Subgroups — Q₈×#127 #36557

Order 256.

1749 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈	C₂	1	6	yes	maximal, complemented
2	#2868	C₂	1	3	yes	maximal, complemented
3	#2542	C₂	1	18	yes	maximal, complemented
4	C₄×#127	C₂	1	3	yes	maximal
5	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈	C₂²	1	4	yes	—
7	D₄×Q₈	—	2	48	no	—
8	C₂×C₄⋊Q₈	C₂²	1	18	yes	—
9	C₂×#127	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Q₈	C₂²	1	36	yes	—
11	C₂×C₄×Q₈	C₂²	1	3	yes	—
12	C₂×C₄×D₄	C₂²	1	18	yes	—
13	#399	C₂²	1	24	yes	—
14	#390	C₂²	1	18	yes	—
15	#384	C₂²	1	3	yes	—
16	#384	C₂²	1	18	yes	—
17	C₄×C₄⋊C₄	C₂²	1	9	yes	—
18	C₄³	C₂²	1	3	yes	—
19	C₂²×Q₈	—	2	48	no	—
20	C₂²×Q₈	C₂³	1	12	yes	—
21	C₂²×Q₈	C₂³	1	1	yes	characteristic
22	C₂²×D₄	C₂³	1	6	yes	—
23	C₂³×C₄	C₂³	1	12	yes	—
24	C₄⋊Q₈	—	2	72	no	—
25	#127	Q₈	1	8	yes	direct product factor
26	C₂²⋊Q₈	—	2	144	no	—
27	C₄×Q₈	D₄	1	24	yes	complemented
28	C₄×D₄	—	2	72	no	—
29	C₂×C₄⋊C₄	C₂³	1	18	yes	—
30	C₂×C₄⋊C₄	C₂³	1	36	yes	—
31	C₂×C₄⋊C₄	C₂³	1	9	yes	—
32	C₂×C₂²⋊C₄	C₂³	1	36	yes	—
33	C₂×C₄²	C₂³	1	9	yes	—
34	C₂×C₄²	C₂³	1	9	yes	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	#95	C₂³	1	6	yes	—
37	C₂⁴	C₂⁴	1	4	yes	—
38	C₂×Q₈	—	2	72	no	—
39	C₂×Q₈	—	4	64	no	—
40	C₂×Q₈	C₂×D₄	1	12	yes	—
41	C₂×D₄	—	2	12	no	—
42	C₂×D₄	—	2	12	no	—
43	C₂×D₄	C₂×Q₈	1	24	yes	complemented
44	C₂²×C₄	—	2	72	no	—
45	C₂²×C₄	C₂⁴	1	12	yes	—
46	C₂²×C₄	C₂⁴	1	9	yes	—
47	C₂²×C₄	C₂⁴	1	3	yes	—
48	C₂²×C₄	C₂⁴	1	3	yes	—
49	C₄⋊C₄	—	2	36	no	—
50	C₄⋊C₄	—	2	72	no	—
51	C₄⋊C₄	C₂×D₄	1	36	yes	—
52	C₂²⋊C₄	—	2	72	no	—
53	C₄²	—	2	18	no	—
54	C₄²	C₂×D₄	1	36	yes	—
55	C₄²	C₂×Q₈	1	4	yes	complemented
56	C₂³	—	2	12	no	—
57	C₂³	—	2	12	no	—
58	C₂³	C₂²×Q₈	1	8	yes	—
59	C₂³	C₂⁵	1	1	yes	characteristic
60	Q₈	—	4	48	no	—
61	Q₈	#127	1	16	yes	direct product factor
62	D₄	—	2	48	no	—
63	C₂×C₄	—	2	18	no	—
64	C₂×C₄	—	2	36	no	—
65	C₂×C₄	—	4	48	no	—
66	C₂×C₄	C₂²×D₄	1	18	yes	—
67	C₂×C₄	C₂²×D₄	1	18	yes	—
68	C₂×C₄	C₂²×D₄	1	6	yes	—
69	C₂×C₄	C₂²×Q₈	1	6	yes	—
70	C₂×C₄	2₋¹⁺⁴	1	6	yes	—
71	C₂²	—	2	24	no	—
72	C₂²	—	4	4	no	—
73	C₂²	—	4	12	no	—
74	C₂²	C₂³×D₄	1	3	yes	—
75	C₂²	C₂³×Q₈	1	1	yes	characteristic
76	C₂²	C₂×2₋¹⁺⁴	1	3	yes	—
77	C₄	—	2	18	no	—
78	C₄	—	4	12	no	—
79	C₄	C₂×#127	1	12	yes	—
80	C₄	D₄×Q₈	1	12	yes	complemented
81	C₂	—	4	8	no	—
82	C₂	C₂²×#127	1	1	yes	characteristic
83	C₂	C₂×D₄×Q₈	1	3	yes	—
84	C₂	#3499	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂×#127	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Q₈	C₂	semidirect
2. #2868	C₂	semidirect
3. #2542	C₂	semidirect
5. C₄²×Q₈	C₂	semidirect
25. #127	Q₈	direct (N × H)
27. C₄×Q₈	D₄	semidirect
43. C₂×D₄	C₂×Q₈	semidirect
55. C₄²	C₂×Q₈	semidirect
61. Q₈	#127	direct (N × H)
80. C₄	D₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈	2
#2868	2
#2542	2
C₄×#127	2
C₄²×Q₈	2

Subgroups — Q8×#127 #36557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#127 #36557