Subgroups — #33302

Order 256.

711 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494	C₂	1	12	yes	maximal
2	#2868	C₂	1	2	yes	maximal
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆	C₂²	1	16	yes	—
5	C₂×C₄⋊Q₈	C₂²	1	12	yes	—
6	#494	—	2	48	no	—
7	C₂×C₄×C₈	C₂²	1	6	yes	—
8	#384	—	2	8	no	—
9	C₄³	C₂²	1	1	yes	characteristic
10	C₂²×Q₈	C₂³	1	8	yes	—
11	C₂×Q₁₆	—	2	96	no	—
12	C₂²×C₈	C₂³	1	4	yes	—
13	C₄⋊Q₈	—	2	48	no	—
14	C₂×C₄⋊C₄	—	2	24	no	—
15	C₂×C₄²	D₄	1	4	yes	—
16	C₂×C₄²	C₂³	1	3	yes	—
17	C₄×C₈	D₄	1	24	yes	—
18	C₂×Q₈	—	2	48	no	—
19	C₂²×C₄	—	2	8	no	—
20	C₂²×C₄	C₂×D₄	1	6	yes	—
21	C₂²×C₄	C₂⁴	1	1	yes	characteristic
22	Q₁₆	—	4	64	no	—
23	C₂×C₈	C₂×D₄	1	24	yes	—
24	C₄⋊C₄	—	4	48	no	—
25	C₄²	Q₁₆	1	16	yes	complemented
26	C₄²	C₂×D₄	1	12	yes	—
27	C₂³	C₂²×D₄	1	1	yes	characteristic
28	Q₈	—	4	32	no	—
29	C₂×C₄	—	4	24	no	—
30	C₂×C₄	#127	1	12	yes	—
31	C₂×C₄	C₂×Q₁₆	1	24	yes	—
32	C₂×C₄	C₂²×D₄	1	6	yes	—
33	C₈	#127	1	16	yes	—
34	C₂²	C₂×#127	1	3	yes	—
35	C₂²	C₂²×Q₁₆	1	4	yes	—
36	C₄	—	8	8	no	—
37	C₄	#494	1	24	yes	complemented
38	C₄	C₂×#127	1	4	yes	—
39	C₂	#2867	1	1	yes	characteristic
40	C₂	C₂×#494	1	6	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
25. C₄²	Q₁₆	semidirect
37. C₄	#494	semidirect

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