Subgroups — #11790176

Order 1152.

553 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63642	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
4	#10591357	C₂	1	2	yes	maximal
5	#10589163	C₂	1	2	yes	maximal
6	#70613	—	3	1	no	maximal
7	#70561	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×Dic₃⋊Dic₃	—	2	2	no	—
14	#63642	—	2	2	no	—
15	C₃×#818	C₄	1	4	yes	complemented
16	C₃×#817	—	2	2	no	—
17	#63579	—	2	2	no	—
18	C₂×C₃×#107	—	3	1	no	—
19	C₂×C₃×#106	—	3	1	no	—
20	C₂×#818	D₃	1	1	yes	characteristic
21	C₂×#817	D₃	1	1	yes	characteristic
22	C₂×#815	—	3	1	no	—
23	C₂×#815	—	3	1	no	—
24	C₂×#815	—	6	1	no	—
25	#4984	—	3	2	no	—
26	#4972	—	3	2	no	—
27	#4529	—	3	2	no	—
28	#4529	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	D₄	1	2	yes	—
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂²×C₃×Dic₃	—	2	2	no	—
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	1	no	—
37	C₄×C₃⋊Dic₃	—	2	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
40	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	2	2	no	—
43	C₃×C₄⋊Dic₃	—	2	2	no	—
44	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
45	Dic₃⋊Dic₃	—	4	4	no	—
46	#1943	—	9	1	no	Sylow
47	C₂²×C₃×C₈	—	2	1	no	—
48	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄⋊C₄	—	3	2	no	—
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₄⋊Dic₃	—	6	2	no	—
53	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
54	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
55	C₂×Dic₃⋊C₄	—	6	2	no	—
56	C₂×C₄×Dic₃	—	3	1	no	—
57	C₂×C₄×Dic₃	—	3	1	no	—
58	C₂×C₄×Dic₃	—	6	1	no	—
59	C₂²×C₃⋊C₈	—	3	1	no	—
60	C₂²×C₃⋊C₈	—	3	1	no	—
61	C₂²×C₃⋊C₈	—	6	1	no	—
62	C₃×#107	—	3	4	no	—
63	C₃×#106	—	6	2	no	—
64	C₃⋊#95	—	6	2	no	—
65	C₃⋊#95	—	6	2	no	—
66	#818	Dic₃	1	4	yes	complemented
67	#817	—	2	2	no	—
68	#815	—	6	4	no	—
69	#815	—	6	2	no	—
70	#815	—	6	2	no	—
71	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×C₄	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	#41	1	1	yes	characteristic
77	C₂×C₃²×C₄	#41	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	—	2	2	no	—
79	C₂×C₃⋊Dic₃	—	2	2	no	—
80	C₂×C₃⋊Dic₃	—	2	2	no	—
81	C₂×C₃×Dic₃	—	2	4	no	—
82	C₂×C₃×Dic₃	—	4	2	no	—
83	C₂×C₃×Dic₃	—	4	2	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	—	4	2	no	—
86	C₂×C₃×Dic₃	—	4	2	no	—
87	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
88	C₃⋊(C₃⋊C₈)	—	2	2	no	—
89	C₂×#107	—	3	1	no	—
90	C₂×#106	—	3	1	no	—
91	C₂×#97	—	9	1	no	—
92	#389	—	9	2	no	—
93	#340	—	9	2	no	—
94	C₂²×C₃×C₄	—	2	1	no	—
95	C₂²×C₃×C₄	—	6	2	no	—
96	C₂²×C₃×C₄	—	6	2	no	—
97	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂²×Dic₃	—	2	2	no	—
100	C₂²×Dic₃	—	2	2	no	—
101	C₂²×Dic₃	—	3	2	no	—
102	C₂²×Dic₃	—	3	2	no	—
103	C₂²×Dic₃	—	6	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	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	—	3	4	no	—
114	C₃×C₄⋊C₄	—	6	2	no	—
115	C₃×C₄⋊C₄	—	6	2	no	—
116	C₃×C₄⋊C₄	—	6	2	no	—
117	C₄⋊Dic₃	—	2	2	no	—
118	C₄⋊Dic₃	—	2	2	no	—
119	C₄⋊Dic₃	—	2	2	no	—
120	C₄⋊Dic₃	—	12	4	no	—
