Subgroups — #11790500

Order 1152.

457 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592800	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
4	#10591130	C₂	1	2	yes	maximal
5	#10589167	C₂	1	2	yes	maximal
6	#70701	—	3	1	no	maximal
7	#70625	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	#64135	C₂²	1	1	yes	characteristic
10	C₂×#3748	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	2	yes	—
13	#63642	—	2	1	no	—
14	C₈×C₃⋊Dic₃	—	2	1	no	—
15	C₃×#819	C₄	1	4	yes	complemented
16	C₃×#818	—	2	1	no	—
17	C₃×#818	—	2	1	no	—
18	#63575	—	2	2	no	—
19	#63573	—	2	2	no	—
20	C₂×C₃×#108	—	3	1	no	—
21	C₂×C₃×#107	—	3	1	no	—
22	#5121	—	3	2	no	—
23	C₂×#819	D₃	1	1	yes	characteristic
24	C₂×#818	D₃	1	1	yes	characteristic
25	#5101	—	3	1	no	—
26	#5101	—	3	1	no	—
27	#5101	—	6	1	no	—
28	#4964	—	3	2	no	—
29	#4554	—	3	2	no	—
30	#4530	—	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	#3757	—	2	1	no	—
38	C₄⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
39	C₄⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
40	C₄×C₃⋊Dic₃	—	2	1	no	—
41	#3748	—	2	1	no	—
42	#3748	Q₈	1	2	yes	—
43	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄⋊Dic₃	—	2	2	no	—
46	C₃×#208	—	2	2	no	—
47	C₃×#208	C₂×C₄	1	4	yes	—
48	C₂×C₃×C₃⋊C₈	—	2	2	no	—
49	#1952	—	9	1	no	Sylow
50	C₂×C₃×M₁₆	—	3	2	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	3	2	no	—
55	#926	—	3	1	no	—
56	#926	—	3	1	no	—
57	#926	—	6	1	no	—
58	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
59	C₂×#208	—	3	1	no	—
60	C₂×#208	—	3	1	no	—
61	C₂×#208	—	6	1	no	—
62	C₂×#208	C₂×D₃	1	2	yes	—
63	C₃×#108	—	3	4	no	—
64	C₃×#107	—	6	1	no	—
65	C₃×#107	—	6	1	no	—
66	#819	Dic₃	1	4	yes	complemented
67	#818	—	2	1	no	—
68	#818	—	2	1	no	—
69	#818	—	6	2	no	—
70	#817	—	6	2	no	—
71	#815	—	6	2	no	—
72	#815	—	6	1	no	—
73	#815	—	6	1	no	—
74	C₈×Dic₃	—	6	2	no	—
75	C₈×Dic₃	—	6	1	no	—
76	C₈×Dic₃	—	6	1	no	—
77	#808	—	6	2	no	—
78	#807	—	6	2	no	—
79	C₂³×C₃²	#41	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₃²×C₄	#41	1	1	yes	characteristic
86	C₂×C₃⋊Dic₃	—	2	1	no	—
87	C₂×C₃⋊Dic₃	—	2	1	no	—
88	C₂×C₃×Dic₃	—	4	2	no	—
89	C₂×C₃×Dic₃	—	8	2	no	—
90	C₃²×C₈	C₄⋊C₄	1	2	yes	—
91	C₃²×C₈	C₄⋊C₄	1	2	yes	—
92	C₃⋊(C₃⋊C₈)	—	2	2	no	—
93	C₃×C₃⋊C₈	—	2	4	no	—
94	C₂×#108	—	3	1	no	—
95	#428	—	9	2	no	—
96	C₂×#107	—	3	1	no	—
97	#405	—	9	1	no	—
98	#343	—	9	2	no	—
99	C₂²×C₃×C₄	—	2	1	no	—
100	C₂²×C₃×C₄	—	6	2	no	—
101	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×Dic₃	—	2	2	no	—
104	C₃×M₁₆	—	3	4	no	—
105	C₃×M₁₆	—	6	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	6	2	no	—
111	C₂×C₃×C₈	Dic₆	1	2	yes	—
112	C₂×C₃×C₈	Dic₆	1	2	yes	—
113	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
114	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
115	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
116	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
117	C₂×C₃×C₈	#67	1	1	yes	characteristic
118	C₂×C₃×C₈	#67	1	1	yes	characteristic
