Subgroups — Q₈×#14556068 #32150966

Order 18176.

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	#22590249	C₂	1	6	yes	maximal, complemented
2	Q₈×C₇₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	#22590096	C₂	1	3	yes	maximal
4	C₄×#14556068	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	3	yes	—
7	C₂×#14556068	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	6	yes	—
9	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	6	yes	—
10	#17985775	C₂²	1	6	yes	—
11	C₄×C₇₁⋊C₁₆	C₂²	1	6	yes	—
12	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₇₁	C₄	1	3	yes	—
14	C₈×Q₈×C₇₁	C₄	1	1	yes	characteristic
15	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₇₁	C₂²	1	3	yes	—
17	C₂×C₄⋊C₈×C₇₁	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
19	#14556068	—	2	6	no	—
20	#14556068	Q₈	1	4	yes	direct product factor
21	C₂×C₇₁⋊C₁₆	C₂³	1	6	yes	—
22	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
23	C₂²×Q₈×C₇₁	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₄×Q₈×C₇₁	C₈	1	3	yes	—
27	C₄×Q₈×C₇₁	C₈	1	1	yes	characteristic
28	C₄×Q₈×C₇₁	C₂×C₄	1	3	yes	—
29	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₁	C₈	1	3	yes	—
31	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₇₁	C₂×C₄	1	3	yes	—
33	C₄⋊C₈×C₇₁	C₂×C₄	1	6	yes	—
34	C₄⋊C₈×C₇₁	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₇₁	C₂³	1	3	yes	—
38	C₄×C₈×C₇₁	C₂³	1	3	yes	—
39	C₇₁⋊C₁₆	—	2	6	no	—
40	C₇₁⋊C₁₆	C₂×Q₈	1	4	yes	complemented
41	C₂×Q₈×C₇₁	—	2	4	no	—
42	C₂×Q₈×C₇₁	C₂×C₈	1	3	yes	—
43	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
46	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	—	2	6	no	—
48	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₇₁	#41	1	2	yes	—
53	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₇₁	C₂×C₈	1	6	yes	—
55	C₄⋊C₄×C₇₁	C₂×C₈	1	3	yes	—
56	C₄⋊C₄×C₇₁	C₂×C₈	1	3	yes	—
57	C₄²×C₇₁	C₂×C₈	1	3	yes	—
58	C₄²×C₇₁	C₂×C₈	1	3	yes	—
59	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
60	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
61	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₇₁	—	2	6	no	—
63	Q₈×C₇₁	M₃₂	1	4	yes	complemented
64	C₂×C₄×C₇₁	—	2	6	no	—
65	C₂×C₄×C₇₁	—	4	3	no	—
66	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
72	C₂×C₄×C₇₁	#131	1	2	yes	—
73	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
74	C₈×C₇₁	—	2	3	no	—
75	C₈×C₇₁	—	2	3	no	—
76	C₈×C₇₁	C₄×Q₈	1	2	yes	—
77	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
79	C₂²×C₇₁	—	2	1	no	—
80	C₂²×C₇₁	—	2	1	no	—
81	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
82	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₇₁	—	2	3	no	—
84	C₄×C₇₁	—	2	3	no	—
85	C₄×C₇₁	—	2	3	no	—
86	C₄×C₇₁	C₈×Q₈	1	2	yes	—
87	C₄×C₇₁	C₂×M₃₂	1	6	yes	—
88	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
90	Q₈×M₃₂	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	2	no	—
92	C₂×C₇₁	C₂×C₈×Q₈	1	1	yes	characteristic
93	C₂×C₇₁	C₂²×M₃₂	1	1	yes	characteristic
94	C₂×C₇₁	#3407	1	1	yes	characteristic
95	C₂×C₈×Q₈	D₇₁	1	1	yes	characteristic
96	#2183	—	71	6	no	—
97	Q₈×C₁₆	—	71	2	no	—
98	#2150	—	71	3	no	—
99	C₄×M₃₂	—	71	3	no	—
100	C₇₁	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	Dic₇₁	1	1	yes	characteristic
102	C₂×M₃₂	—	71	1	no	—
103	C₂×M₃₂	—	71	3	no	—
104	C₈×Q₈	Dic₇₁	1	3	yes	—
105	C₈×Q₈	Dic₇₁	1	1	yes	characteristic
