Subgroups — #33313

Order 256.

431 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500	C₂	1	2	yes	maximal
2	C₂×#498	C₂	1	2	yes	maximal
3	#2862	C₂	1	2	yes	maximal
4	#1846	C₂	1	8	yes	maximal
5	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈	C₂²	1	4	yes	—
7	#500	—	2	8	no	—
8	#498	—	2	8	no	—
9	C₂×#107	C₂²	1	8	yes	—
10	C₂×#106	C₂²	1	8	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	4	yes	—
14	#391	—	2	8	no	—
15	#382	C₂²	1	8	yes	—
16	C₄³	C₂²	1	1	yes	characteristic
17	C₂²×Q₈	—	2	4	no	—
18	C₂²×C₈	C₂³	1	2	yes	—
19	C₂²×C₈	C₂³	1	2	yes	—
20	C₄⋊Q₈	—	2	8	no	—
21	C₄⋊Q₈	—	2	8	no	—
22	C₂×C₄⋊C₄	—	2	4	no	—
23	C₂×C₄⋊C₄	C₂³	1	8	yes	—
24	C₂×C₄²	D₄	1	4	yes	—
25	C₂×C₄²	C₂³	1	1	yes	characteristic
26	C₂×C₄²	C₂³	1	2	yes	—
27	#107	—	2	16	no	—
28	#106	—	2	16	no	—
29	C₄×C₈	D₄	1	4	yes	—
30	C₄×C₈	D₄	1	4	yes	—
31	C₄×C₈	Q₈	1	16	yes	complemented
32	#95	—	2	16	no	—
33	C₂×Q₈	—	4	8	no	—
34	C₂×Q₈	—	4	16	no	—
35	C₂²×C₄	—	2	8	no	—
36	C₂²×C₄	C₂×D₄	1	4	yes	—
37	C₂²×C₄	C₂×D₄	1	2	yes	—
38	C₂²×C₄	C₂⁴	1	1	yes	characteristic
39	C₂×C₈	C₂×D₄	1	4	yes	—
40	C₂×C₈	C₂×D₄	1	4	yes	—
41	C₂×C₈	C₂×Q₈	1	8	yes	—
42	C₂×C₈	C₂×Q₈	1	8	yes	—
43	C₄⋊C₄	—	2	8	no	—
44	C₄⋊C₄	—	2	8	no	—
45	C₄⋊C₄	—	4	8	no	—
46	C₄²	—	2	8	no	—
47	C₄²	C₂×D₄	1	2	yes	—
48	C₄²	C₂×D₄	1	2	yes	—
49	C₄²	C₂×Q₈	1	8	yes	—
50	C₂³	C₂²×D₄	1	1	yes	characteristic
51	Q₈	—	8	16	no	—
52	C₂×C₄	—	2	8	no	—
53	C₂×C₄	—	4	8	no	—
54	C₂×C₄	—	4	16	no	—
55	C₂×C₄	#127	1	4	yes	—
56	C₂×C₄	C₄⋊Q₈	1	8	yes	—
57	C₂×C₄	#135	1	8	yes	—
58	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₄	C₂²×Q₈	1	4	yes	—
61	C₈	C₄⋊Q₈	1	8	yes	—
62	C₈	C₄⋊Q₈	1	8	yes	—
63	C₂²	C₂×#127	1	1	yes	characteristic
64	C₂²	C₂×C₄⋊Q₈	1	2	yes	—
65	C₂²	C₂×#135	1	4	yes	—
66	C₄	—	2	8	no	—
67	C₄	—	8	8	no	—
68	C₄	#495	1	8	yes	—
69	C₄	C₂×C₄⋊Q₈	1	2	yes	—
70	C₄	C₂×C₄⋊Q₈	1	2	yes	—
71	C₂	#2868	1	1	yes	characteristic
72	C₂	C₂×#495	1	2	yes	—
73	C₂	#3160	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
31. C₄×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500	2
C₂×#498	2
#2862	2
#1846	2
C₄²×C₈	2