Subgroups — #11776900

Order 1152.

431 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592623	C₂	1	1	yes	maximal, characteristic
2	C₃×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589099	C₂	1	4	yes	maximal, complemented
5	#70491	—	3	1	no	maximal
6	#69731	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#3748	—	2	1	no	—
9	C₂×C₃×#208	—	2	1	no	—
10	C₂²×C₃×C₃⋊C₈	C₄	1	2	yes	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
12	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
13	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
14	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃×C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
16	C₃×C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃×#423	—	3	1	no	—
19	C₂×C₃×C₄×C₈	—	3	1	no	—
20	#4924	—	3	1	no	—
21	#4924	—	3	1	no	—
22	#4924	—	6	1	no	—
23	#4924	D₃	1	1	yes	characteristic
24	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
25	#4479	—	3	4	no	—
26	#4458	—	3	4	no	—
27	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃²×C₄²	D₄	1	1	yes	characteristic
32	C₃²×C₄²	Q₈	1	1	yes	characteristic
33	C₃²×C₄²	C₂³	1	1	yes	characteristic
34	#3748	—	4	2	no	—
35	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
36	C₃×#208	—	4	2	no	—
37	C₂×C₃×C₃⋊C₈	—	2	4	no	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	#1567	—	9	1	no	Sylow
42	C₂×C₃×M₁₆	—	6	1	no	—
43	C₂²×C₃×C₈	—	3	2	no	—
44	C₂×C₃×C₄²	—	2	1	no	—
45	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₂×#208	—	2	1	no	—
48	C₂×#208	—	6	2	no	—
49	C₂×#208	—	6	1	no	—
50	C₂×#208	—	6	1	no	—
51	C₂²×C₃⋊C₈	Dic₃	1	2	yes	—
52	C₃×C₄⋊C₈	—	3	2	no	—
53	C₃×C₄⋊C₈	—	6	1	no	—
54	C₃×C₄×C₈	—	3	1	no	—
55	C₃×C₄×C₈	—	3	1	no	—
56	C₃×C₄×C₈	—	3	2	no	—
57	C₄⋊(C₃⋊C₈)	—	2	1	no	—
58	C₄⋊(C₃⋊C₈)	—	3	2	no	—
59	C₄⋊(C₃⋊C₈)	—	3	2	no	—
60	C₄⋊(C₃⋊C₈)	—	6	2	no	—
61	C₄⋊(C₃⋊C₈)	—	6	2	no	—
62	C₄⋊(C₃⋊C₈)	—	6	1	no	—
63	C₄⋊(C₃⋊C₈)	—	6	1	no	—
64	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
65	C₄×C₃⋊C₈	Dic₃	1	1	yes	characteristic
66	C₄×C₃⋊C₈	Dic₃	1	1	yes	characteristic
67	C₄×C₃⋊C₈	C₂×D₃	1	2	yes	—
68	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	—	2	2	no	—
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	D₈	1	2	yes	—
75	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₃⋊(C₃⋊C₈)	—	4	2	no	—
80	C₃×C₃⋊C₈	—	2	2	no	—
81	C₃×C₃⋊C₈	—	4	2	no	—
82	C₃×C₃⋊C₈	M₁₆	1	4	yes	complemented
83	#423	—	3	1	no	—
84	#423	—	9	1	no	—
85	C₂×C₄×C₈	—	3	1	no	—
86	#335	—	9	4	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
91	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	#67	1	1	yes	characteristic
96	C₃×M₁₆	—	12	2	no	—
97	C₂×C₃×C₈	—	3	2	no	—
98	C₂×C₃×C₈	—	3	2	no	—
99	C₂×C₃×C₈	—	6	4	no	—
100	C₂×C₃×C₈	—	6	2	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	Dic₆	1	1	yes	characteristic
106	C₃×C₄²	Dic₆	1	1	yes	characteristic
107	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
108	C₃×C₄²	D₁₂	1	1	yes	characteristic
109	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
110	C₃×C₄²	#67	1	1	yes	characteristic
111	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
112	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
113	#208	—	4	2	no	—
114	#208	—	12	4	no	—
115	#208	—	12	2	no	—
116	#208	—	12	2	no	—
117	C₂×C₃⋊C₈	—	2	4	no	—
118	C₂×C₃⋊C₈	—	2	2	no	—
119	C₂×C₃⋊C₈	—	6	4	no	—
120	C₂×C₃⋊C₈	—	6	2	no	—
121	C₂×C₃⋊C₈	—	6	2	no	—
122	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
123	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
124	C₂²×C₃²	—	2	1	no	—
125	C₂²×C₃²	—	2	1	no	—
126	C₂²×C₃²	#107	1	2	yes	—
127	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₃²×C₄	—	2	1	no	—
129	C₃²×C₄	—	2	1	no	—
130	C₃²×C₄	#107	1	2	yes	—
131	C₃²×C₄	#108	1	2	yes	—
132	C₃²×C₄	C₂×M₁₆	1	2	yes	—
133	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
134	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×M₁₆	—	6	1	no	—
136	C₂×M₁₆	—	18	1	no	—
137	C₂²×C₈	—	3	2	no	—
138	C₂×C₄²	D₃²	1	1	yes	characteristic
139	C₄⋊C₈	—	3	2	no	—
140	C₄⋊C₈	—	6	1	no	—
