Subgroups — #14772775

Order 2816.

642 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#4318	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13069391	C₂	1	3	yes	maximal
4	#13069389	C₂	1	3	yes	maximal
5	#13068753	C₂	1	3	yes	maximal
6	C₄×#66886	C₂	1	3	yes	maximal
7	Q₈×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₁₁×#446	—	2	3	no	—
10	C₈×Q₈×C₁₁	—	2	1	no	—
11	C₁₁×#423	C₂²	1	3	yes	—
12	C₄×C₁₁×M₁₆	C₂²	1	3	yes	—
13	C₂×#66886	C₂²	1	3	yes	—
14	C₂×#66886	C₂²	1	3	yes	—
15	#10622637	—	2	3	no	—
16	Q₈×C₁₁⋊C₈	—	2	1	no	—
17	Q₈×C₁₁⋊C₈	C₄	1	6	yes	—
18	Q₈×C₁₁⋊C₈	C₄	1	2	yes	—
19	#10622537	C₂²	1	3	yes	—
20	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	3	yes	—
21	C₄×#4318	C₂²	1	3	yes	—
22	C₂×C₄×C₁₁⋊C₈	C₂²	1	3	yes	—
23	#10622196	C₂²	1	6	yes	—
24	#10622196	C₂²	1	3	yes	—
25	#10622196	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₁₁	Q₈	1	1	yes	characteristic
27	C₂×C₁₁×M₁₆	C₂³	1	3	yes	—
28	C₂×C₁₁×M₁₆	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₁₁	C₂×C₄	1	3	yes	—
30	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₁	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₁	D₄	1	3	yes	—
33	C₂×C₁₁×C₄⋊C₄	Q₈	1	3	yes	—
34	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
35	C₂×#4318	C₂³	1	3	yes	—
36	C₂×#4318	C₂³	1	1	yes	characteristic
37	C₂²×C₁₁⋊C₈	C₂³	1	3	yes	—
38	C₂²×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
39	C₁₁×C₄⋊C₈	—	2	3	no	—
40	C₁₁×C₄⋊C₈	—	2	3	no	—
41	C₁₁×#97	—	2	3	no	—
42	C₄×C₈×C₁₁	—	2	3	no	—
43	#66886	—	2	6	no	—
44	#66886	—	2	6	no	—
45	C₄⋊(C₁₁⋊C₈)	—	2	3	no	—
46	C₄⋊(C₁₁⋊C₈)	—	2	3	no	—
47	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	6	yes	—
48	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	6	yes	—
49	#66867	—	2	3	no	—
50	C₄×C₁₁⋊C₈	—	2	3	no	—
51	C₄×C₁₁⋊C₈	C₂×C₄	1	6	yes	—
52	C₄×C₁₁⋊C₈	C₂×C₄	1	6	yes	—
53	#22746	—	11	1	no	maximal, Sylow
54	C₂×Q₈×C₁₁	—	2	4	no	—
55	C₂×Q₈×C₁₁	C₄⋊C₄	1	3	yes	—
56	C₂×Q₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
58	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
59	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
60	C₁₁×M₁₆	—	2	3	no	—
61	C₁₁×M₁₆	—	2	1	no	—
62	C₁₁×M₁₆	—	2	3	no	—
63	C₁₁×M₁₆	#41	1	2	yes	—
64	C₂×C₈×C₁₁	—	2	1	no	—
65	C₂×C₈×C₁₁	—	2	3	no	—
66	C₁₁×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
67	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
68	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
69	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
70	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
71	C₄²×C₁₁	C₂×D₄	1	3	yes	—
72	C₄²×C₁₁	C₂×D₄	1	3	yes	—
73	#4318	—	2	3	no	—
74	#4318	—	2	1	no	—
75	#4318	—	2	3	no	—
76	#4318	#41	1	2	yes	—
77	C₂×C₁₁⋊C₈	—	2	1	no	—
78	C₂×C₁₁⋊C₈	—	2	3	no	—
