Subgroups — #32874

Order 256.

344 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2032	C₂	1	8	yes	maximal, complemented
2	#2030	C₂	1	4	yes	maximal, complemented
3	#1782	C₂	1	3	yes	maximal
4	C₂²×#41	C₂²	1	1	yes	characteristic
5	#543	C₂²	1	4	yes	—
6	C₂×#99	C₂²	1	12	yes	—
7	#381	C₂²	1	6	yes	—
8	#342	C₂²	1	12	yes	—
9	C₂×#41	—	4	6	no	—
10	C₂²×Q₈	Q₈	1	1	yes	characteristic, complemented
11	C₂²×D₄	Q₈	1	1	yes	characteristic, complemented
12	C₂²×D₄	C₂³	1	2	yes	—
13	C₂³×C₄	Q₈	1	2	yes	complemented
14	C₂³×C₄	C₂³	1	1	yes	characteristic
15	C₂²⋊Q₈	—	4	12	no	—
16	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
17	#99	—	4	12	no	—
18	#95	—	4	6	no	—
19	C₂⁴	C₂×Q₈	1	2	yes	—
20	C₂⁴	C₂⁴	1	1	yes	characteristic
21	#41	—	4	16	no	—
22	C₂×Q₈	—	4	3	no	—
23	C₂×Q₈	—	4	4	no	—
24	C₂×D₄	—	2	12	no	—
25	C₂×D₄	—	4	3	no	—
26	C₂²×C₄	—	2	6	no	—
27	C₂²×C₄	—	2	12	no	—
28	C₂²×C₄	—	4	6	no	—
29	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₄	C₂×Q₈	1	2	yes	—
31	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
32	C₄⋊C₄	—	4	12	no	—
33	C₂²⋊C₄	—	2	24	no	—
34	C₂³	—	2	6	no	—
35	C₂³	—	8	1	no	—
36	C₂³	—	8	2	no	—
37	C₂³	C₂²×Q₈	1	1	yes	characteristic
38	C₂³	H(5,𝔽₂)	1	6	yes	—
39	Q₈	—	4	4	no	—
40	Q₈	—	16	4	no	—
41	D₄	—	4	4	no	—
42	D₄	—	4	8	no	—
43	C₂×C₄	—	2	3	no	—
44	C₂×C₄	—	2	3	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	4	8	no	—
47	C₂×C₄	—	4	4	no	—
48	C₂×C₄	—	4	12	no	—
49	C₂×C₄	—	8	12	no	—
50	C₂²	—	4	6	no	—
51	C₂²	—	4	1	no	—
52	C₂²	—	4	2	no	—
53	C₂²	—	8	6	no	—
54	C₂²	#543	1	3	yes	—
55	C₂²	H(3,𝔽₄)	1	4	yes	—
56	C₄	—	4	1	no	—
57	C₄	—	4	1	no	—
58	C₄	—	4	2	no	—
59	C₄	—	8	12	no	—
60	C₂	—	2	4	no	—
61	C₂	—	2	3	no	—
62	C₂	—	8	2	no	—
63	C₂	#2840	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2840
commutator, Frattini, cosocle	C₂⁴	C₂⁴
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2032	C₂	semidirect
2. #2030	C₂	semidirect
10. C₂²×Q₈	Q₈	semidirect
11. C₂²×D₄	Q₈	semidirect
13. C₂³×C₄	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2032	2
#2030	2
#1782	2