Subgroups — Q₈×#14556062 #32016292

Order 18176.

838 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₇₁	C₂	1	2	yes	maximal
2	#22589099	C₂	1	6	yes	maximal
3	#22588500	C₂	1	3	yes	maximal
4	C₄×#14556062	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₇₁	—	2	2	no	—
7	C₂×#14556062	C₂²	1	3	yes	—
8	C₂×#14556062	C₂²	1	6	yes	—
9	C₂×#14556062	C₂²	1	1	yes	characteristic
10	#17985898	—	2	6	no	—
11	#17985869	C₂²	1	6	yes	—
12	C₄×C₄⋊Dic₇₁	C₂²	1	6	yes	—
13	#17985762	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₇₁	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
18	Q₈×Dic₇₁	—	4	8	no	—
19	C₂×C₄⋊Dic₇₁	—	2	6	no	—
20	C₂×C₄⋊Dic₇₁	—	2	6	no	—
21	C₂×C₄⋊Dic₇₁	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₇₁	C₂³	1	2	yes	—
23	C₂×C₄×Dic₇₁	—	2	6	no	—
24	#14556062	—	2	6	no	—
25	#14556062	—	4	6	no	—
26	#14556062	Q₈	1	4	yes	direct product factor
27	C₇₁⋊#95	—	2	6	no	—
28	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₇₁	C₂×C₄	1	4	yes	—
32	C₄×Q₈×C₇₁	Q₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₇₁	D₄	1	3	yes	—
34	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
35	C₄⋊C₈×C₇₁	C₂×C₄	1	12	yes	—
36	C₄×C₈×C₇₁	C₂×C₄	1	12	yes	—
37	C₂²×Dic₇₁	—	2	2	no	—
38	C₂²×Dic₇₁	—	2	6	no	—
39	C₄⋊Dic₇₁	—	2	2	no	—
40	C₄⋊Dic₇₁	—	2	6	no	—
41	C₄⋊Dic₇₁	—	2	6	no	—
42	C₄⋊Dic₇₁	—	4	12	no	—
43	C₄⋊Dic₇₁	—	8	6	no	—
44	C₄⋊Dic₇₁	C₂×Q₈	1	4	yes	—
45	C₄×Dic₇₁	—	4	12	no	—
46	C₂×Q₈×C₇₁	C₄⋊C₄	1	4	yes	—
47	C₂×Q₈×C₇₁	D₈	1	4	yes	—
48	C₂×Q₈×C₇₁	Q₁₆	1	4	yes	—
49	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₇₁	—	2	3	no	—
53	C₂×C₈×C₇₁	—	2	3	no	—
54	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
55	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₇₁	#41	1	2	yes	—
58	C₄⋊C₄×C₇₁	C₄⋊C₄	1	12	yes	—
59	C₄²×C₇₁	C₂²×C₄	1	6	yes	—
60	C₄²×C₇₁	C₂×Q₈	1	6	yes	—
61	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₇₁	#107	1	16	yes	complemented
63	C₂×C₄×C₇₁	—	2	3	no	—
64	C₂×C₄×C₇₁	—	2	3	no	—
65	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
67	C₂×C₄×C₇₁	C₂×D₈	1	6	yes	—
68	C₂×C₄×C₇₁	C₂×Q₁₆	1	6	yes	—
69	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
73	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
74	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
75	C₂×Dic₇₁	—	2	4	no	—
76	C₂×Dic₇₁	—	4	2	no	—
77	C₂×Dic₇₁	—	4	2	no	—
78	C₂×Dic₇₁	—	4	6	no	—
79	C₂×Dic₇₁	—	4	6	no	—
80	C₂×Dic₇₁	—	4	6	no	—
81	C₈×C₇₁	—	2	6	no	—
82	C₈×C₇₁	C₄×Q₈	1	4	yes	complemented
83	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₇₁	D₄×Q₈	1	1	yes	characteristic
85	C₂²×C₇₁	#550	1	1	yes	characteristic
86	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
87	C₂²×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
88	C₂²×C₇₁	#576	1	1	yes	characteristic
89	C₂²×C₇₁	#578	1	1	yes	characteristic
90	C₄×C₇₁	—	2	6	no	—
91	C₄×C₇₁	C₂×#107	1	12	yes	—
92	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
93	C₄×C₇₁	#520	1	1	yes	characteristic
94	C₄×C₇₁	#557	1	1	yes	characteristic
95	C₄×C₇₁	Q₈²	1	1	yes	characteristic
96	Dic₇₁	—	4	4	no	—
97	Dic₇₁	—	8	6	no	—
98	Q₈×#107	—	71	1	no	maximal, Sylow
99	C₂×C₇₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₇₁	C₂²×#107	1	1	yes	characteristic
101	C₂×C₇₁	#2913	1	1	yes	characteristic
102	C₂×C₇₁	Q₈×D₈	1	1	yes	characteristic
103	C₂×C₇₁	Q₈×Q₁₆	1	1	yes	characteristic
104	C₂×C₇₁	#3391	1	1	yes	characteristic
105	C₂×C₇₁	#3395	1	1	yes	characteristic
106	C₂×C₈×Q₈	D₇₁	1	1	yes	characteristic
107	Q₈×C₄⋊C₄	—	71	2	no	—
108	#1947	—	71	6	no	—
109	#1845	—	71	3	no	—
110	C₄×#107	—	71	3	no	—
111	C₇₁	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	—	142	2	no	—
113	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
114	C₈×Q₈	Dic₇₁	1	8	yes	complemented
115	C₂×#107	—	71	6	no	—
116	C₂×#107	—	71	1	no	—
117	C₂×#107	—	71	3	no	—
118	C₂×C₄⋊C₈	C₂×D₇₁	1	3	yes	—
