Subgroups — #11777017

Order 1152.

405 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×#4924	C₂	1	1	yes	maximal, characteristic
4	#10589099	C₂	1	4	yes	maximal, complemented
5	#70750	—	3	1	no	maximal
6	#70494	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	—	2	1	no	—
9	C₂×C₃×#208	—	2	1	no	—
10	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
12	C₃×#830	C₄	1	2	yes	—
13	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
14	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
15	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃×#431	—	3	1	no	—
18	C₃×#423	—	3	1	no	—
19	#4925	D₃	1	1	yes	characteristic
20	#4924	D₃	1	1	yes	characteristic
21	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
24	#4479	—	3	4	no	—
25	#4458	—	3	4	no	—
26	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
29	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃²×C₄²	D₄	1	1	yes	characteristic
31	C₃²×C₄²	Q₈	1	1	yes	characteristic
32	C₃²×C₄²	C₂³	1	1	yes	characteristic
33	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	4	2	no	—
35	C₃×#208	—	4	2	no	—
36	C₂×C₃×C₃⋊C₈	—	2	2	no	—
37	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
38	#1571	—	9	1	no	Sylow
39	C₂×C₃×M₁₆	—	6	1	no	—
40	C₂×C₃×C₄²	—	2	1	no	—
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×#208	—	2	1	no	—
44	C₂²×C₃⋊C₈	—	6	2	no	—
45	C₂²×C₃⋊C₈	—	6	1	no	—
46	C₂²×C₃⋊C₈	—	6	1	no	—
47	C₃×C₄⋊C₈	—	3	2	no	—
48	C₃×C₄⋊C₈	—	3	2	no	—
49	C₃×C₄⋊C₈	—	6	1	no	—
50	C₃×C₂²⋊C₈	—	3	2	no	—
51	C₃×C₄×C₈	—	3	2	no	—
52	#830	Dic₃	1	2	yes	—
53	C₄⋊(C₃⋊C₈)	—	2	1	no	—
54	C₄⋊(C₃⋊C₈)	—	3	2	no	—
55	C₄⋊(C₃⋊C₈)	—	3	2	no	—
56	C₄⋊(C₃⋊C₈)	—	6	2	no	—
57	C₄⋊(C₃⋊C₈)	—	6	2	no	—
58	C₄⋊(C₃⋊C₈)	—	6	1	no	—
59	C₄⋊(C₃⋊C₈)	—	6	1	no	—
60	C₄⋊(C₃⋊C₈)	Dic₃	1	2	yes	—
61	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
62	C₄×C₃⋊C₈	C₂×D₃	1	2	yes	—
63	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	—	4	1	no	—
66	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	D₈	1	2	yes	—
70	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
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×Q₈	1	1	yes	characteristic
74	C₃⋊(C₃⋊C₈)	—	4	2	no	—
75	C₃×C₃⋊C₈	—	2	4	no	—
76	C₃×C₃⋊C₈	—	4	2	no	—
77	#431	—	3	1	no	—
78	#423	—	3	1	no	—
79	C₂×C₄⋊C₈	—	9	1	no	—
80	#335	—	9	4	no	—
81	C₂²×C₃×C₄	—	2	1	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
85	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	#67	1	1	yes	characteristic
90	C₃×M₁₆	—	12	2	no	—
91	C₂×C₃×C₈	—	3	4	no	—
92	C₂×C₃×C₈	—	6	2	no	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	Dic₆	1	1	yes	characteristic
98	C₃×C₄²	Dic₆	1	1	yes	characteristic
99	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
100	C₃×C₄²	D₁₂	1	1	yes	characteristic
101	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
102	C₃×C₄²	#67	1	1	yes	characteristic
103	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
104	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
105	#208	—	4	2	no	—
106	C₂×C₃⋊C₈	—	2	2	no	—
107	C₂×C₃⋊C₈	—	6	4	no	—
108	C₂×C₃⋊C₈	—	6	2	no	—
109	C₂×C₃⋊C₈	—	6	2	no	—
110	C₂×C₃⋊C₈	—	12	4	no	—
111	C₂×C₃⋊C₈	—	12	2	no	—
112	C₂×C₃⋊C₈	—	12	2	no	—
113	C₂×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
114	C₂²×C₃²	—	2	1	no	—
115	C₂²×C₃²	—	2	1	no	—
116	C₂²×C₃²	#107	1	2	yes	—
117	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₃²×C₄	—	2	1	no	—
119	C₃²×C₄	—	2	1	no	—
120	C₃²×C₄	—	2	1	no	—
121	C₃²×C₄	#107	1	2	yes	—
122	C₃²×C₄	#131	1	2	yes	—
123	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
124	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×M₁₆	—	6	1	no	—
126	C₂²×C₈	—	18	1	no	—
127	C₂×C₄²	D₃²	1	1	yes	characteristic
128	C₄⋊C₈	—	3	2	no	—