121	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
122	Dic₃⋊C₄	—	12	4	no	—
123	C₄×Dic₃	—	6	2	no	—
124	C₄×Dic₃	—	6	2	no	—
125	C₄×Dic₃	—	6	1	no	—
126	C₄×Dic₃	—	6	1	no	—
127	C₄×Dic₃	—	6	1	no	—
128	C₄×Dic₃	—	6	1	no	—
129	C₂×C₃⋊C₈	—	3	2	no	—
130	C₂×C₃⋊C₈	—	3	2	no	—
131	C₂×C₃⋊C₈	—	6	2	no	—
132	C₂×C₃⋊C₈	—	6	2	no	—
133	C₂×C₃⋊C₈	—	6	2	no	—
134	C₂×C₃⋊C₈	—	6	1	no	—
135	C₂×C₃⋊C₈	—	6	1	no	—
136	C₂×C₃⋊C₈	—	6	1	no	—
137	C₂×C₃⋊C₈	—	6	1	no	—
138	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
139	C₂²×C₃²	#121	1	1	yes	characteristic
140	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₃²	AGL(1,ℤ/8)	1	2	yes	—
142	C₂²×C₃²	#137	1	2	yes	—
143	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
145	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
146	C₃²×C₄	#125	1	1	yes	characteristic
147	C₃⋊Dic₃	—	4	2	no	—
148	C₃×Dic₃	—	4	4	no	—
149	C₃×Dic₃	—	8	2	no	—
150	C₂²×C₈	—	9	1	no	—
151	C₂²×C₈	D₃²	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	3	2	no	—
153	C₂×C₄⋊C₄	—	3	2	no	—
154	C₂×C₄⋊C₄	—	18	2	no	—
155	C₂×C₄²	—	9	1	no	—
156	#107	—	3	4	no	—
157	#106	—	6	2	no	—
158	#97	—	18	2	no	—
159	#95	—	18	2	no	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
162	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	6	4	no	—
170	C₂×C₃×C₄	—	12	2	no	—
171	C₂×C₃×C₄	—	12	2	no	—
172	C₂×C₃×C₄	—	12	2	no	—
173	C₂×C₃×C₄	—	12	2	no	—
174	C₂×C₃×C₄	—	12	2	no	—
175	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
179	C₂×C₃×C₄	#237	1	1	yes	characteristic
180	C₂×C₃×C₄	#237	1	1	yes	characteristic
181	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
182	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
183	C₂×C₃×C₄	#239	1	1	yes	characteristic
184	C₂×C₃×C₄	#239	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
187	C₂×Dic₃	—	2	4	no	—
188	C₂×Dic₃	—	4	2	no	—
189	C₂×Dic₃	—	4	2	no	—
190	C₂×Dic₃	—	4	2	no	—
191	C₂×Dic₃	—	4	2	no	—
192	C₂×Dic₃	—	4	2	no	—
193	C₂×Dic₃	—	6	4	no	—
194	C₂×Dic₃	—	6	2	no	—
195	C₂×Dic₃	—	6	2	no	—
196	C₂×Dic₃	—	6	2	no	—
197	C₂×Dic₃	—	6	2	no	—
198	C₂×Dic₃	—	6	2	no	—
199	C₂×Dic₃	—	6	2	no	—
200	C₂×Dic₃	—	12	2	no	—
201	C₂×Dic₃	—	12	2	no	—
202	C₃×C₈	—	2	4	no	—
203	C₃×C₈	Dic₃⋊C₄	1	4	yes	complemented
204	C₃×C₈	C₄⋊Dic₃	1	4	yes	complemented
205	C₃⋊C₈	—	6	4	no	—
206	C₃⋊C₈	—	6	2	no	—
207	C₃⋊C₈	—	6	2	no	—
208	C₂×C₃²	#389	1	1	yes	characteristic
209	C₂×C₃²	#441	1	1	yes	characteristic
210	C₂×C₃²	#442	1	1	yes	characteristic
211	C₂×C₃²	#468	1	2	yes	—
212	C₂×C₃²	#501	1	2	yes	—
213	C₂²×C₄	—	6	2	no	—
214	C₂²×C₄	—	6	2	no	—
215	C₂²×C₄	—	9	2	no	—
216	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
217	C₂×C₈	—	9	2	no	—
218	C₂×C₈	—	18	1	no	—
219	C₂×C₈	—	18	1	no	—
220	C₂×C₈	D₃×Dic₃	1	2	yes	—
221	C₂×C₈	#626	1	2	yes	—
222	C₂×C₈	#627	1	2	yes	—
223	C₄⋊C₄	—	3	4	no	—
224	C₄⋊C₄	—	6	2	no	—
225	C₄⋊C₄	—	6	2	no	—
226	C₄⋊C₄	—	6	2	no	—
227	C₄⋊C₄	—	36	4	no	—
228	C₄²	—	18	1	no	—
229	C₄²	—	18	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	#888	1	1	yes	characteristic
236	C₂²×C₃	#888	1	1	yes	characteristic