119	C₃×C₄⋊C₄	—	6	2	no	—
120	C₃×C₄⋊C₄	—	6	2	no	—
121	#217	—	6	2	no	—
122	#217	—	6	1	no	—
123	#217	—	6	1	no	—
124	C₄⋊Dic₃	—	2	2	no	—
125	C₄⋊Dic₃	—	2	2	no	—
126	C₄⋊Dic₃	—	3	1	no	—
127	C₄⋊Dic₃	—	3	1	no	—
128	C₄⋊Dic₃	—	3	1	no	—
129	C₄⋊Dic₃	—	3	1	no	—
130	C₄⋊Dic₃	—	6	1	no	—
131	C₄⋊Dic₃	—	6	1	no	—
132	C₄×Dic₃	—	6	2	no	—
133	C₄×Dic₃	—	6	1	no	—
134	C₄×Dic₃	—	6	1	no	—
135	#208	—	2	2	no	—
136	#208	—	3	2	no	—
137	#208	—	3	2	no	—
138	#208	—	6	2	no	—
139	#208	—	6	2	no	—
140	#208	—	6	1	no	—
141	#208	—	6	1	no	—
142	#208	C₂×Dic₃	1	4	yes	—
143	C₂×C₃⋊C₈	—	2	2	no	—
144	C₂×C₃⋊C₈	—	6	2	no	—
145	C₂×C₃⋊C₈	—	6	1	no	—
146	C₂×C₃⋊C₈	—	6	1	no	—
147	C₂²×C₃²	—	2	1	no	—
148	C₂²×C₃²	—	2	1	no	—
149	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
150	C₂²×C₃²	#121	1	1	yes	characteristic
151	C₂²×C₃²	#125	1	1	yes	characteristic
152	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
154	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
155	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
156	C₃⋊Dic₃	—	4	1	no	—
157	C₃⋊Dic₃	—	4	1	no	—
158	C₃×Dic₃	—	8	2	no	—
159	C₂×M₁₆	—	3	2	no	—
160	C₂×M₁₆	—	9	1	no	—
161	C₂²×C₈	D₃²	1	1	yes	characteristic
162	#117	—	9	1	no	—
163	C₂×C₄⋊C₄	—	3	2	no	—
164	#108	—	3	4	no	—
165	#107	—	6	1	no	—
166	#107	—	6	1	no	—
167	#107	—	18	2	no	—
168	#106	—	18	2	no	—
169	#97	—	18	1	no	—
170	C₄×C₈	—	18	1	no	—
171	C₂³×C₃	—	2	1	no	—
172	C₂³×C₃	#237	1	1	yes	characteristic
173	C₂³×C₃	#239	1	1	yes	characteristic
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	12	2	no	—
181	C₂×C₃×C₄	—	24	2	no	—
182	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
186	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	#237	1	1	yes	characteristic
189	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
190	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
191	C₂×C₃×C₄	#239	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
194	C₂×Dic₃	—	4	2	no	—
195	C₂×Dic₃	—	6	2	no	—
196	C₂×Dic₃	—	6	2	no	—
197	C₂×Dic₃	—	6	1	no	—
198	C₂×Dic₃	—	6	1	no	—
199	C₂×Dic₃	—	6	1	no	—
200	C₂×Dic₃	—	6	1	no	—
201	C₂×Dic₃	—	8	2	no	—
202	C₃×C₈	—	2	2	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	6	4	no	—
205	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
206	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
207	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
208	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
209	C₃⋊C₈	—	2	4	no	—
210	C₃⋊C₈	—	6	4	no	—
211	C₃⋊C₈	—	6	2	no	—
212	C₃⋊C₈	—	6	2	no	—
213	C₂×C₃²	—	2	1	no	—
214	C₂×C₃²	—	2	1	no	—
215	C₂×C₃²	#389	1	1	yes	characteristic
216	C₂×C₃²	#472	1	1	yes	characteristic
217	C₂×C₃²	#473	1	1	yes	characteristic
218	C₂²×C₄	—	6	2	no	—
219	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
220	M₁₆	—	3	4	no	—
221	M₁₆	—	6	2	no	—
222	M₁₆	—	9	2	no	—
223	M₁₆	—	18	1	no	—
224	C₂×C₈	—	6	2	no	—
225	C₂×C₈	—	18	1	no	—
226	C₂×C₈	D₃×Dic₃	1	2	yes	—
227	C₂×C₈	#626	1	1	yes	characteristic
228	C₂×C₈	#626	1	1	yes	characteristic
