Subgroups — #33310

Order 256.

503 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

Complements

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

N	H ≅ G/N	type
26. C₄×C₈	Q₈	semidirect
51. C₈	C₄⋊Q₈	semidirect
57. C₄	#498	semidirect

Maximal subgroups

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

subgroup	index
C₂×#498	2
#2868	2
#1844	2
C₄²×C₈	2