Subgroups — #2225

Order 128.

65 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#367	C₂	1	2	yes	maximal
4	C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#358	C₂	1	2	yes	maximal
6	C₂×Q₁₆	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈	—	2	1	no	—
8	C₄×Q₈	—	2	1	no	—
9	C₂×C₁₆	C₂²	1	2	yes	—
10	#107	—	2	1	no	—
11	#107	C₂²	1	2	yes	—
12	#107	C₂²	1	1	yes	characteristic
13	#103	—	2	1	no	—
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₈	—	2	1	no	—
16	C₂×Q₈	—	4	1	no	—
17	Q₁₆	—	2	1	no	—
18	Q₁₆	Q₈	1	2	yes	complemented
19	C₂×C₈	D₄	1	1	yes	characteristic
20	C₂×C₈	C₂³	1	1	yes	characteristic
21	C₄⋊C₄	—	2	1	no	—
22	C₄⋊C₄	—	2	2	no	—
23	C₄⋊C₄	—	4	1	no	—
24	C₄⋊C₄	—	4	1	no	—
25	C₄²	—	4	1	no	—
26	C₄²	D₄	1	1	yes	characteristic
27	C₁₆	—	2	2	no	—
28	Q₈	—	2	2	no	—
29	Q₈	—	4	1	no	—
30	Q₈	—	8	2	no	—
31	C₂×C₄	—	4	1	no	—
32	C₂×C₄	—	4	1	no	—
33	C₂×C₄	—	4	2	no	—
34	C₂×C₄	D₈	1	1	yes	characteristic
35	C₂×C₄	D₈	1	1	yes	characteristic
36	C₂×C₄	C₂×D₄	1	1	yes	characteristic
37	C₈	—	2	1	no	—
38	C₈	C₂×Q₈	1	1	yes	characteristic
39	C₈	#41	1	1	yes	characteristic
40	C₂²	C₂×D₈	1	1	yes	characteristic
41	C₄	—	2	1	no	—
42	C₄	—	4	2	no	—
43	C₄	—	8	2	no	—
44	C₄	—	8	1	no	—
45	C₄	SD₃₂	1	2	yes	—
46	C₄	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄	#137	1	1	yes	characteristic
48	C₂	#474	1	1	yes	characteristic
49	C₂	C₂×SD₃₂	1	1	yes	characteristic
50	C₂	#510	1	1	yes	characteristic

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

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

N	H ≅ G/N	type
18. Q₁₆	Q₈	semidirect

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