Subgroups — Q₈×#66876 #14907287

Order 2816.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070525	C₂	1	6	yes	maximal, complemented
2	Q₈×C₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	#13070372	C₂	1	3	yes	maximal
4	C₄×#66876	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₁	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₁₁	C₄	1	3	yes	—
8	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₁₁	C₂²	1	3	yes	—
11	C₂×C₁₁×C₄⋊C₈	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
13	C₂×#66876	C₂²	1	3	yes	—
14	C₂×#66876	C₂²	1	1	yes	characteristic
15	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	6	yes	—
16	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	6	yes	—
17	#10622099	C₂²	1	6	yes	—
18	C₄×C₁₁⋊C₁₆	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₁	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₁	C₈	1	3	yes	—
23	C₄×Q₈×C₁₁	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₁₁	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₁×C₄⋊C₄	C₈	1	3	yes	—
27	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
28	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
29	C₁₁×C₄⋊C₈	C₂×C₄	1	6	yes	—
30	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
32	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₁₁	C₂³	1	3	yes	—
34	C₄×C₈×C₁₁	C₂³	1	3	yes	—
35	#66876	—	2	6	no	—
36	#66876	Q₈	1	4	yes	direct product factor
37	C₂×C₁₁⋊C₁₆	C₂³	1	6	yes	—
38	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
39	Q₈×M₃₂	—	11	1	no	maximal, Sylow
40	C₂×Q₈×C₁₁	—	2	4	no	—
41	C₂×Q₈×C₁₁	C₂×C₈	1	3	yes	—
42	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₁	—	2	6	no	—
47	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₁	#41	1	2	yes	—
52	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
53	C₁₁×C₄⋊C₄	C₂×C₈	1	6	yes	—
54	C₁₁×C₄⋊C₄	C₂×C₈	1	3	yes	—
55	C₁₁×C₄⋊C₄	C₂×C₈	1	3	yes	—
56	C₄²×C₁₁	C₂×C₈	1	3	yes	—
57	C₄²×C₁₁	C₂×C₈	1	3	yes	—
58	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
59	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
60	C₁₁⋊C₁₆	—	2	6	no	—
61	C₁₁⋊C₁₆	C₂×Q₈	1	4	yes	complemented
62	C₂×C₈×Q₈	D₁₁	1	1	yes	characteristic
63	#2183	—	11	6	no	—
64	Q₈×C₁₆	—	11	2	no	—
65	#2150	—	11	3	no	—
66	C₄×M₃₂	—	11	3	no	—
67	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₁₁	—	2	6	no	—
69	Q₈×C₁₁	M₃₂	1	4	yes	complemented
70	C₂×C₄×C₁₁	—	2	6	no	—
71	C₂×C₄×C₁₁	—	4	3	no	—
72	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₁₁	#131	1	2	yes	—
79	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₁₁	—	2	3	no	—
81	C₈×C₁₁	—	2	3	no	—
82	C₈×C₁₁	C₄×Q₈	1	2	yes	—
83	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×Q₈	Dic₁₁	1	1	yes	characteristic
86	C₂×M₃₂	—	11	1	no	—
87	C₂×M₃₂	—	11	3	no	—
88	C₈×Q₈	Dic₁₁	1	3	yes	—
89	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
90	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
91	C₈×Q₈	C₂×D₁₁	1	3	yes	—
92	C₂×C₄⋊C₈	Dic₁₁	1	3	yes	—
93	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
94	C₄⋊C₁₆	—	11	6	no	—
95	C₄⋊C₁₆	—	11	6	no	—
96	#346	—	11	6	no	—
97	C₄×C₁₆	—	11	6	no	—
98	C₂²×C₁₁	—	2	1	no	—
99	C₂²×C₁₁	—	2	1	no	—
100	C₂²×C₁₁	C₈×Q₈	1	2	yes	—
101	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
102	C₄×C₁₁	—	2	3	no	—
103	C₄×C₁₁	—	2	3	no	—
104	C₄×C₁₁	—	2	3	no	—
105	C₄×C₁₁	C₈×Q₈	1	2	yes	—
106	C₄×C₁₁	C₂×M₃₂	1	6	yes	—
107	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