79	C₂×C₁₁⋊C₈	—	2	3	no	—
80	C₂×C₁₁⋊C₈	—	2	3	no	—
81	C₂×C₁₁⋊C₈	—	2	1	no	—
82	C₂×C₁₁⋊C₈	—	2	1	no	—
83	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₁₁⋊C₈	C₂²×C₄	1	6	yes	—
85	Q₈×M₁₆	—	11	1	no	—
86	Q₈×M₁₆	D₁₁	1	1	yes	characteristic
87	C₂×C₈×Q₈	—	11	1	no	—
88	#1950	—	11	3	no	—
89	#1950	—	11	3	no	—
90	#1837	—	11	3	no	—
91	C₄×#108	—	11	3	no	—
92	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
93	Q₈×C₁₁	—	2	6	no	—
94	Q₈×C₁₁	#108	1	4	yes	complemented
95	C₂×C₄×C₁₁	—	2	6	no	—
96	C₂×C₄×C₁₁	—	4	3	no	—
97	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
98	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
99	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
100	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
101	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
106	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
107	C₈×C₁₁	—	2	2	no	—
108	C₈×C₁₁	—	4	3	no	—
109	C₁₁⋊C₈	—	2	6	no	—
110	C₁₁⋊C₈	—	2	2	no	—
111	C₁₁⋊C₈	—	2	2	no	—
112	C₁₁⋊C₈	—	4	3	no	—
113	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
114	#446	—	2	3	no	—
115	#446	—	22	3	no	—
116	C₈×Q₈	—	2	1	no	—
117	C₈×Q₈	—	11	6	no	—
118	C₈×Q₈	—	11	2	no	—
119	C₈×Q₈	—	22	1	no	—
120	C₂×#108	—	11	3	no	—
121	C₂×#108	—	11	3	no	—
122	#423	—	11	3	no	—
123	#423	C₂×D₁₁	1	3	yes	—
124	C₂×C₄⋊C₈	—	11	3	no	—
125	C₄×M₁₆	—	11	3	no	—
126	C₄×M₁₆	C₂×D₁₁	1	3	yes	—
127	C₂×C₄×C₈	—	11	3	no	—
128	#341	—	11	3	no	—
129	#341	—	11	6	no	—
130	#341	—	11	1	no	—
131	C₂²×C₁₁	—	2	1	no	—
132	C₂²×C₁₁	—	2	1	no	—
133	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₁₁	#557	1	1	yes	characteristic
135	C₂²×C₁₁	#557	1	1	yes	characteristic
136	C₄×C₁₁	—	2	3	no	—
137	C₄×C₁₁	—	2	3	no	—
138	C₄×C₁₁	—	2	3	no	—
139	C₄×C₁₁	C₂×#108	1	6	yes	—
140	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
141	C₄×C₁₁	#519	1	1	yes	characteristic
142	C₄×C₁₁	#548	1	1	yes	characteristic
143	C₄×C₁₁	#548	1	1	yes	characteristic
144	C₂²×Q₈	Dic₂₂	1	1	yes	characteristic
145	C₂×M₁₆	—	11	1	no	—
146	C₂×M₁₆	—	11	3	no	—
147	C₂×M₁₆	C₂²×D₁₁	1	3	yes	—
148	C₂×M₁₆	C₂²×D₁₁	1	1	yes	characteristic
149	C₂²×C₈	—	11	1	no	—
150	C₂²×C₈	—	11	3	no	—
151	C₄×Q₈	C₄×D₁₁	1	3	yes	—
152	C₄×Q₈	C₄×D₁₁	1	1	yes	characteristic
153	C₄×Q₈	C₂²⋊D₁₁	1	1	yes	characteristic
154	C₄×Q₈	C₂²⋊D₁₁	1	3	yes	—
155	C₂×C₄⋊C₄	Dic₂₂	1	3	yes	—
156	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
157	#108	—	22	6	no	—
158	#108	—	22	6	no	—
159	C₄⋊C₈	—	2	3	no	—
160	C₄⋊C₈	—	2	3	no	—
161	C₄⋊C₈	—	11	6	no	—
162	C₄⋊C₈	—	11	6	no	—
163	C₄⋊C₈	—	22	3	no	—
164	C₄⋊C₈	—	22	3	no	—
165	#97	—	2	3	no	—
166	#97	—	22	3	no	—
167	C₄×C₈	—	2	3	no	—
168	C₄×C₈	—	11	6	no	—
169	C₄×C₈	—	11	6	no	—
170	C₄×C₈	—	22	3	no	—
171	C₂×C₁₁	—	2	1	no	—
172	C₂×C₁₁	—	2	1	no	—