141	C₄⋊C₈	—	9	2	no	—
142	C₄⋊C₈	—	18	1	no	—
143	C₄×C₈	—	3	1	no	—
144	C₄×C₈	—	3	1	no	—
145	C₄×C₈	—	3	2	no	—
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
148	C₂³×C₃	C₄⋊Dic₃	1	1	yes	characteristic
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
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	—	4	2	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	D₂₄	1	2	yes	—
164	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
165	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
166	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
171	C₂×C₃×C₄	#213	1	2	yes	—
172	C₂×C₃×C₄	#216	1	2	yes	—
173	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
179	C₃×C₈	—	3	4	no	—
180	C₃×C₈	—	6	2	no	—
181	C₃×C₈	—	12	2	no	—
182	C₃⋊C₈	—	2	2	no	—
183	C₃⋊C₈	—	4	2	no	—
184	C₃⋊C₈	—	12	4	no	—
185	C₃⋊C₈	—	12	2	no	—
186	C₃⋊C₈	—	12	2	no	—
187	C₃⋊C₈	#208	1	4	yes	complemented
188	C₂×C₃²	—	2	2	no	—
189	C₂×C₃²	#423	1	1	yes	characteristic
190	C₂×C₃²	C₂×#107	1	1	yes	characteristic
191	C₂×C₃²	C₂×#108	1	1	yes	characteristic
192	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
193	C₂²×C₄	#626	1	1	yes	characteristic
194	C₂²×C₄	#627	1	1	yes	characteristic
195	M₁₆	—	12	2	no	—
196	M₁₆	—	36	2	no	—
197	C₂×C₈	—	3	2	no	—
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	6	4	no	—
200	C₂×C₈	—	6	2	no	—
201	C₂×C₈	—	18	2	no	—
202	C₄²	D₃×Dic₃	1	1	yes	characteristic
203	C₄²	#626	1	1	yes	characteristic
204	C₄²	#627	1	1	yes	characteristic
205	C₄²	C₂×D₃²	1	1	yes	characteristic
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	4	1	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	#807	1	2	yes	—
215	C₂²×C₃	#818	1	2	yes	—
216	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
217	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	4	1	no	—
228	C₃×C₄	—	4	1	no	—
229	C₃×C₄	#807	1	2	yes	—
230	C₃×C₄	#818	1	2	yes	—
231	C₃×C₄	#819	1	2	yes	—
232	C₃×C₄	#822	1	2	yes	—
233	C₃×C₄	C₂×D₂₄	1	1	yes	characteristic
234	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
235	C₃×C₄	D₃×M₁₆	1	2	yes	—
236	C₃×C₄	C₂×#208	1	2	yes	—
237	C₃×C₄	C₂×#213	1	1	yes	characteristic
238	C₃×C₄	C₂×#216	1	1	yes	characteristic
239	C₃²	#1567	1	1	yes	characteristic, complemented, Sylow
240	C₂³	Dic₃⋊Dic₃	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	#3714	1	2	yes	—
244	C₂×C₄	#3719	1	2	yes	—
245	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
246	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
247	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
248	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
249	C₂×C₄	C₂×#626	1	1	yes	characteristic
250	C₂×C₄	C₂×#627	1	1	yes	characteristic
251	C₈	—	3	4	no	—
252	C₈	—	6	2	no	—
253	C₈	—	12	2	no	—
254	C₈	—	36	2	no	—
255	C₂×C₃	—	2	2	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	#4924	1	1	yes	characteristic
262	C₂×C₃	C₂×#807	1	1	yes	characteristic
263	C₂×C₃	C₂×#818	1	1	yes	characteristic
264	C₂×C₃	C₂×#819	1	1	yes	characteristic
265	C₂×C₃	#5118	1	1	yes	characteristic
266	C₂×C₃	C₂×#822	1	1	yes	characteristic
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	#63575	1	2	yes	—
270	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	#63575	1	2	yes	—
274	C₄	#63577	1	2	yes	—
275	C₄	D₃×#208	1	2	yes	—
276	C₄	C₂×#3714	1	1	yes	characteristic
277	C₄	C₂×#3719	1	1	yes	characteristic
278	C₃	—	2	1	no	—
279	C₃	#69731	1	1	yes	characteristic, complemented
280	C₃	#70491	1	1	yes	characteristic, complemented
281	C₂	—	2	2	no	—
282	C₂	#10591115	1	1	yes	characteristic
283	C₂	C₂×#63575	1	1	yes	characteristic
284	C₂	C₂×#63577	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1567
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #10589099	C₂	semidirect
82. C₃×C₃⋊C₈	M₁₆	semidirect
187. C₃⋊C₈	#208	semidirect
239. C₃²	#1567	semidirect
279. C₃	#69731	semidirect
280. C₃	#70491	semidirect

Maximal subgroups

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

subgroup	index
#10592623	2
C₃×#4924	2
C₂×C₃×C₄×C₃⋊C₈	2
#10589099	2
#70491	3
#69731	3