119	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
120	#389	—	142	6	no	—
121	#384	—	71	6	no	—
122	C₄×C₄⋊C₄	—	71	6	no	—
123	#340	—	71	6	no	—
124	C₂²×Q₈	D₂₈₄	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
126	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
127	C₄×Q₈	—	284	8	no	—
128	C₄×Q₈	Dic₁₄₂	1	4	yes	—
129	C₄×Q₈	C₂×Dic₇₁	1	4	yes	—
130	C₂×C₄⋊C₄	—	71	2	no	—
131	C₂×C₄⋊C₄	—	71	6	no	—
132	C₂×C₄⋊C₄	—	142	6	no	—
133	C₂×C₄⋊C₄	—	142	6	no	—
134	C₂×C₄⋊C₄	D₂₈₄	1	3	yes	—
135	C₂×C₄²	—	142	6	no	—
136	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
137	#107	—	284	6	no	—
138	#107	—	71	4	no	—
139	#107	—	142	6	no	—
140	C₄⋊C₈	C₂×Dic₇₁	1	12	yes	—
141	C₄×C₈	C₂×Dic₇₁	1	12	yes	—
142	#95	—	142	6	no	—
143	C₂×Q₈	C₄⋊Dic₇₁	1	4	yes	—
144	C₂×Q₈	D₅₆₈	1	4	yes	—
145	C₂×Q₈	Dic₂₈₄	1	4	yes	—
146	C₂²×C₄	—	142	6	no	—
147	C₂²×C₄	—	142	2	no	—
148	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
149	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
150	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
151	C₂×C₈	—	2	3	no	—
152	C₂×C₈	—	2	3	no	—
153	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
154	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
155	C₂×C₈	Q₈×D₇₁	1	2	yes	—
156	C₂×C₈	#11765866	1	2	yes	—
157	C₄⋊C₄	—	284	12	no	—
158	C₄⋊C₄	—	568	6	no	—
159	C₄⋊C₄	—	71	4	no	—
160	C₄⋊C₄	—	142	6	no	—
161	C₄⋊C₄	—	142	6	no	—
162	C₄⋊C₄	—	142	2	no	—
163	C₄⋊C₄	C₄⋊Dic₇₁	1	12	yes	—
164	C₄²	—	284	12	no	—
165	C₄²	C₂²×Dic₇₁	1	6	yes	—
166	C₄²	C₂×Dic₁₄₂	1	6	yes	—
167	C₂³	C₂²×D₂₈₄	1	1	yes	characteristic
168	Q₈	#14556062	1	16	yes	direct product factor
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	284	2	no	—
172	C₂×C₄	—	284	2	no	—
173	C₂×C₄	—	284	6	no	—
174	C₂×C₄	—	284	6	no	—
175	C₂×C₄	—	284	6	no	—
176	C₂×C₄	—	142	4	no	—
177	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
179	C₂×C₄	C₂×D₅₆₈	1	6	yes	—
180	C₂×C₄	C₂×Dic₂₈₄	1	6	yes	—
181	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
182	C₂×C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
184	C₂×C₄	C₂×#11765866	1	1	yes	characteristic
185	C₂×C₄	#14556174	1	1	yes	characteristic
186	C₂×C₄	#14556175	1	1	yes	characteristic
187	C₈	—	2	6	no	—
188	C₈	Q₈×Dic₇₁	1	4	yes	complemented
189	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
190	C₂²	Q₈×D₂₈₄	1	1	yes	characteristic
191	C₂²	#17986671	1	1	yes	characteristic
192	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
193	C₂²	C₂²×Dic₂₈₄	1	1	yes	characteristic
194	C₂²	#17986804	1	1	yes	characteristic
195	C₂²	#17986816	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	284	4	no	—
198	C₄	—	568	6	no	—
199	C₄	C₂×#14556062	1	12	yes	—
200	C₄	C₂×Q₈×Dic₇₁	1	1	yes	characteristic
201	C₄	#17986482	1	1	yes	characteristic
202	C₄	#17986728	1	1	yes	characteristic
203	C₄	Q₈×Dic₁₄₂	1	1	yes	characteristic
204	C₂	Q₈×C₄⋊Dic₇₁	1	1	yes	characteristic
205	C₂	C₂²×#14556062	1	1	yes	characteristic
206	C₂	#22596940	1	1	yes	characteristic
207	C₂	Q₈×D₅₆₈	1	1	yes	characteristic
208	C₂	Q₈×Dic₂₈₄	1	1	yes	characteristic
209	C₂	#22601315	1	1	yes	characteristic
210	C₂	#22601346	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂×C₈×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×D₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₁
hypocenter	C₇₁	Q₈×#107
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₇₁	C₄	semidirect
26. #14556062	Q₈	direct (N × H)
62. Q₈×C₇₁	#107	semidirect
82. C₈×C₇₁	C₄×Q₈	semidirect
111. C₇₁	Q₈×#107	semidirect
114. C₈×Q₈	Dic₇₁	semidirect
168. Q₈	#14556062	direct (N × H)
188. C₈	Q₈×Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₇₁	2
#22589099	2
#22588500	2
C₄×#14556062	2
C₂×C₈×Q₈×C₇₁	2
Q₈×#107	71

Subgroups — Q8×#14556062 #32016292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14556062 #32016292