173	C₂×C₁₁	#2351	1	1	yes	characteristic
174	C₂×C₁₁	C₂²×#108	1	1	yes	characteristic
175	C₂×C₁₁	#2916	1	1	yes	characteristic
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	Dic₁₁⋊C₄	1	3	yes	—
178	C₂×Q₈	Dic₁₁⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
180	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
181	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
182	M₁₆	—	2	1	no	—
183	M₁₆	—	2	3	no	—
184	M₁₆	—	2	3	no	—
185	M₁₆	—	11	2	no	—
186	M₁₆	—	22	1	no	—
187	M₁₆	—	22	3	no	—
188	M₁₆	—	22	3	no	—
189	M₁₆	#4343	1	2	yes	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	3	no	—
192	C₂×C₈	—	11	2	no	—
193	C₂×C₈	—	11	6	no	—
194	C₂×C₈	—	22	3	no	—
195	C₂×C₈	—	22	1	no	—
196	C₂×C₈	—	22	3	no	—
197	C₂×C₈	—	22	1	no	—
198	C₂×C₈	—	22	1	no	—
199	C₂×C₈	—	22	3	no	—
200	C₄⋊C₄	Dic₁₁⋊C₄	1	6	yes	—
201	C₄⋊C₄	Dic₁₁⋊C₄	1	3	yes	—
202	C₄⋊C₄	Dic₁₁⋊C₄	1	3	yes	—
203	C₄²	C₂×C₄×D₁₁	1	3	yes	—
204	C₄²	C₂×C₄×D₁₁	1	3	yes	—
205	C₄²	C₂×C₂²⋊D₁₁	1	3	yes	—
206	C₄²	C₂×C₂²⋊D₁₁	1	3	yes	—
207	C₁₁	#22746	1	1	yes	characteristic, complemented, Sylow
208	C₂³	C₂²×Dic₂₂	1	1	yes	characteristic
209	Q₈	—	2	6	no	—
210	Q₈	#66886	1	4	yes	complemented
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂×Dic₁₁⋊C₄	1	3	yes	—
214	C₂×C₄	C₂×Dic₁₁⋊C₄	1	3	yes	—
215	C₂×C₄	C₂×Dic₁₁⋊C₄	1	3	yes	—
216	C₂×C₄	C₂×Dic₁₁⋊C₄	1	3	yes	—
217	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
218	C₂×C₄	C₂×#4343	1	1	yes	characteristic
219	C₂×C₄	#67040	1	1	yes	characteristic
220	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
221	C₂×C₄	#67043	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	3	no	—
225	C₈	—	22	2	no	—
226	C₈	—	22	6	no	—
227	C₈	—	22	2	no	—
228	C₈	—	44	3	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#10623100	1	1	yes	characteristic
232	C₂²	#10623101	1	1	yes	characteristic
233	C₂²	C₂²×Dic₁₁⋊C₄	1	1	yes	characteristic
234	C₄	—	2	3	no	—
235	C₄	—	2	3	no	—
236	C₄	—	2	3	no	—
237	C₄	C₂×#66886	1	6	yes	—
238	C₄	#10623103	1	1	yes	characteristic
239	C₄	C₄×#4343	1	1	yes	characteristic
240	C₄	#10623348	1	1	yes	characteristic
241	C₄	#10623362	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	#13071534	1	1	yes	characteristic
245	C₂	C₂²×#66886	1	1	yes	characteristic
246	C₂	#13077225	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	Q₈×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₁	C₂²×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	Q₈×M₁₆	D₁₁
hypocenter	C₁₁	#22746
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
94. Q₈×C₁₁	#108	semidirect
207. C₁₁	#22746	semidirect
210. Q₈	#66886	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#4318	2
C₂×Q₈×C₁₁⋊C₈	2
#13069391	2
#13069389	2
#13068753	2
C₄×#66886	2
Q₈×C₁₁×M₁₆	2
#22746	11