Subgroups — #33958

Order 256.

304 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3391	C₂	1	2	yes	maximal, complemented
2	#2246	C₂	1	3	yes	maximal, complemented
3	#2206	C₂	1	6	yes	maximal, complemented
4	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₁₆	C₂	1	3	yes	maximal, complemented
6	#550	—	2	2	no	—
7	C₂×D₁₆	C₂²	1	3	yes	—
8	C₂×D₁₆	C₂²	1	6	yes	—
9	#493	C₂²	1	6	yes	—
10	#459	—	2	6	no	—
11	C₈×Q₈	C₂²	1	1	yes	characteristic
12	C₄×D₈	C₂²	1	6	yes	—
13	#366	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	C₂²	1	3	yes	—
15	#357	C₂²	1	6	yes	—
16	C₄×C₁₆	C₂²	1	3	yes	—
17	C₂×#41	—	4	2	no	—
18	C₂×D₈	—	2	6	no	—
19	C₂×D₈	C₂³	1	6	yes	—
20	#127	—	2	6	no	—
21	#121	—	4	6	no	—
22	C₄×Q₈	D₄	1	1	yes	characteristic
23	C₄×D₄	—	2	6	no	—
24	D₁₆	—	2	6	no	—
25	D₁₆	—	4	6	no	—
26	C₂×C₁₆	C₂³	1	3	yes	—
27	C₂×C₁₆	C₂³	1	1	yes	characteristic
28	#107	C₂³	1	2	yes	—
29	C₄⋊C₈	D₄	1	3	yes	—
30	#102	—	2	6	no	—
31	C₄×C₈	C₂³	1	3	yes	—
32	#41	—	8	8	no	—
33	C₂×Q₈	D₈	1	1	yes	characteristic
34	C₂×D₄	—	2	6	no	—
35	C₂×D₄	—	4	6	no	—
36	C₂×D₄	—	4	6	no	—
37	C₂²×C₄	—	4	6	no	—
38	D₈	—	2	6	no	—
39	D₈	—	2	6	no	—
40	D₈	—	8	6	no	—
41	C₂×C₈	C₂×D₄	1	3	yes	—
42	C₂×C₈	C₂⁴	1	1	yes	characteristic
43	C₄⋊C₄	—	2	2	no	—
44	C₄⋊C₄	D₈	1	3	yes	—
45	C₂²⋊C₄	—	4	6	no	—
46	C₄²	C₂×D₄	1	3	yes	—
47	C₁₆	—	2	3	no	—
48	C₁₆	#41	1	2	yes	complemented
49	C₂³	—	4	6	no	—
50	Q₈	D₁₆	1	4	yes	complemented
51	D₄	—	4	6	no	—
52	D₄	—	4	6	no	—
53	D₄	—	8	12	no	—
54	D₄	—	16	6	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	8	12	no	—
57	C₂×C₄	C₂×D₈	1	3	yes	—
58	C₂×C₄	C₂×D₈	1	3	yes	—
59	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
60	C₈	—	2	3	no	—
61	C₈	C₂×#41	1	1	yes	characteristic
62	C₈	H(5,𝔽₂)	1	1	yes	characteristic
63	C₂²	—	8	6	no	—
64	C₂²	—	8	6	no	—
65	C₂²	—	8	6	no	—
66	C₂²	C₂²×D₈	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	8	2	no	—
69	C₄	C₂×D₁₆	1	6	yes	—
70	C₄	#550	1	1	yes	characteristic
71	C₄	#576	1	1	yes	characteristic
72	C₂	—	16	6	no	—
73	C₂	#3391	1	1	yes	characteristic
74	C₂	C₂²×D₁₆	1	1	yes	characteristic
75	C₂	#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3391	C₂	semidirect
2. #2246	C₂	semidirect
3. #2206	C₂	semidirect
4. Q₈×C₁₆	C₂	semidirect
5. C₄×D₁₆	C₂	semidirect
48. C₁₆	#41	semidirect
50. Q₈	D₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3391	2
#2246	2
#2206	2
Q₈×C₁₆	2
C₄×D₁₆	2