Subgroups — #11786233

Order 1152.

625 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63645	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10591349	C₂	1	2	yes	maximal
5	C₂×#63575	C₂	1	2	yes	maximal
6	#70606	—	3	1	no	maximal
7	#69765	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Dic₃⋊Dic₃	—	2	4	no	—
14	#63645	—	2	2	no	—
15	C₃×#818	—	2	2	no	—
16	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
17	#63575	—	2	4	no	—
18	C₂×C₃×#107	—	3	1	no	—
19	C₂×C₃×C₄×C₈	—	3	1	no	—
20	C₂×#818	—	3	1	no	—
21	C₂×#818	—	3	1	no	—
22	C₂×#818	—	3	2	no	—
23	C₂×#818	—	6	1	no	—
24	C₂×#818	D₃	1	1	yes	characteristic
25	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
26	#4975	—	3	2	no	—
27	C₂×#807	—	3	2	no	—
28	#4935	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂²×C₃×Dic₃	D₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₂×C₃²×C₈	Q₈	1	2	yes	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₃×C₄⋊Dic₃	—	2	2	no	—
39	C₃×C₄⋊Dic₃	—	2	2	no	—
40	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
41	C₃×C₄×Dic₃	Q₈	1	2	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
45	Dic₃⋊Dic₃	—	4	8	no	—
46	#1845	—	9	1	no	Sylow
47	C₂²×C₃×C₈	—	2	1	no	—
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄²	—	3	1	no	—
53	C₂×C₄⋊Dic₃	—	3	2	no	—
54	C₂×C₄⋊Dic₃	—	3	2	no	—
55	C₂×C₄⋊Dic₃	—	6	2	no	—
56	C₂×C₄⋊Dic₃	—	6	4	no	—
57	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
58	C₂×Dic₃⋊C₄	—	6	4	no	—
59	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×#107	—	6	2	no	—
62	C₃×C₄×C₈	—	3	4	no	—
63	#818	—	2	2	no	—
64	#818	—	6	4	no	—
65	#818	—	6	4	no	—
66	#818	—	6	2	no	—
67	#818	—	6	2	no	—
68	C₈×Dic₃	Dic₃	1	4	yes	complemented
69	#807	—	6	4	no	—
70	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃⋊Dic₃	—	4	2	no	—
78	C₂×C₃⋊Dic₃	—	4	4	no	—
79	C₂×C₃×Dic₃	—	4	2	no	—
80	C₂×C₃×Dic₃	—	4	4	no	—
81	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
82	C₂×C₃×Dic₃	D₈	1	4	yes	—
83	C₂×C₃×Dic₃	Q₁₆	1	4	yes	—
84	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
85	C₃×C₃⋊C₈	C₄⋊C₄	1	4	yes	complemented
86	C₂×#107	—	3	1	no	—
87	C₂×#107	—	9	2	no	—
88	C₂×#107	—	9	1	no	—
89	C₂×C₄×C₈	—	3	1	no	—
90	#384	—	9	2	no	—
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	—	3	2	no	—
93	C₂²×C₃×C₄	—	6	2	no	—
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂²×Dic₃	—	2	2	no	—
97	C₂²×Dic₃	—	6	2	no	—
98	C₂²×Dic₃	—	6	2	no	—
99	C₂²×Dic₃	—	12	2	no	—
100	C₂²×Dic₃	D₁₂	1	2	yes	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	3	2	no	—
105	C₂×C₃×C₈	—	3	2	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	Dic₆	1	2	yes	—
108	C₂×C₃×C₈	Dic₆	1	2	yes	—
109	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
110	C₂×C₃×C₈	D₁₂	1	2	yes	—
111	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
112	C₂×C₃×C₈	#67	1	2	yes	—
113	C₃×C₄⋊C₄	—	6	2	no	—
114	C₃×C₄⋊C₄	—	6	2	no	—
115	C₃×C₄²	—	3	2	no	—
116	C₃×C₄²	—	3	2	no	—
117	C₄⋊Dic₃	—	2	2	no	—
118	C₄⋊Dic₃	—	2	2	no	—
119	C₄⋊Dic₃	—	6	4	no	—
