Subgroups — #11790499

Order 1152.

457 conjugacy classes of subgroups.

The subgroup lattice has 316 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×#819	C₂	1	1	yes	maximal, characteristic
3	C₃×#5101	C₂	1	1	yes	maximal, characteristic
4	#10591131	C₂	1	2	yes	maximal
5	#10589168	C₂	1	2	yes	maximal
6	#70701	—	3	1	no	maximal
7	#69894	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₃×#926	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	#63645	—	2	1	no	—
14	#63645	—	2	1	no	—
15	C₃×#819	C₄	1	4	yes	complemented
16	C₃×#815	—	2	1	no	—
17	C₃×C₈×Dic₃	—	2	1	no	—
18	#63575	—	2	2	no	—
19	#63573	—	2	2	no	—
20	C₂×C₃×#108	—	3	1	no	—
21	C₃×#405	—	3	1	no	—
22	#5121	—	3	2	no	—
23	C₂×#819	D₃	1	1	yes	characteristic
24	C₂×#818	—	3	1	no	—
25	C₂×#818	—	3	1	no	—
26	C₂×#818	—	6	1	no	—
27	#5101	D₃	1	1	yes	characteristic
28	#4995	—	3	2	no	—
29	#4554	—	3	2	no	—
30	#4554	—	3	2	no	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃⋊Dic₃	—	2	2	no	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	Q₈	1	2	yes	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
39	C₃×#217	—	2	1	no	—
40	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
41	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
42	C₃×C₄×Dic₃	—	2	1	no	—
43	C₃×#208	—	2	2	no	—
44	C₃×#208	—	2	1	no	—
45	C₃×#208	C₂×C₄	1	4	yes	—
46	C₃×#208	Q₈	1	2	yes	—
47	C₂×C₃×C₃⋊C₈	—	2	2	no	—
48	C₂×C₃×C₃⋊C₈	—	2	1	no	—
49	#1952	—	9	1	no	Sylow
50	C₂×C₃×M₁₆	—	3	2	no	—
51	C₂×C₃×M₁₆	—	3	1	no	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₃×#117	—	3	1	no	—
56	#926	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄⋊Dic₃	—	3	2	no	—
58	C₂×C₄⋊Dic₃	—	3	2	no	—
59	C₂×C₄⋊Dic₃	—	6	2	no	—
60	C₂×#208	C₂×D₃	1	1	yes	characteristic
61	C₂×#208	C₂×D₃	1	2	yes	—
62	C₃×#108	—	3	4	no	—
63	C₃×#97	—	6	1	no	—
64	C₃×C₄×C₈	—	6	1	no	—
65	#819	Dic₃	1	4	yes	complemented
66	#818	—	6	2	no	—
67	#818	—	6	2	no	—
68	#818	—	6	1	no	—
69	#818	—	6	1	no	—
70	#818	—	6	1	no	—
71	#818	—	6	1	no	—
72	#818	—	6	2	no	—
73	#817	—	6	2	no	—
74	#815	—	2	1	no	—
75	C₈×Dic₃	—	2	1	no	—
76	#808	—	6	2	no	—
77	#807	—	6	2	no	—
78	C₂³×C₃²	#41	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₃²×C₄	#41	1	1	yes	characteristic
85	C₂×C₃⋊Dic₃	—	4	2	no	—
86	C₂×C₃⋊Dic₃	—	8	2	no	—
87	C₂×C₃×Dic₃	—	2	1	no	—
88	C₂×C₃×Dic₃	—	2	1	no	—
89	C₃²×C₈	C₄⋊C₄	1	2	yes	—
90	C₃²×C₈	C₄⋊C₄	1	2	yes	—
91	C₃×C₃⋊C₈	—	2	4	no	—
92	C₃×C₃⋊C₈	—	2	2	no	—
93	C₂×#108	—	3	1	no	—
94	#428	—	9	2	no	—
95	C₂×#107	—	9	1	no	—
96	#405	—	3	1	no	—
97	#343	—	9	2	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	6	2	no	—
102	C₂²×Dic₃	—	6	2	no	—
103	C₂²×Dic₃	—	12	2	no	—
104	C₃×M₁₆	—	3	4	no	—
105	C₃×M₁₆	—	3	2	no	—
106	C₃×M₁₆	—	6	2	no	—
107	C₃×M₁₆	—	6	1	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	6	2	no	—
113	C₂×C₃×C₈	—	6	1	no	—
114	C₂×C₃×C₈	Dic₆	1	2	yes	—
115	C₂×C₃×C₈	Dic₆	1	2	yes	—
116	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
