Subgroups — #33962

Order 256.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3395	C₂	1	2	yes	maximal
2	#2247	C₂	1	3	yes	maximal
3	#2209	C₂	1	6	yes	maximal
4	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂	C₂	1	3	yes	maximal
6	#557	—	2	2	no	—
7	C₂×Q₃₂	C₂²	1	3	yes	—
8	C₂×Q₃₂	C₂²	1	6	yes	—
9	#494	C₂²	1	6	yes	—
10	#462	—	2	6	no	—
11	C₈×Q₈	C₂²	1	1	yes	characteristic
12	C₄×Q₁₆	C₂²	1	6	yes	—
13	#366	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	C₂²	1	3	yes	—
15	#358	C₂²	1	6	yes	—
16	C₄×C₁₆	C₂²	1	3	yes	—
17	C₂×Q₁₆	—	2	6	no	—
18	C₂×Q₁₆	C₂³	1	6	yes	—
19	C₄⋊Q₈	—	2	6	no	—
20	#125	—	4	6	no	—
21	C₄×Q₈	—	2	6	no	—
22	C₄×Q₈	—	4	2	no	—
23	C₄×Q₈	D₄	1	1	yes	characteristic
24	Q₃₂	—	2	6	no	—
25	Q₃₂	—	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	#103	—	2	6	no	—
31	C₄×C₈	C₂³	1	3	yes	—
32	C₂×Q₈	—	2	6	no	—
33	C₂×Q₈	D₈	1	1	yes	characteristic
34	Q₁₆	—	2	6	no	—
35	Q₁₆	—	2	6	no	—
36	Q₁₆	—	8	6	no	—
37	C₂×C₈	C₂×D₄	1	3	yes	—
38	C₂×C₈	C₂⁴	1	1	yes	characteristic
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	—	4	6	no	—
41	C₄⋊C₄	—	4	6	no	—
42	C₄⋊C₄	—	4	6	no	—
43	C₄⋊C₄	D₈	1	3	yes	—
44	C₄²	—	4	6	no	—
45	C₄²	C₂×D₄	1	3	yes	—
46	C₁₆	—	2	3	no	—
47	C₁₆	#41	1	2	yes	—
48	Q₈	—	4	6	no	—
49	Q₈	—	4	6	no	—
50	Q₈	Q₃₂	1	4	yes	complemented
51	C₂×C₄	—	4	2	no	—
52	C₂×C₄	—	4	6	no	—
53	C₂×C₄	C₂×D₈	1	3	yes	—
54	C₂×C₄	C₂×D₈	1	3	yes	—
55	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
56	C₈	—	2	3	no	—
57	C₈	C₂×#41	1	1	yes	characteristic
58	C₈	H(5,𝔽₂)	1	1	yes	characteristic
59	C₂²	C₂²×D₈	1	1	yes	characteristic
60	C₄	—	2	3	no	—
61	C₄	—	8	6	no	—
62	C₄	—	8	2	no	—
63	C₄	C₂×Q₃₂	1	6	yes	—
64	C₄	#550	1	1	yes	characteristic
65	C₄	#576	1	1	yes	characteristic
66	C₂	#3391	1	1	yes	characteristic
67	C₂	C₂²×Q₃₂	1	1	yes	characteristic
68	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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
50. Q₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#3395	2
#2247	2
#2209	2
Q₈×C₁₆	2
C₄×Q₃₂	2