Subgroups — Q₈×#66883 #14772666

Order 2816.

658 conjugacy classes of subgroups.

The subgroup lattice has 208 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	2	yes	maximal
2	#13069392	C₂	1	6	yes	maximal
3	#13068788	C₂	1	3	yes	maximal
4	C₄×#66883	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₁	C₄	1	6	yes	—
8	C₈×Q₈×C₁₁	C₄	1	2	yes	—
9	C₂×C₁₁×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
11	C₂×#66883	C₂²	1	3	yes	—
12	C₂×#66883	C₂²	1	6	yes	—
13	C₂×#66883	C₂²	1	1	yes	characteristic
14	#10622637	—	2	6	no	—
15	Q₈×C₁₁⋊C₈	—	2	2	no	—
16	#10622537	C₂²	1	6	yes	—
17	C₄×#4318	C₂²	1	6	yes	—
18	#10622190	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₁	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₁	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₁	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₁	D₄	1	3	yes	—
26	C₂×C₁₁×C₄⋊C₄	Q₈	1	3	yes	—
27	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
28	C₂×#4318	C₂³	1	6	yes	—
29	C₂×#4318	C₂³	1	2	yes	—
30	C₁₁×C₄⋊C₈	C₂×C₄	1	6	yes	—
31	C₁₁×C₄⋊C₈	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₁₁	C₂×C₄	1	6	yes	—
33	C₄×C₈×C₁₁	C₂×C₄	1	6	yes	—
34	#66883	—	2	6	no	—
35	#66883	—	4	6	no	—
36	#66883	Q₈	1	4	yes	direct product factor
37	C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
38	C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
39	#66867	—	2	6	no	—
40	C₄×C₁₁⋊C₈	—	2	6	no	—
41	Q₈×#108	—	11	1	no	maximal, Sylow
42	C₂×Q₈×C₁₁	—	2	4	no	—
43	C₂×Q₈×C₁₁	C₄⋊C₄	1	3	yes	—
44	C₂×Q₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
47	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₁₁	—	2	3	no	—
49	C₂×C₈×C₁₁	—	2	3	no	—
50	C₂×C₈×C₁₁	C₂²×C₄	1	6	yes	—
51	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₁	#41	1	2	yes	—
54	C₁₁×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
55	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
56	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
57	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
58	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
59	C₄²×C₁₁	C₂×D₄	1	3	yes	—
60	C₄²×C₁₁	C₂×D₄	1	3	yes	—
61	#4318	—	2	6	no	—
62	#4318	—	2	6	no	—
63	#4318	—	2	2	no	—
64	#4318	C₂×Q₈	1	4	yes	—
65	C₂×C₁₁⋊C₈	—	2	2	no	—
66	C₂×C₁₁⋊C₈	—	2	6	no	—
67	Q₈×M₁₆	—	11	2	no	—
68	C₂×C₈×Q₈	D₁₁	1	1	yes	characteristic
69	#1950	—	11	6	no	—
70	#1848	—	11	3	no	—
71	C₄×#108	—	11	3	no	—
72	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
73	Q₈×C₁₁	—	2	6	no	—
74	Q₈×C₁₁	#108	1	4	yes	complemented
75	C₂×C₄×C₁₁	—	2	6	no	—
76	C₂×C₄×C₁₁	—	4	3	no	—
77	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
79	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
80	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
86	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
87	C₈×C₁₁	—	2	6	no	—
88	C₈×C₁₁	C₄×Q₈	1	4	yes	—
89	C₁₁⋊C₈	—	2	4	no	—
90	C₁₁⋊C₈	—	4	6	no	—
91	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
92	#446	—	22	6	no	—
93	C₈×Q₈	—	22	2	no	—
94	C₈×Q₈	Dic₁₁	1	6	yes	—
95	C₈×Q₈	Dic₁₁	1	2	yes	—
96	C₂×#108	—	11	1	no	—
97	C₂×#108	—	11	3	no	—
98	C₂×#108	—	11	6	no	—
99	#423	—	11	6	no	—
100	C₂×C₄⋊C₈	C₂×D₁₁	1	3	yes	—
101	C₄×M₁₆	—	11	6	no	—
102	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
103	#341	—	11	6	no	—
104	C₂²×C₁₁	—	2	1	no	—
105	C₂²×C₁₁	—	2	1	no	—
106	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₁	#557	1	1	yes	characteristic
108	C₂²×C₁₁	Q₈²	1	1	yes	characteristic
109	C₄×C₁₁	—	2	3	no	—
110	C₄×C₁₁	—	2	3	no	—
111	C₄×C₁₁	—	2	3	no	—
