Subgroups — #1633

Order 128.

103 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²	C₂	1	1	yes	maximal, characteristic
2	#494	C₂	1	1	yes	maximal, characteristic
3	#477	C₂	1	2	yes	maximal
4	#333	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊C₈	C₂	1	2	yes	maximal
6	C₂×Q₁₆	—	2	2	no	—
7	C₄⋊Q₈	—	2	2	no	—
8	C₄⋊Q₈	—	2	2	no	—
9	C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₄×Q₈	—	2	2	no	—
12	C₄×Q₈	C₂²	1	2	yes	—
13	#107	—	2	2	no	—
14	C₄⋊C₈	C₂²	1	2	yes	—
15	#103	—	2	2	no	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈	—	2	2	no	—
18	C₂×Q₈	—	2	1	no	—
19	C₂×Q₈	D₄	1	2	yes	—
20	C₂×Q₈	D₄	1	2	yes	—
21	Q₁₆	—	4	4	no	—
22	C₂×C₈	—	2	1	no	—
23	C₂×C₈	—	2	2	no	—
24	C₄⋊C₄	—	2	4	no	—
25	C₄⋊C₄	—	2	2	no	—
26	C₄⋊C₄	—	2	2	no	—
27	C₄⋊C₄	—	2	2	no	—
28	C₄⋊C₄	D₄	1	2	yes	—
29	C₄²	—	2	2	no	—
30	C₄²	—	2	2	no	—
31	C₄²	C₂³	1	1	yes	characteristic
32	Q₈	—	2	2	no	—
33	Q₈	—	4	2	no	—
34	Q₈	—	4	2	no	—
35	Q₈	—	8	2	no	—
36	Q₈	Q₁₆	1	4	yes	complemented
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄	C₂×D₄	1	2	yes	—
43	C₈	—	2	2	no	—
44	C₈	—	4	2	no	—
45	C₂²	#120	1	1	yes	characteristic
46	C₄	—	2	4	no	—
47	C₄	—	2	1	no	—
48	C₄	—	4	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	8	1	no	—
51	C₄	C₂×Q₁₆	1	2	yes	—
52	C₄	#137	1	2	yes	—
53	C₂	#451	1	2	yes	—
54	C₂	#453	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#120
commutator, Frattini, cosocle	C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

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

N	H ≅ G/N	type
36. Q₈	Q₁₆	semidirect

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