Subgroups — #1994

Order 128.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic
2	#423	C₂	1	1	yes	maximal, characteristic
3	C₄³	C₂	1	1	yes	maximal, characteristic
4	#339	C₂	1	4	yes	maximal
5	C₂×M₁₆	C₂²	1	2	yes	—
6	C₄⋊Q₈	C₄	1	2	yes	complemented
7	#125	C₄	1	2	yes	complemented
8	C₂²⋊Q₈	—	2	2	no	—
9	C₄×Q₈	—	2	2	no	—
10	#117	C₂²	1	2	yes	—
11	C₂×C₄²	—	2	2	no	—
12	C₂×C₄²	C₂²	1	2	yes	—
13	C₂×C₄²	C₂²	1	1	yes	characteristic
14	C₄⋊C₈	—	2	2	no	—
15	C₂×Q₈	—	2	2	no	—
16	C₂²×C₄	—	2	2	no	—
17	C₂²×C₄	D₄	1	2	yes	—
18	C₂²×C₄	C₂³	1	1	yes	characteristic
19	M₁₆	—	2	4	no	—
20	C₂×C₈	—	2	2	no	—
21	C₄⋊C₄	—	2	4	no	—
22	C₄⋊C₄	—	2	2	no	—
23	C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₂²⋊C₄	—	2	2	no	—
25	C₄²	—	2	8	no	—
26	C₄²	—	2	2	no	—
27	C₄²	—	2	2	no	—
28	C₄²	C₂×C₄	1	1	yes	characteristic
29	C₄²	C₂×C₄	1	1	yes	characteristic
30	C₄²	D₄	1	2	yes	—
31	C₄²	Q₈	1	4	yes	complemented
32	C₂³	C₂×D₄	1	1	yes	characteristic
33	Q₈	—	4	4	no	—
34	C₂×C₄	—	2	8	no	—
35	C₂×C₄	—	2	4	no	—
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄	C₂×Q₈	1	2	yes	—
43	C₂×C₄	#41	1	2	yes	—
44	C₈	—	4	2	no	—
45	C₂²	—	2	2	no	—
46	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²	C₂²⋊Q₈	1	2	yes	—
48	C₄	—	2	8	no	—
49	C₄	—	2	2	no	—
50	C₄	—	4	4	no	—
51	C₄	#104	1	4	yes	—
52	C₄	C₄×Q₈	1	2	yes	—
53	C₄	#124	1	1	yes	characteristic
54	C₄	C₄⋊Q₈	1	1	yes	characteristic
55	C₂	—	2	2	no	—
56	C₂	#391	1	1	yes	characteristic
57	C₂	C₂×#104	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₂×C₂²⋊C₄
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
6. C₄⋊Q₈	C₄	semidirect
7. #125	C₄	semidirect
31. C₄²	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#423	2
C₄³	2
#339	2