112	C₄×C₁₁	C₂×#108	1	6	yes	—
113	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₄×C₁₁	#520	1	1	yes	characteristic
115	C₄×C₁₁	D₄×Q₈	1	1	yes	characteristic
116	C₄×C₁₁	#550	1	1	yes	characteristic
117	C₂²×Q₈	Dic₂₂	1	1	yes	characteristic
118	C₂×M₁₆	—	11	2	no	—
119	C₂×M₁₆	—	11	6	no	—
120	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
121	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
122	C₄×Q₈	D₄₄	1	1	yes	characteristic
123	C₄×Q₈	D₄₄	1	3	yes	—
124	C₄×Q₈	C₂×Dic₁₁	1	3	yes	—
125	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
126	C₂×C₄⋊C₄	Dic₂₂	1	3	yes	—
127	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
128	#108	—	11	4	no	—
129	#108	—	22	6	no	—
130	#108	—	44	6	no	—
131	C₄⋊C₈	—	22	6	no	—
132	C₄⋊C₈	—	22	6	no	—
133	C₄⋊C₈	C₂×Dic₁₁	1	6	yes	—
134	C₄⋊C₈	C₂×Dic₁₁	1	6	yes	—
135	#97	—	22	6	no	—
136	C₄×C₈	—	22	6	no	—
137	C₄×C₈	C₂×Dic₁₁	1	6	yes	—
138	C₄×C₈	C₂×Dic₁₁	1	6	yes	—
139	C₂×C₁₁	—	2	1	no	—
140	C₂×C₁₁	—	2	1	no	—
141	C₂×C₁₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₁₁	C₂²×#108	1	1	yes	characteristic
143	C₂×C₁₁	#2916	1	1	yes	characteristic
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	C₄⋊Dic₁₁	1	3	yes	—
146	C₂×Q₈	C₄⋊Dic₁₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
148	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
149	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
150	M₁₆	—	11	4	no	—
151	M₁₆	—	22	2	no	—
152	M₁₆	—	22	6	no	—
153	M₁₆	—	22	6	no	—
154	C₂×C₈	—	2	3	no	—
155	C₂×C₈	—	2	3	no	—
156	C₂×C₈	—	22	2	no	—
157	C₂×C₈	—	22	6	no	—
158	C₂×C₈	Q₈×D₁₁	1	2	yes	—
159	C₂×C₈	#4343	1	2	yes	—
160	C₂×C₈	C₂²×Dic₁₁	1	6	yes	—
161	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
162	C₄⋊C₄	C₄⋊Dic₁₁	1	6	yes	—
163	C₄⋊C₄	C₄⋊Dic₁₁	1	3	yes	—
164	C₄⋊C₄	C₄⋊Dic₁₁	1	3	yes	—
165	C₄²	C₂×D₄₄	1	3	yes	—
166	C₄²	C₂×D₄₄	1	3	yes	—
167	C₄²	C₂²×Dic₁₁	1	3	yes	—
168	C₄²	C₂²×Dic₁₁	1	3	yes	—
169	C₁₁	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂²×Dic₂₂	1	1	yes	characteristic
171	Q₈	—	2	6	no	—
172	Q₈	#66883	1	4	yes	direct product factor
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
179	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
181	C₂×C₄	C₂×#4343	1	1	yes	characteristic
182	C₂×C₄	#67042	1	1	yes	characteristic
183	C₂×C₄	#67043	1	1	yes	characteristic
184	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
185	C₈	—	2	6	no	—
186	C₈	—	22	4	no	—
187	C₈	—	44	6	no	—
188	C₈	Q₈×Dic₁₁	1	4	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	Q₈×Dic₂₂	1	1	yes	characteristic
192	C₂²	#10623101	1	1	yes	characteristic
193	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×#66883	1	6	yes	—
198	C₄	Q₈×D₄₄	1	1	yes	characteristic
199	C₄	#10623108	1	1	yes	characteristic
200	C₄	C₂×Q₈×Dic₁₁	1	1	yes	characteristic
201	C₄	#10623358	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	Q₈×C₄⋊Dic₁₁	1	1	yes	characteristic
205	C₂	C₂²×#66883	1	1	yes	characteristic
206	C₂	#13077227	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₈×Q₈×C₁₁	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₁	C₂²×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	Q₈×#108
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
36. #66883	Q₈	direct (N × H)
74. Q₈×C₁₁	#108	semidirect
169. C₁₁	Q₈×#108	semidirect
172. Q₈	#66883	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#4318	2
#13069392	2
#13068788	2
C₄×#66883	2
C₂×C₈×Q₈×C₁₁	2
Q₈×#108	11

Subgroups — Q8×#66883 #14772666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#66883 #14772666