Subgroups — #353039

Order 512.

906 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33310	C₂	1	6	yes	maximal
2	#8813	C₂	1	8	yes	maximal
3	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
4	C₂×#498	—	2	12	no	—
5	#2868	C₂²	1	4	yes	—
6	#1844	—	2	12	no	—
7	#1844	C₂²	1	24	yes	—
8	C₄²×C₈	C₂²	1	3	yes	—
9	C₂×C₈²	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈	—	2	12	no	—
11	#498	—	4	48	no	—
12	C₂×#106	—	2	48	no	—
13	C₂×C₄×C₈	D₄	1	12	yes	—
14	C₂×C₄×C₈	C₂³	1	6	yes	—
15	#384	—	2	12	no	—
16	#384	C₂³	1	8	yes	—
17	C₄³	C₂³	1	1	yes	characteristic
18	C₈²	Q₈	1	16	yes	complemented
19	C₂²×Q₈	—	4	4	no	—
20	C₂²×C₈	C₂×D₄	1	12	yes	—
21	C₄⋊Q₈	—	4	48	no	—
22	C₂×C₄⋊C₄	—	2	24	no	—
23	C₂×C₄⋊C₄	—	4	12	no	—
24	C₂×C₄²	C₂×D₄	1	6	yes	—
25	C₂×C₄²	C₂⁴	1	1	yes	characteristic
26	#106	—	4	96	no	—
27	C₄×C₈	SD₁₆	1	48	yes	—
28	C₄×C₈	C₂×Q₈	1	24	yes	—
29	C₂×Q₈	—	8	24	no	—
30	C₂²×C₄	—	4	8	no	—
31	C₂²×C₄	#127	1	4	yes	—
32	C₂²×C₄	C₂²×D₄	1	3	yes	—
33	C₂×C₈	C₄⋊Q₈	1	24	yes	—
34	C₂×C₈	C₂×SD₁₆	1	48	yes	—
35	C₄⋊C₄	—	4	48	no	—
36	C₄⋊C₄	—	8	24	no	—
37	C₄²	C₂×SD₁₆	1	24	yes	—
38	C₄²	C₂²×Q₈	1	4	yes	—
39	C₂³	C₂×#127	1	1	yes	characteristic
40	Q₈	—	16	16	no	—
41	C₂×C₄	—	8	24	no	—
42	C₂×C₄	#492	1	24	yes	—
43	C₂×C₄	C₂×C₄⋊Q₈	1	6	yes	—
44	C₂×C₄	C₂²×SD₁₆	1	12	yes	—
45	C₈	#498	1	48	yes	complemented
46	C₂²	#2868	1	1	yes	characteristic
47	C₂²	C₂×#492	1	6	yes	—
48	C₄	—	16	8	no	—
49	C₄	#1705	1	16	yes	—
50	C₄	C₂×#498	1	12	yes	—
51	C₂	C₂×#1705	1	4	yes	—
52	C₂	#33310	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

N	H ≅ G/N	type
18. C₈²	Q₈	semidirect
45. C₈	#498	semidirect

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