117	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
118	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
120	C₂×C₃×C₈	#67	1	1	yes	characteristic
121	C₂×C₃×C₈	#67	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	3	1	no	—
123	C₃×C₄⋊C₄	—	3	1	no	—
124	C₃×C₂²⋊C₄	—	6	1	no	—
125	C₃×C₄²	—	6	1	no	—
126	#217	—	2	1	no	—
127	C₄⋊Dic₃	—	6	4	no	—
128	C₄⋊Dic₃	—	6	2	no	—
129	C₄⋊Dic₃	—	6	2	no	—
130	C₄⋊Dic₃	—	6	2	no	—
131	C₄⋊Dic₃	—	6	2	no	—
132	C₄⋊Dic₃	—	12	2	no	—
133	C₄⋊Dic₃	D₁₂	1	1	yes	characteristic
134	C₄⋊Dic₃	D₁₂	1	1	yes	characteristic
135	C₄×Dic₃	—	2	1	no	—
136	#208	—	2	1	no	—
137	#208	—	2	2	no	—
138	#208	Dic₆	1	2	yes	—
139	#208	C₂×Dic₃	1	4	yes	—
140	C₂×C₃⋊C₈	—	2	2	no	—
141	C₂×C₃⋊C₈	—	2	1	no	—
142	C₂²×C₃²	—	2	1	no	—
143	C₂²×C₃²	—	2	1	no	—
144	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₃²	#121	1	1	yes	characteristic
146	C₂²×C₃²	#125	1	1	yes	characteristic
147	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
149	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
150	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
151	C₃⋊Dic₃	—	8	2	no	—
152	C₃×Dic₃	—	4	1	no	—
153	C₃×Dic₃	—	4	1	no	—
154	C₂×M₁₆	—	3	1	no	—
155	C₂×M₁₆	—	3	2	no	—
156	C₂²×C₈	D₃²	1	1	yes	characteristic
157	#117	—	3	1	no	—
158	C₂×C₄⋊C₄	—	9	2	no	—
159	#108	—	3	4	no	—
160	#107	—	18	2	no	—
161	#107	—	18	1	no	—
162	#107	—	18	1	no	—
163	#106	—	18	2	no	—
164	#97	—	6	1	no	—
165	C₄×C₈	—	6	1	no	—
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	#235	1	1	yes	characteristic
168	C₂³×C₃	#239	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	6	1	no	—
176	C₂×C₃×C₄	—	6	1	no	—
177	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
183	C₂×C₃×C₄	#235	1	1	yes	characteristic
184	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	#237	1	1	yes	characteristic
186	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
189	C₂×Dic₃	—	2	1	no	—
190	C₂×Dic₃	—	2	1	no	—
191	C₂×Dic₃	—	12	4	no	—
192	C₂×Dic₃	—	12	2	no	—
193	C₂×Dic₃	—	12	2	no	—
194	C₂×Dic₃	—	24	4	no	—
195	C₂×Dic₃	—	24	2	no	—
196	C₂×Dic₃	—	24	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	—	6	4	no	—
200	C₃×C₈	—	6	2	no	—
201	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
202	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
203	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
204	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
205	C₃⋊C₈	—	2	2	no	—
206	C₃⋊C₈	—	2	4	no	—
207	C₂×C₃²	—	2	1	no	—
208	C₂×C₃²	—	2	1	no	—
209	C₂×C₃²	#389	1	1	yes	characteristic
210	C₂×C₃²	#472	1	1	yes	characteristic
211	C₂×C₃²	#473	1	1	yes	characteristic
212	C₂²×C₄	—	18	2	no	—
213	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
214	M₁₆	—	3	2	no	—
215	M₁₆	—	3	4	no	—
216	M₁₆	—	6	1	no	—
217	M₁₆	—	6	2	no	—
218	C₂×C₈	—	6	2	no	—
219	C₂×C₈	—	6	1	no	—
220	C₂×C₈	D₃×Dic₃	1	2	yes	—
221	C₂×C₈	#626	1	1	yes	characteristic
222	C₂×C₈	#626	1	1	yes	characteristic
