Subgroups — Q₈×2₋¹⁺⁴ #62997

Order 256.

3425 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3503	C₂	1	30	yes	maximal, complemented
2	#3502	C₂	1	15	yes	maximal, complemented
3	Q₈×#41	C₂	1	10	yes	maximal, complemented
4	C₂×Q₈²	C₂	1	5	yes	maximal, complemented
5	C₄×2₋¹⁺⁴	C₂	1	3	yes	maximal
6	C₂×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
7	Q₈²	C₂²	1	10	yes	—
8	Q₈²	C₂²	1	60	yes	—
9	#557	C₂²	1	30	yes	—
10	#557	C₂²	1	60	yes	—
11	#554	C₂²	1	90	yes	—
12	D₄×Q₈	C₂²	1	10	yes	—
13	D₄×Q₈	C₂²	1	60	yes	—
14	#544	C₂²	1	90	yes	—
15	#533	C₂²	1	90	yes	—
16	C₂×C₄⋊Q₈	C₂²	1	45	yes	—
17	#520	C₂²	1	30	yes	—
18	#519	C₂²	1	15	yes	—
19	C₄×#41	C₂²	1	30	yes	—
20	C₂×C₄×Q₈	C₂²	1	15	yes	—
21	C₂×C₄×Q₈	C₂²	1	15	yes	—
22	2₋¹⁺⁴	Q₈	1	16	yes	direct product factor
23	C₂×#41	C₂³	1	10	yes	—
24	C₂²×Q₈	C₂³	1	5	yes	—
25	C₂²×Q₈	C₂³	1	30	yes	—
26	C₂²×Q₈	C₂³	1	5	yes	—
27	C₄⋊Q₈	C₂³	1	90	yes	—
28	C₄⋊Q₈	C₂³	1	180	yes	—
29	C₄⋊Q₈	C₂³	1	30	yes	—
30	#125	C₂³	1	180	yes	—
31	C₂²⋊Q₈	C₂³	1	60	yes	—
32	C₂²⋊Q₈	C₂³	1	180	yes	—
33	C₄×Q₈	C₂³	1	10	yes	—
34	C₄×Q₈	C₂³	1	30	yes	—
35	C₄×Q₈	C₂³	1	60	yes	—
36	C₄×Q₈	C₂³	1	90	yes	—
37	C₄×Q₈	C₂³	1	30	yes	—
38	C₄×Q₈	C₂³	1	60	yes	—
39	C₄×D₄	C₂³	1	30	yes	—
40	C₄×D₄	C₂³	1	90	yes	—
41	#117	C₂³	1	90	yes	—
42	C₂×C₄⋊C₄	C₂³	1	45	yes	—
43	C₂×C₄⋊C₄	C₂³	1	45	yes	—
44	C₂×C₄²	C₂³	1	45	yes	—
45	#41	C₂×Q₈	1	80	yes	complemented
46	C₂×Q₈	—	2	20	no	—
47	C₂×Q₈	—	8	30	no	—
48	C₂×Q₈	C₂×Q₈	1	40	yes	complemented
49	C₂×Q₈	C₂⁴	1	15	yes	—
50	C₂×Q₈	C₂⁴	1	60	yes	—
51	C₂×Q₈	C₂⁴	1	10	yes	—
52	C₂×Q₈	C₂⁴	1	1	yes	characteristic
53	C₂×D₄	C₂⁴	1	10	yes	—
54	C₂²×C₄	C₂⁴	1	15	yes	—
55	C₂²×C₄	C₂⁴	1	45	yes	—
56	C₂²×C₄	C₂⁴	1	15	yes	—
57	C₄⋊C₄	C₂⁴	1	30	yes	—
58	C₄⋊C₄	C₂⁴	1	90	yes	—
59	C₄⋊C₄	C₂⁴	1	90	yes	—
60	C₄⋊C₄	C₂⁴	1	30	yes	—
61	C₄⋊C₄	C₂⁴	1	60	yes	—
62	C₂²⋊C₄	C₂⁴	1	60	yes	—
63	C₄²	C₂⁴	1	30	yes	—
64	C₄²	C₂⁴	1	90	yes	—
65	C₂³	C₂⁵	1	5	yes	—
66	Q₈	—	2	30	no	—
67	Q₈	—	4	60	no	—
68	Q₈	C₂²×Q₈	1	40	yes	—
69	Q₈	2₋¹⁺⁴	1	4	yes	direct product factor
70	D₄	C₂²×Q₈	1	40	yes	—
71	C₂×C₄	—	2	30	no	—
72	C₂×C₄	—	4	45	no	—
73	C₂×C₄	C₂²×Q₈	1	60	yes	—
74	C₂×C₄	C₂⁵	1	15	yes	—
75	C₂×C₄	C₂⁵	1	30	yes	—
76	C₂×C₄	C₂⁵	1	10	yes	—
77	C₂×C₄	C₂⁵	1	3	yes	—
78	C₂²	—	2	5	no	—
79	C₂²	—	2	5	no	—
80	C₂²	C₂³×Q₈	1	10	yes	—
81	C₂²	C₂⁶	1	1	yes	characteristic
82	C₄	—	2	15	no	—
83	C₄	—	2	30	no	—
84	C₄	C₂³×Q₈	1	20	yes	—
85	C₄	C₂×2₋¹⁺⁴	1	6	yes	—
86	C₂	—	2	10	no	—
87	C₂	C₂⁴×Q₈	1	1	yes	characteristic
88	C₂	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
89	C₂	H(7,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
prime radical	C₂×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3503	C₂	semidirect
2. #3502	C₂	semidirect
3. Q₈×#41	C₂	semidirect
4. C₂×Q₈²	C₂	semidirect
22. 2₋¹⁺⁴	Q₈	direct (N × H)
45. #41	C₂×Q₈	semidirect
48. C₂×Q₈	C₂×Q₈	semidirect
69. Q₈	2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
#3503	2
#3502	2
Q₈×#41	2
C₂×Q₈²	2
C₄×2₋¹⁺⁴	2

Subgroups — Q8×2−1+4 #62997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×2₋¹⁺⁴ #62997