129	C₄⋊C₈	—	3	2	no	—
130	C₄⋊C₈	—	6	1	no	—
131	C₄⋊C₈	—	9	2	no	—
132	C₄⋊C₈	—	18	1	no	—
133	C₂²⋊C₈	—	3	2	no	—
134	C₄×C₈	—	3	2	no	—
135	C₂³×C₃	—	2	1	no	—
136	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
137	C₂³×C₃	C₄⋊Dic₃	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	4	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	4	1	no	—
150	C₂×C₃×C₄	—	4	1	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	D₂₄	1	2	yes	—
153	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
154	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
156	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
158	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	#213	1	2	yes	—
161	C₂×C₃×C₄	#216	1	2	yes	—
162	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
168	C₃×C₈	—	6	4	no	—
169	C₃×C₈	—	12	2	no	—
170	C₃⋊C₈	—	2	4	no	—
171	C₃⋊C₈	—	4	2	no	—
172	C₃⋊C₈	—	12	4	no	—
173	C₃⋊C₈	—	12	2	no	—
174	C₃⋊C₈	—	12	2	no	—
175	C₂×C₃²	—	2	2	no	—
176	C₂×C₃²	#424	1	1	yes	characteristic
177	C₂×C₃²	C₂×#107	1	1	yes	characteristic
178	C₂×C₃²	#430	1	1	yes	characteristic
179	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
180	C₂²×C₄	#626	1	1	yes	characteristic
181	C₂²×C₄	#627	1	1	yes	characteristic
182	M₁₆	—	12	2	no	—
183	C₂×C₈	—	3	4	no	—
184	C₂×C₈	—	6	2	no	—
185	C₂×C₈	—	18	2	no	—
186	C₂×C₈	—	36	2	no	—
187	C₄²	D₃×Dic₃	1	1	yes	characteristic
188	C₄²	#626	1	1	yes	characteristic
189	C₄²	#627	1	1	yes	characteristic
190	C₄²	C₂×D₃²	1	1	yes	characteristic
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	#807	1	2	yes	—
200	C₂²×C₃	#818	1	2	yes	—
201	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
202	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	#807	1	2	yes	—
217	C₃×C₄	#818	1	2	yes	—
218	C₃×C₄	C₂×D₂₄	1	1	yes	characteristic
219	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
220	C₃×C₄	#907	1	2	yes	—
221	C₃×C₄	C₂×#213	1	1	yes	characteristic
222	C₃×C₄	C₂×#216	1	1	yes	characteristic
223	C₃×C₄	#948	1	2	yes	—
224	C₃²	#1571	1	1	yes	characteristic, complemented, Sylow
225	C₂³	Dic₃⋊Dic₃	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	#3714	1	2	yes	—
229	C₂×C₄	#3719	1	2	yes	—
230	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
231	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
232	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
233	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
234	C₂×C₄	C₂×#626	1	1	yes	characteristic
235	C₂×C₄	C₂×#627	1	1	yes	characteristic
236	C₈	—	6	4	no	—
237	C₈	—	12	2	no	—
238	C₈	—	36	2	no	—
239	C₂×C₃	—	2	4	no	—
240	C₂×C₃	—	2	2	no	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	C₂×#807	1	1	yes	characteristic
246	C₂×C₃	#4999	1	1	yes	characteristic
247	C₂×C₃	C₂×#818	1	1	yes	characteristic
248	C₂×C₃	#5119	1	1	yes	characteristic
249	C₂×C₃	#5124	1	1	yes	characteristic
250	C₂×C₃	#5125	1	1	yes	characteristic
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	#63575	1	2	yes	—
254	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	#63575	1	2	yes	—
259	C₄	#63815	1	2	yes	—
260	C₄	C₂×#3714	1	1	yes	characteristic
261	C₄	C₂×#3719	1	1	yes	characteristic
262	C₃	—	2	1	no	—
263	C₃	#70494	1	1	yes	characteristic, complemented
264	C₃	#70750	1	1	yes	characteristic, complemented
265	C₂	—	2	2	no	—
266	C₂	#10591117	1	1	yes	characteristic
267	C₂	C₂×#63575	1	1	yes	characteristic
268	C₂	#10591141	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	#1571
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
224. C₃²	#1571	semidirect
263. C₃	#70494	semidirect
264. C₃	#70750	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×#4925	2
C₃×#4924	2
#10589099	2
#70750	3
#70494	3