Subgroups — #60191

Order 256.

905 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3458	C₂	1	1	yes	maximal, characteristic
2	#2961	C₂	1	12	yes	maximal, complemented
3	#2960	C₂	1	12	yes	maximal, complemented
4	#2905	C₂	1	3	yes	maximal
5	C₂×#405	C₂	1	3	yes	maximal
6	C₂²×M₁₆	C₂²	1	3	yes	—
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	#544	C₄	1	12	yes	—
9	#543	C₄	1	4	yes	—
10	#533	C₂²	1	12	yes	—
11	C₂×#117	C₂²	1	3	yes	—
12	#446	C₂²	1	48	yes	complemented
13	C₈×Q₈	C₂²	1	16	yes	complemented
14	#424	C₂²	1	24	yes	—
15	#423	C₂²	1	12	yes	—
16	#405	C₂²	1	12	yes	—
17	#405	C₂²	1	12	yes	—
18	C₂×#97	C₂²	1	6	yes	—
19	C₂×C₄×C₈	C₂²	1	6	yes	—
20	C₂³×C₄	C₂³	1	1	yes	characteristic
21	C₂×M₁₆	—	2	12	no	—
22	C₂×M₁₆	C₂³	1	12	yes	—
23	C₂²×C₈	Q₈	1	8	yes	complemented
24	C₂²×C₈	C₂³	1	6	yes	—
25	C₂²×C₈	C₂³	1	3	yes	—
26	C₂²×C₈	C₂³	1	3	yes	—
27	C₄⋊Q₈	C₂×C₄	1	24	yes	—
28	#125	C₂×C₄	1	24	yes	—
29	C₂²⋊Q₈	C₂×C₄	1	48	yes	—
30	C₄×Q₈	C₂³	1	16	yes	—
31	#117	C₂×C₄	1	12	yes	—
32	#117	C₂³	1	12	yes	—
33	C₂×C₄⋊C₄	C₂×C₄	1	6	yes	—
34	C₂×C₂²⋊C₄	C₂×C₄	1	6	yes	—
35	C₂×C₄²	C₂³	1	6	yes	—
36	C₄⋊C₈	C₂³	1	48	yes	—
37	#97	C₂³	1	24	yes	—
38	C₄×C₈	C₂³	1	12	yes	—
39	C₄×C₈	C₂³	1	12	yes	—
40	C₂⁴	C₂²×C₄	1	1	yes	characteristic
41	C₂×Q₈	C₂²×C₄	1	16	yes	—
42	C₂²×C₄	C₂²×C₄	1	12	yes	—
43	C₂²×C₄	C₂²×C₄	1	3	yes	—
44	C₂²×C₄	C₂×Q₈	1	4	yes	—
45	C₂²×C₄	#41	1	4	yes	—
46	C₂²×C₄	C₂⁴	1	3	yes	—
47	M₁₆	—	2	24	no	—
48	C₂×C₈	—	2	12	no	—
49	C₂×C₈	C₂×Q₈	1	12	yes	—
50	C₂×C₈	C₂×Q₈	1	12	yes	—
51	C₂×C₈	C₂⁴	1	12	yes	—
52	C₂×C₈	C₂⁴	1	3	yes	—
53	C₂×C₈	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄	C₂²×C₄	1	24	yes	—
55	C₄⋊C₄	C₂²×C₄	1	48	yes	—
56	C₂²⋊C₄	C₂²×C₄	1	24	yes	—
57	C₄²	C₂²×C₄	1	12	yes	—
58	C₄²	C₂⁴	1	12	yes	—
59	C₂³	—	4	2	no	—
60	C₂³	C₄×Q₈	1	4	yes	—
61	C₂³	C₂³×C₄	1	3	yes	—
62	Q₈	—	4	16	no	—
63	C₂×C₄	—	4	12	no	—
64	C₂×C₄	C₄×Q₈	1	12	yes	—
65	C₂×C₄	C₂³×C₄	1	24	yes	—
66	C₂×C₄	C₂³×C₄	1	3	yes	—
67	C₂×C₄	C₂²×Q₈	1	6	yes	—
68	C₂×C₄	C₂×#41	1	6	yes	—
69	C₂×C₄	C₂⁵	1	1	yes	characteristic
70	C₈	—	2	12	no	—
71	C₈	C₂²×Q₈	1	6	yes	—
72	C₈	C₂²×Q₈	1	2	yes	—
73	C₂²	—	2	6	no	—
74	C₂²	—	4	4	no	—
75	C₂²	C₂×C₄×Q₈	1	6	yes	—
76	C₂²	C₂⁴×C₄	1	1	yes	characteristic
77	C₄	—	2	24	no	—
78	C₄	C₂×C₄×Q₈	1	6	yes	—
79	C₄	C₂³×Q₈	1	1	yes	characteristic
80	C₄	C₂²×#41	1	1	yes	characteristic
81	C₂	—	2	6	no	—
82	C₂	C₂²×C₄×Q₈	1	1	yes	characteristic
83	C₂	#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2961	C₂	semidirect
3. #2960	C₂	semidirect
12. #446	C₂²	semidirect
13. C₈×Q₈	C₂²	semidirect
23. C₂²×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3458	2
#2961	2
#2960	2
#2905	2
C₂×#405	2