — C3×#10596200 #13179480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596200 #13179480