Subgroups — #6132641

Order 512.

1390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17053	C₂	1	28	yes	maximal
2	#14690	C₂	1	3	yes	maximal
3	C₂×C₄³	C₂²	1	1	yes	characteristic
4	#1446	C₂²	1	56	yes	—
5	#1443	C₂²	1	56	yes	—
6	#1434	C₂²	1	42	yes	—
7	C₂³×Q₈	C₂³	1	8	yes	—
8	C₂²×C₄²	C₂³	1	7	yes	—
9	C₂×#95	C₂³	1	84	yes	—
10	C₂×#95	C₂³	1	56	yes	—
11	C₄³	Q₈	1	8	yes	complemented
12	C₂²×Q₈	—	4	56	no	—
13	C₂³×C₄	C₂⁴	1	24	yes	—
14	C₂³×C₄	C₂⁴	1	7	yes	—
15	C₂×C₄²	—	4	14	no	—
16	C₂×C₄²	C₂×Q₈	1	28	yes	—
17	#95	—	4	84	no	—
18	#95	—	4	56	no	—
19	C₂⁴	C₂⁵	1	1	yes	characteristic
20	C₂×Q₈	—	8	112	no	—
21	C₂²×C₄	—	2	42	no	—
22	C₂²×C₄	—	2	168	no	—
23	C₂²×C₄	C₂²×Q₈	1	14	yes	—
24	C₄²	—	4	28	no	—
25	C₂³	C₂³×Q₈	1	1	yes	characteristic
26	C₂³	#585	1	14	yes	—
27	Q₈	—	8	64	no	—
28	C₂×C₄	—	2	42	no	—
29	C₂×C₄	—	4	28	no	—
30	C₂×C₄	—	4	168	no	—
31	C₂²	#3458	1	7	yes	—
32	C₂²	#3568	1	28	yes	—
33	C₄	—	4	14	no	—
34	C₄	—	8	24	no	—
35	C₂	#37563	1	7	yes	—
36	C₂	#53389	1	8	yes	—

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

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

N	H ≅ G/N	type
11. C₄³	Q₈	semidirect

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

subgroup	index
#17053	2
#14690	2