106	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
107	C₈×Q₈	C₂×D₇₁	1	3	yes	—
108	C₂×C₄⋊C₈	Dic₇₁	1	3	yes	—
109	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
110	C₄⋊C₁₆	—	71	6	no	—
111	C₄⋊C₁₆	—	71	6	no	—
112	#346	—	71	6	no	—
113	C₄×C₁₆	—	71	6	no	—
114	C₂²×Q₈	C₇₁⋊C₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
116	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	C₄×Q₈	C₇₁⋊C₈	1	3	yes	—
118	C₄×Q₈	C₇₁⋊C₈	1	1	yes	characteristic
119	C₄×Q₈	C₂×Dic₇₁	1	3	yes	—
120	C₄×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₇₁⋊C₈	1	3	yes	—
122	C₂×C₄²	C₂×Dic₇₁	1	3	yes	—
123	M₃₂	—	71	4	no	—
124	M₃₂	—	142	6	no	—
125	C₂×C₁₆	—	71	2	no	—
126	C₂×C₁₆	—	71	6	no	—
127	C₄⋊C₈	C₂×Dic₇₁	1	3	yes	—
128	C₄⋊C₈	C₂×Dic₇₁	1	6	yes	—
129	C₄⋊C₈	C₂×Dic₇₁	1	3	yes	—
130	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
131	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
132	C₄×C₈	C₂²×D₇₁	1	3	yes	—
133	C₄×C₈	C₂²×D₇₁	1	3	yes	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₂×C₇₁⋊C₈	1	3	yes	—
136	C₂×Q₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
138	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
139	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
140	C₂×C₈	—	2	6	no	—
141	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
142	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
143	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₇₁	1	2	yes	—
145	C₂×C₈	#11765866	1	2	yes	—
146	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
147	C₄⋊C₄	C₂×C₇₁⋊C₈	1	6	yes	—
148	C₄⋊C₄	C₂×C₇₁⋊C₈	1	3	yes	—
149	C₄⋊C₄	C₂×C₇₁⋊C₈	1	3	yes	—
150	C₄²	C₂×C₇₁⋊C₈	1	3	yes	—
151	C₄²	C₂×C₇₁⋊C₈	1	3	yes	—
152	C₄²	C₂²×Dic₇₁	1	3	yes	—
153	C₄²	C₂²×Dic₇₁	1	3	yes	—
154	C₁₆	—	71	4	no	—
155	C₁₆	—	142	6	no	—
156	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	#14556068	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	Q₈×Dic₇₁	1	2	yes	—
162	C₂×C₄	C₂²×C₇₁⋊C₈	1	3	yes	—
163	C₂×C₄	C₂²×C₇₁⋊C₈	1	3	yes	—
164	C₂×C₄	C₂²×C₇₁⋊C₈	1	3	yes	—
165	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂²×C₇₁⋊C₈	1	3	yes	—
167	C₂×C₄	#14556135	1	2	yes	—
168	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	2	3	no	—
171	C₈	Q₈×Dic₇₁	1	2	yes	—
172	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
173	C₈	C₂×#11765866	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	Q₈×C₇₁⋊C₈	1	2	yes	—
177	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	Q₈×C₇₁⋊C₈	1	2	yes	—
182	C₄	C₂×#14556068	1	6	yes	—
183	C₄	C₂×Q₈×Dic₇₁	1	1	yes	characteristic
184	C₄	C₂×#14556135	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×Q₈×C₇₁⋊C₈	1	1	yes	characteristic
187	C₂	C₂²×#14556068	1	1	yes	characteristic
188	C₂	#22601443	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₇₁	C₂
prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₁
hypocenter	C₇₁	Q₈×M₃₂
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #22590249	C₂	semidirect
2. Q₈×C₇₁⋊C₁₆	C₂	semidirect
20. #14556068	Q₈	direct (N × H)
40. C₇₁⋊C₁₆	C₂×Q₈	semidirect
63. Q₈×C₇₁	M₃₂	semidirect
100. C₇₁	Q₈×M₃₂	semidirect
158. Q₈	#14556068	direct (N × H)

Maximal subgroups

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

subgroup	index
#22590249	2
Q₈×C₇₁⋊C₁₆	2
#22590096	2
C₄×#14556068	2
C₂×C₈×Q₈×C₇₁	2
Q₈×M₃₂	71

Subgroups — Q8×#14556068 #32150966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14556068 #32150966