223	C₂×C₈	#627	1	2	yes	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	18	2	no	—
227	C₄⋊C₄	—	18	2	no	—
228	C₂²⋊C₄	—	6	1	no	—
229	C₄²	—	6	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
240	C₂²×C₃	#870	1	1	yes	characteristic
241	C₂²×C₃	#890	1	1	yes	characteristic
242	C₂²×C₃	#895	1	1	yes	characteristic
243	C₂²×C₃	#930	1	1	yes	characteristic
244	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	12	1	no	—
250	C₃×C₄	—	12	1	no	—
251	C₃×C₄	#869	1	1	yes	characteristic
252	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
253	C₃×C₄	#888	1	1	yes	characteristic
254	C₃×C₄	#897	1	1	yes	characteristic
255	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
256	C₃×C₄	#924	1	1	yes	characteristic
257	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
258	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
259	Dic₃	—	4	1	no	—
260	Dic₃	—	4	1	no	—
261	Dic₃	—	24	4	no	—
262	Dic₃	—	24	2	no	—
263	Dic₃	—	24	2	no	—
264	C₃²	#1952	1	1	yes	characteristic, complemented, Sylow
265	C₂³	#3799	1	1	yes	characteristic
266	C₂×C₄	—	6	1	no	—
267	C₂×C₄	—	6	1	no	—
268	C₂×C₄	—	36	2	no	—
269	C₂×C₄	—	72	2	no	—
270	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
271	C₂×C₄	#3795	1	1	yes	characteristic
272	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
273	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#626	1	1	yes	characteristic
275	C₂×C₄	C₂×#627	1	1	yes	characteristic
276	C₈	—	6	2	no	—
277	C₈	—	6	4	no	—
278	C₈	Dic₃⋊Dic₃	1	2	yes	—
279	C₈	Dic₃⋊Dic₃	1	2	yes	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	—	2	2	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	#4896	1	1	yes	characteristic
290	C₂×C₃	#4897	1	1	yes	characteristic
291	C₂×C₃	#4928	1	1	yes	characteristic
292	C₂×C₃	#4984	1	1	yes	characteristic
293	C₂×C₃	#5142	1	1	yes	characteristic
294	C₂×C₃	#5143	1	1	yes	characteristic
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	Dic₃×Dic₆	1	1	yes	characteristic
298	C₂²	#63869	1	1	yes	characteristic
299	C₂²	#63909	1	1	yes	characteristic
300	C₄	—	12	1	no	—
301	C₄	—	12	1	no	—
302	C₄	—	72	2	no	—
303	C₄	#63857	1	1	yes	characteristic
304	C₄	Dic₃×D₁₂	1	1	yes	characteristic
305	C₄	#63918	1	1	yes	characteristic
306	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
307	C₃	—	2	1	no	—
308	C₃	#69894	1	1	yes	characteristic, complemented
309	C₃	#70701	1	1	yes	characteristic, complemented
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	#10590927	1	1	yes	characteristic
313	C₂	#10590932	1	1	yes	characteristic
314	C₂	#10591352	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63909
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1952
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
15. C₃×#819	C₄	semidirect
65. #819	Dic₃	semidirect
264. C₃²	#1952	semidirect
308. C₃	#69894	semidirect
309. C₃	#70701	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×#819	2
C₃×#5101	2
#10591131	2
#10589168	2
#70701	3
#69894	3