229	C₂×C₈	#627	1	2	yes	—
230	C₄⋊C₄	—	6	2	no	—
231	C₄⋊C₄	—	6	2	no	—
232	C₄⋊C₄	—	9	1	no	—
233	C₄⋊C₄	—	9	1	no	—
234	C₂²⋊C₄	—	18	1	no	—
235	C₄²	—	18	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	4	1	no	—
244	C₂²×C₃	—	4	1	no	—
245	C₂²×C₃	#887	1	1	yes	characteristic
246	C₂²×C₃	#890	1	1	yes	characteristic
247	C₂²×C₃	#890	1	1	yes	characteristic
248	C₂²×C₃	#895	1	1	yes	characteristic
249	C₂²×C₃	#938	1	1	yes	characteristic
250	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	24	2	no	—
256	C₃×C₄	#888	1	1	yes	characteristic
257	C₃×C₄	#888	1	1	yes	characteristic
258	C₃×C₄	#893	1	1	yes	characteristic
259	C₃×C₄	#897	1	1	yes	characteristic
260	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
261	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
262	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
263	C₃×C₄	#946	1	1	yes	characteristic
264	Dic₃	—	8	2	no	—
265	Dic₃	—	12	2	no	—
266	Dic₃	—	12	2	no	—
267	Dic₃	—	12	1	no	—
268	Dic₃	—	12	1	no	—
269	Dic₃	—	12	1	no	—
270	Dic₃	—	12	1	no	—
271	C₃²	#1952	1	1	yes	characteristic, complemented, Sylow
272	C₂³	#3796	1	1	yes	characteristic
273	C₂×C₄	—	12	2	no	—
274	C₂×C₄	—	18	1	no	—
275	C₂×C₄	—	18	1	no	—
276	C₂×C₄	—	24	2	no	—
277	C₂×C₄	#3796	1	1	yes	characteristic
278	C₂×C₄	#3797	1	1	yes	characteristic
279	C₂×C₄	#3802	1	1	yes	characteristic
280	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
281	C₂×C₄	C₂×#626	1	1	yes	characteristic
282	C₂×C₄	C₂×#627	1	1	yes	characteristic
283	C₈	—	6	4	no	—
284	C₈	—	18	2	no	—
285	C₈	Dic₃⋊Dic₃	1	2	yes	—
286	C₈	Dic₃⋊Dic₃	1	2	yes	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	4	1	no	—
295	C₂×C₃	—	4	1	no	—
296	C₂×C₃	#4896	1	1	yes	characteristic
297	C₂×C₃	#4897	1	1	yes	characteristic
298	C₂×C₃	#4972	1	1	yes	characteristic
299	C₂×C₃	#4984	1	1	yes	characteristic
300	C₂×C₃	#5160	1	1	yes	characteristic
301	C₂×C₃	#5192	1	1	yes	characteristic
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	#63843	1	1	yes	characteristic
305	C₂²	#63872	1	1	yes	characteristic
306	C₂²	#63916	1	1	yes	characteristic
307	C₄	—	24	2	no	—
308	C₄	—	36	1	no	—
309	C₄	—	36	1	no	—
310	C₄	#63871	1	1	yes	characteristic
311	C₄	#63898	1	1	yes	characteristic
312	C₄	#63919	1	1	yes	characteristic
313	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
314	C₃	—	2	1	no	—
315	C₃	#70625	1	1	yes	characteristic, complemented
316	C₃	#70701	1	1	yes	characteristic, complemented
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	#10590929	1	1	yes	characteristic
320	C₂	#10590933	1	1	yes	characteristic
321	C₂	#10591357	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63916
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
66. #819	Dic₃	semidirect
271. C₃²	#1952	semidirect
315. C₃	#70625	semidirect
316. C₃	#70701	semidirect

Maximal subgroups

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

subgroup	index
#10592800	2
C₂×C₃×#819	2
C₂×C₃×#818	2
#10591130	2
#10589167	2
#70701	3
#70625	3