109	C₂²×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₁₁	1	3	yes	—
111	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
112	C₄×Q₈	C₁₁⋊C₈	1	3	yes	—
113	C₄×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
114	C₄×Q₈	C₂×Dic₁₁	1	3	yes	—
115	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₁₁⋊C₈	1	3	yes	—
117	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
118	M₃₂	—	11	4	no	—
119	M₃₂	—	22	6	no	—
120	C₂×C₁₆	—	11	2	no	—
121	C₂×C₁₆	—	11	6	no	—
122	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
123	C₄⋊C₈	C₂×Dic₁₁	1	6	yes	—
124	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
125	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
126	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
127	C₄×C₈	C₂²×D₁₁	1	3	yes	—
128	C₄×C₈	C₂²×D₁₁	1	3	yes	—
129	C₂×C₁₁	—	2	2	no	—
130	C₂×C₁₁	C₂×C₈×Q₈	1	1	yes	characteristic
131	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
132	C₂×C₁₁	#3407	1	1	yes	characteristic
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	C₂×C₁₁⋊C₈	1	3	yes	—
135	C₂×Q₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₁₁⋊C₈	1	3	yes	—
137	C₂²×C₄	C₂×C₁₁⋊C₈	1	3	yes	—
138	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
139	C₂×C₈	—	2	6	no	—
140	C₂×C₈	Q₈×D₁₁	1	2	yes	—
141	C₂×C₈	#4343	1	2	yes	—
142	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
143	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
144	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
145	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₁₁⋊C₈	1	6	yes	—
147	C₄⋊C₄	C₂×C₁₁⋊C₈	1	3	yes	—
148	C₄⋊C₄	C₂×C₁₁⋊C₈	1	3	yes	—
149	C₄²	C₂×C₁₁⋊C₈	1	3	yes	—
150	C₄²	C₂×C₁₁⋊C₈	1	3	yes	—
151	C₄²	C₂²×Dic₁₁	1	3	yes	—
152	C₄²	C₂²×Dic₁₁	1	3	yes	—
153	C₁₆	—	11	4	no	—
154	C₁₆	—	22	6	no	—
155	C₁₁	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
156	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	#66876	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
162	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
163	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
164	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₁₁⋊C₈	1	3	yes	—
166	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
167	C₂×C₄	#67001	1	2	yes	—
168	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	2	3	no	—
171	C₈	Q₈×Dic₁₁	1	2	yes	—
172	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
173	C₈	C₂×#4343	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	Q₈×C₁₁⋊C₈	1	2	yes	—
177	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	Q₈×C₁₁⋊C₈	1	2	yes	—
182	C₄	C₂×#66876	1	6	yes	—
183	C₄	C₂×Q₈×Dic₁₁	1	1	yes	characteristic
184	C₄	C₂×#67001	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×Q₈×C₁₁⋊C₈	1	1	yes	characteristic
187	C₂	C₂²×#66876	1	1	yes	characteristic
188	C₂	#13081719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₁₁	C₂
prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #13070525	C₂	semidirect
2. Q₈×C₁₁⋊C₁₆	C₂	semidirect
36. #66876	Q₈	direct (N × H)
61. C₁₁⋊C₁₆	C₂×Q₈	semidirect
69. Q₈×C₁₁	M₃₂	semidirect
155. C₁₁	Q₈×M₃₂	semidirect
158. Q₈	#66876	direct (N × H)

Maximal subgroups

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

subgroup	index
#13070525	2
Q₈×C₁₁⋊C₁₆	2
#13070372	2
C₄×#66876	2
C₂×C₈×Q₈×C₁₁	2
Q₈×M₃₂	11

Subgroups — Q8×#66876 #14907287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#66876 #14907287