Subgroups — #1997

Order 128.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#423	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#343	C₂	1	4	yes	maximal
5	C₂²×Q₈	C₄	1	2	yes	—
6	C₂×M₁₆	C₂²	1	2	yes	—
7	C₂×M₁₆	C₂²	1	2	yes	—
8	C₄⋊Q₈	—	2	1	no	—
9	C₄⋊Q₈	—	2	1	no	—
10	C₄⋊Q₈	—	2	1	no	—
11	C₄⋊Q₈	—	2	1	no	—
12	C₂×C₄⋊C₄	C₄	1	2	yes	—
13	C₂×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	C₄⋊C₈	—	2	1	no	—
16	C₄⋊C₈	—	2	1	no	—
17	#97	—	2	1	no	—
18	C₄×C₈	—	2	1	no	—
19	C₂×Q₈	—	2	2	no	—
20	C₂×Q₈	—	2	2	no	—
21	C₂×Q₈	—	4	4	no	—
22	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄	C₂×C₄	1	4	yes	—
25	C₂²×C₄	C₂³	1	1	yes	characteristic
26	M₁₆	—	2	2	no	—
27	M₁₆	—	2	4	no	—
28	M₁₆	Q₈	1	4	yes	complemented
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	—	2	2	no	—
31	C₄⋊C₄	—	2	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	2	2	no	—
34	C₄⋊C₄	—	2	2	no	—
35	C₄²	D₄	1	1	yes	characteristic
36	C₄²	D₄	1	1	yes	characteristic
37	C₄²	D₄	1	1	yes	characteristic
38	C₄²	D₄	1	1	yes	characteristic
39	C₂³	C₂²×C₄	1	1	yes	characteristic
40	Q₈	—	4	4	no	—
41	Q₈	—	4	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	—	4	4	no	—
46	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄	C₂×Q₈	1	2	yes	—
53	C₂×C₄	#41	1	2	yes	—
54	C₈	—	2	4	no	—
55	C₈	—	4	2	no	—
56	C₂²	—	2	1	no	—
57	C₂²	—	2	1	no	—
58	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²	C₄×Q₈	1	2	yes	—
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	—	4	4	no	—
63	C₄	#100	1	2	yes	—
64	C₄	#101	1	2	yes	—
65	C₄	C₂²⋊Q₈	1	2	yes	—
66	C₄	#124	1	1	yes	characteristic
67	C₄	C₄⋊Q₈	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	#391	1	1	yes	characteristic
70	C₂	C₂×#100	1	1	yes	characteristic
71	C₂	C₂×#101	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
28. M₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈	2
#423	2
C₄×M₁₆	2
#343	2