Subgroups — #33728

Order 256.

241 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160	C₂	1	2	yes	maximal
2	#2255	C₂	1	8	yes	maximal, complemented
3	#2158	C₂	1	4	yes	maximal
4	C₂×#346	C₂	1	1	yes	maximal, characteristic
5	#533	—	2	2	no	—
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	#500	C₂²	1	4	yes	—
8	#499	—	2	2	no	—
9	#499	C₂²	1	4	yes	—
10	#498	—	2	2	no	—
11	#427	—	2	2	no	—
12	#427	C₂²	1	4	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#367	C₂²	1	8	yes	—
15	#366	C₂²	1	8	yes	—
16	#346	C₂²	1	4	yes	—
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	C₄⋊Q₈	—	2	4	no	—
20	#125	—	2	4	no	—
21	C₂²⋊Q₈	—	4	4	no	—
22	C₄×Q₈	—	4	4	no	—
23	#117	—	2	4	no	—
24	C₂×C₄²	D₄	1	1	yes	characteristic
25	C₂×C₁₆	Q₈	1	8	yes	complemented
26	C₂×C₁₆	C₂³	1	4	yes	—
27	#107	—	2	4	no	—
28	#107	C₂³	1	8	yes	—
29	#106	—	2	4	no	—
30	#106	—	2	4	no	—
31	C₄×C₈	D₄	1	2	yes	—
32	C₄×C₈	C₂³	1	2	yes	—
33	C₂×Q₈	—	4	4	no	—
34	C₂²×C₄	D₈	1	2	yes	—
35	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈	Q₁₆	1	4	yes	—
37	C₂×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈	C₂×D₄	1	2	yes	—
39	C₂×C₈	C₂×Q₈	1	4	yes	—
40	C₂×C₈	C₂⁴	1	1	yes	characteristic
41	C₄⋊C₄	—	2	8	no	—
42	C₄⋊C₄	—	4	8	no	—
43	C₄⋊C₄	—	4	4	no	—
44	C₂²⋊C₄	—	4	4	no	—
45	C₄²	—	4	4	no	—
46	C₄²	D₈	1	2	yes	—
47	C₄²	C₂×D₄	1	2	yes	—
48	C₁₆	C₂×Q₈	1	8	yes	—
49	C₂³	C₂×D₈	1	1	yes	characteristic
50	Q₈	—	8	8	no	—
51	C₂×C₄	—	4	2	no	—
52	C₂×C₄	—	4	8	no	—
53	C₂×C₄	C₄⋊Q₈	1	2	yes	—
54	C₂×C₄	C₂×D₈	1	4	yes	—
55	C₂×C₄	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄	C₂×Q₁₆	1	2	yes	—
57	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
58	C₈	C₄⋊Q₈	1	2	yes	—
59	C₈	C₂×Q₁₆	1	4	yes	—
60	C₈	C₂²×Q₈	1	2	yes	—
61	C₂²	—	2	2	no	—
62	C₂²	#500	1	2	yes	—
63	C₂²	C₂²×D₈	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	—	8	8	no	—
66	C₄	#500	1	2	yes	—
67	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₄	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂	—	2	2	no	—
70	C₂	C₂×#500	1	1	yes	characteristic
71	C₂	#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂²	C₂²×D₈
prime radical	C₂³	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2255	C₂	semidirect
25. C₂×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3160	2
#2255	2
#2158	2
C₂×#346	2