237	C₂²×C₃	#893	1	1	yes	characteristic
238	C₂²×C₃	#895	1	1	yes	characteristic
239	C₂²×C₃	#911	1	2	yes	—
240	C₂²×C₃	#914	1	2	yes	—
241	C₂²×C₃	#915	1	2	yes	—
242	C₂²×C₃	#918	1	2	yes	—
243	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
244	C₂²×C₃	#938	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	4	no	—
250	C₃×C₄	—	24	2	no	—
251	C₃×C₄	#887	1	1	yes	characteristic
252	C₃×C₄	#890	1	1	yes	characteristic
253	C₃×C₄	#890	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	C₂×C₄⋊Dic₃	1	1	yes	characteristic
257	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
258	C₃×C₄	#946	1	1	yes	characteristic
259	Dic₃	—	4	4	no	—
260	Dic₃	—	8	2	no	—
261	Dic₃	—	12	4	no	—
262	Dic₃	—	12	2	no	—
263	Dic₃	—	12	2	no	—
264	C₃²	#1943	1	1	yes	characteristic, complemented, Sylow
265	C₂³	#3802	1	1	yes	characteristic
266	C₂×C₄	—	6	4	no	—
267	C₂×C₄	—	12	2	no	—
268	C₂×C₄	—	12	2	no	—
269	C₂×C₄	—	12	2	no	—
270	C₂×C₄	—	12	2	no	—
271	C₂×C₄	—	12	2	no	—
272	C₂×C₄	—	18	2	no	—
273	C₂×C₄	—	18	2	no	—
274	C₂×C₄	—	18	2	no	—
275	C₂×C₄	#3796	1	1	yes	characteristic
276	C₂×C₄	#3796	1	1	yes	characteristic
277	C₂×C₄	#3797	1	1	yes	characteristic
278	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#626	1	1	yes	characteristic
280	C₂×C₄	C₂×#627	1	1	yes	characteristic
281	C₈	—	18	2	no	—
282	C₈	Dic₃⋊Dic₃	1	4	yes	complemented
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	2	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	#4853	1	1	yes	characteristic
289	C₂×C₃	#4856	1	2	yes	—
290	C₂×C₃	#4869	1	1	yes	characteristic
291	C₂×C₃	#4876	1	2	yes	—
292	C₂×C₃	#4886	1	2	yes	—
293	C₂×C₃	#4972	1	1	yes	characteristic
294	C₂×C₃	#4984	1	1	yes	characteristic
295	C₂×C₃	#5152	1	1	yes	characteristic
296	C₂×C₃	#5174	1	2	yes	—
297	C₂×C₃	#5184	1	1	yes	characteristic
298	C₂²	#63798	1	2	yes	—
299	C₂²	#63811	1	2	yes	—
300	C₂²	#63898	1	1	yes	characteristic
301	C₂²	#63916	1	1	yes	characteristic
302	C₂²	#63919	1	1	yes	characteristic
303	C₄	—	12	4	no	—
304	C₄	—	24	2	no	—
305	C₄	—	36	2	no	—
306	C₄	#63843	1	1	yes	characteristic
307	C₄	#63871	1	1	yes	characteristic
308	C₄	#63872	1	1	yes	characteristic
309	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
310	C₃	—	2	1	no	—
311	C₃	#70561	1	1	yes	characteristic, complemented
312	C₃	#70613	1	1	yes	characteristic, complemented
313	C₂	#10590781	1	1	yes	characteristic
314	C₂	#10590819	1	1	yes	characteristic
315	C₂	#10590845	1	2	yes	—
316	C₂	#10590903	1	2	yes	—
317	C₂	#10591357	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3802
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	#1943
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
15. C₃×#818	C₄	semidirect
66. #818	Dic₃	semidirect
87. C₃²×C₈	C₄⋊C₄	semidirect
203. C₃×C₈	Dic₃⋊C₄	semidirect
204. C₃×C₈	C₄⋊Dic₃	semidirect
264. C₃²	#1943	semidirect
282. C₈	Dic₃⋊Dic₃	semidirect
311. C₃	#70561	semidirect
312. C₃	#70613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63642	2
C₂×C₃×#818	2
C₂×C₃×#817	2
#10591357	2
#10589163	2
#70613	3
#70561	3