Subgroups — #1966

Order 128.

146 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	C₂×#101	C₂	1	4	yes	maximal
4	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈	C₂²	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	2	no	—
11	C₂×C₄⋊C₄	C₄	1	4	yes	—
12	C₂×C₄²	C₂²	1	1	yes	characteristic
13	C₄⋊C₈	—	2	2	no	—
14	#101	—	2	8	no	—
15	#97	—	2	1	no	—
16	C₄×C₈	—	2	1	no	—
17	C₂×Q₈	—	2	4	no	—
18	C₂×Q₈	—	2	2	no	—
19	C₂×Q₈	—	2	2	no	—
20	C₂×Q₈	—	2	4	no	—
21	C₂²×C₄	C₂×C₄	1	4	yes	—
22	C₂²×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	M₁₆	—	2	2	no	—
25	M₁₆	—	2	4	no	—
26	M₁₆	D₄	1	4	yes	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	—	2	2	no	—
29	C₄⋊C₄	—	2	4	no	—
30	C₄⋊C₄	—	2	4	no	—
31	C₄²	D₄	1	1	yes	characteristic
32	C₄²	D₄	1	1	yes	characteristic
33	C₄²	D₄	1	2	yes	—
34	C₂³	C₂²×C₄	1	1	yes	characteristic
35	Q₈	—	4	4	no	—
36	Q₈	—	4	2	no	—
37	Q₈	—	4	2	no	—
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	—	2	4	no	—
40	C₂×C₄	—	2	8	no	—
41	C₂×C₄	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄	C₂×D₄	1	2	yes	—
45	C₂×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄	#41	1	2	yes	—
47	C₈	—	2	4	no	—
48	C₈	—	4	2	no	—
49	C₂²	—	2	2	no	—
50	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²	C₄×D₄	1	2	yes	—
52	C₄	—	2	2	no	—
53	C₄	—	4	4	no	—
54	C₄	#101	1	4	yes	complemented
55	C₄	#121	1	2	yes	—
56	C₄	#124	1	1	yes	characteristic
57	C₄	#127	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	#390	1	1	yes	characteristic
60	C₂	C₂×#101	1	2	yes	—

	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
54. C₄	#101	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
C₂×#101	2
C₄×M₁₆	2