120	C₄⋊Dic₃	—	6	2	no	—
121	C₄⋊Dic₃	—	6	2	no	—
122	C₄⋊Dic₃	—	6	2	no	—
123	C₄⋊Dic₃	—	6	2	no	—
124	C₄⋊Dic₃	—	12	2	no	—
125	C₄⋊Dic₃	—	12	8	no	—
126	Dic₃⋊C₄	—	12	8	no	—
127	C₄×Dic₃	Dic₆	1	2	yes	—
128	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
129	C₂×C₃⋊C₈	Dic₆	1	2	yes	—
130	C₂×C₃⋊C₈	D₁₂	1	2	yes	—
131	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
132	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₃²	#127	1	1	yes	characteristic
134	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
135	C₂²×C₃²	C₂×D₈	1	2	yes	—
136	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
137	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
140	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
141	C₃⋊Dic₃	—	8	2	no	—
142	C₃×Dic₃	—	8	2	no	—
143	C₃×Dic₃	#107	1	8	yes	complemented
144	C₂²×C₈	—	3	1	no	—
145	C₂²×C₈	D₃²	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	3	2	no	—
147	C₂×C₄⋊C₄	—	9	2	no	—
148	C₂×C₄⋊C₄	—	18	4	no	—
149	C₂×C₄²	—	3	1	no	—
150	#107	—	6	2	no	—
151	#107	—	18	4	no	—
152	#107	—	18	2	no	—
153	C₄×C₈	—	3	4	no	—
154	C₂³×C₃	—	2	1	no	—
155	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
156	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	3	4	no	—
164	C₂×C₃×C₄	—	3	4	no	—
165	C₂×C₃×C₄	—	3	4	no	—
166	C₂×C₃×C₄	—	12	2	no	—
167	C₂×C₃×C₄	—	12	4	no	—
168	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
177	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
180	C₂×Dic₃	—	4	4	no	—
181	C₂×Dic₃	—	4	2	no	—
182	C₂×Dic₃	—	12	4	no	—
183	C₂×Dic₃	—	12	2	no	—
184	C₂×Dic₃	—	12	4	no	—
185	C₂×Dic₃	—	12	4	no	—
186	C₂×Dic₃	—	12	2	no	—
187	C₂×Dic₃	—	24	4	no	—
188	C₂×Dic₃	D₂₄	1	4	yes	—
189	C₂×Dic₃	Dic₁₂	1	4	yes	—
190	C₂×Dic₃	C₄⋊Dic₃	1	4	yes	—
191	C₃×C₈	—	2	4	no	—
192	C₃×C₈	—	3	4	no	—
193	C₃×C₈	Dic₃⋊C₄	1	4	yes	complemented
194	C₃×C₈	C₄⋊Dic₃	1	4	yes	complemented
195	C₃⋊C₈	C₄⋊Dic₃	1	4	yes	complemented
196	C₂×C₃²	#384	1	1	yes	characteristic
197	C₂×C₃²	C₂×#107	1	2	yes	—
198	C₂×C₃²	#493	1	1	yes	characteristic
199	C₂×C₃²	#494	1	1	yes	characteristic
200	C₂×C₃²	#500	1	2	yes	—
201	C₂²×C₄	—	3	2	no	—
202	C₂²×C₄	—	6	2	no	—
203	C₂²×C₄	—	18	2	no	—
204	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
205	C₂×C₈	—	3	2	no	—
206	C₂×C₈	—	3	2	no	—
207	C₂×C₈	—	3	2	no	—
208	C₂×C₈	D₃×Dic₃	1	2	yes	—
209	C₂×C₈	#626	1	2	yes	—
210	C₂×C₈	#627	1	2	yes	—
211	C₄⋊C₄	—	6	2	no	—
212	C₄⋊C₄	—	6	2	no	—
213	C₄⋊C₄	—	18	2	no	—
214	C₄⋊C₄	—	18	2	no	—
215	C₄⋊C₄	—	36	8	no	—
216	C₄²	—	3	2	no	—
217	C₄²	—	3	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	#869	1	1	yes	characteristic
224	C₂²×C₃	#874	1	1	yes	characteristic
225	C₂²×C₃	#888	1	1	yes	characteristic
226	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
227	C₂²×C₃	C₂×D₂₄	1	2	yes	—
228	C₂²×C₃	C₂×Dic₁₂	1	2	yes	—
229	C₂²×C₃	D₃×D₈	1	2	yes	—
230	C₂²×C₃	D₃×Q₁₆	1	2	yes	—
231	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
232	C₂²×C₃	#940	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	3	8	no	—
238	C₃×C₄	—	24	2	no	—
239	C₃×C₄	#869	1	1	yes	characteristic
240	C₃×C₄	#869	1	1	yes	characteristic
241	C₃×C₄	#888	1	1	yes	characteristic
242	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
243	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
244	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
245	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
246	C₃×C₄	#944	1	1	yes	characteristic
247	Dic₃	—	8	2	no	—
248	Dic₃	—	24	4	no	—
249	Dic₃	—	24	2	no	—
250	Dic₃	—	24	2	no	—
251	Dic₃	#818	1	8	yes	complemented
252	C₃²	#1845	1	1	yes	characteristic, complemented, Sylow
253	C₂³	D₃×D₁₂	1	1	yes	characteristic
254	C₂×C₄	—	3	4	no	—
255	C₂×C₄	—	3	4	no	—
256	C₂×C₄	—	3	4	no	—
257	C₂×C₄	—	12	4	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	—	36	4	no	—
260	C₂×C₄	—	36	2	no	—
261	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
262	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
263	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
264	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#626	1	1	yes	characteristic
266	C₂×C₄	C₂×#627	1	1	yes	characteristic
267	C₈	—	3	4	no	—
268	C₈	Dic₃⋊Dic₃	1	4	yes	complemented
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	2	no	—
274	C₂×C₃	#4682	1	2	yes	—
275	C₂×C₃	#4695	1	1	yes	characteristic
276	C₂×C₃	#4699	1	1	yes	characteristic
277	C₂×C₃	#4874	1	2	yes	—
278	C₂×C₃	D₃×#107	1	2	yes	—
279	C₂×C₃	#4935	1	1	yes	characteristic
280	C₂×C₃	#4975	1	1	yes	characteristic
281	C₂×C₃	C₂×#818	1	2	yes	—
282	C₂×C₃	#5151	1	1	yes	characteristic
283	C₂×C₃	#5183	1	1	yes	characteristic
284	C₂²	D₃×D₂₄	1	2	yes	—
285	C₂²	D₃×Dic₁₂	1	2	yes	—
286	C₂²	D₃×C₄⋊Dic₃	1	1	yes	characteristic
287	C₂²	#63911	1	1	yes	characteristic
288	C₂²	#63918	1	1	yes	characteristic
289	C₄	—	3	8	no	—
290	C₄	—	24	2	no	—
291	C₄	—	72	2	no	—
292	C₄	#63866	1	1	yes	characteristic
293	C₄	D₃×C₄⋊Dic₃	1	1	yes	characteristic
294	C₄	#63918	1	1	yes	characteristic
295	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
296	C₃	—	2	1	no	—
297	C₃	#69765	1	1	yes	characteristic, complemented
298	C₃	#70606	1	1	yes	characteristic, complemented
299	C₂	D₃×#818	1	2	yes	—
300	C₂	#10590880	1	1	yes	characteristic
301	C₂	#10590891	1	2	yes	—
302	C₂	#10590896	1	1	yes	characteristic
303	C₂	#10591349	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₁₂
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1845
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
16. C₃×C₈×Dic₃	C₄	semidirect
68. C₈×Dic₃	Dic₃	semidirect
84. C₃²×C₈	C₄⋊C₄	semidirect
85. C₃×C₃⋊C₈	C₄⋊C₄	semidirect
143. C₃×Dic₃	#107	semidirect
193. C₃×C₈	Dic₃⋊C₄	semidirect
194. C₃×C₈	C₄⋊Dic₃	semidirect
195. C₃⋊C₈	C₄⋊Dic₃	semidirect
251. Dic₃	#818	semidirect
252. C₃²	#1845	semidirect
268. C₈	Dic₃⋊Dic₃	semidirect
297. C₃	#69765	semidirect
298. C₃	#70606	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63645	2
C₂×C₃×#818	2
C₂×C₃×C₈×Dic₃	2
#10591349	2
C₂×#63575	2
#70606	3
#69765	3