Subgroups — #3152

Order 128.

276 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137	C₂	1	4	yes	maximal, complemented
2	#532	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#497	C₂	1	4	yes	maximal, complemented
5	#494	C₂	1	2	yes	maximal, complemented
6	#492	C₂	1	2	yes	maximal, complemented
7	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	2	yes	—
9	C₂²×Q₈	C₂²	1	2	yes	—
10	#137	—	2	16	no	—
11	C₂×Q₁₆	C₂²	1	8	yes	—
12	C₂×SD₁₆	C₂²	1	8	yes	—
13	C₂×M₁₆	C₂²	1	2	yes	—
14	C₄⋊Q₈	—	2	2	no	—
15	C₄⋊Q₈	C₂²	1	2	yes	—
16	C₄⋊Q₈	C₂²	1	2	yes	—
17	C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	#127	C₂²	1	1	yes	characteristic
19	#124	C₂²	1	2	yes	—
20	#121	—	2	2	no	—
21	C₄×D₄	—	2	2	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄²	C₂²	1	1	yes	characteristic
24	#97	C₂²	1	2	yes	—
25	C₄×C₈	C₂²	1	2	yes	—
26	#41	—	2	8	no	—
27	C₂×Q₈	—	2	2	no	—
28	C₂×Q₈	—	2	8	no	—
29	C₂×Q₈	C₂³	1	2	yes	—
30	C₂×Q₈	C₂³	1	4	yes	—
31	C₂×D₄	—	2	2	no	—
32	C₂×D₄	—	2	2	no	—
33	C₂×D₄	C₂³	1	2	yes	—
34	C₂²×C₄	—	2	2	no	—
35	C₂²×C₄	—	2	2	no	—
36	C₂²×C₄	D₄	1	2	yes	—
37	C₂²×C₄	C₂³	1	1	yes	characteristic
38	Q₁₆	—	2	16	no	—
39	SD₁₆	—	2	16	no	—
40	M₁₆	D₄	1	8	yes	complemented
41	C₂×C₈	C₂³	1	4	yes	—
42	C₄⋊C₄	—	2	4	no	—
43	C₄⋊C₄	—	2	4	no	—
44	C₄⋊C₄	—	2	2	no	—
45	C₂²⋊C₄	—	2	4	no	—
46	C₄²	D₄	1	2	yes	—
47	C₄²	C₂³	1	1	yes	characteristic
48	C₄²	C₂³	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	C₂³	C₂×D₄	1	1	yes	characteristic
51	Q₈	—	2	8	no	—
52	Q₈	—	2	4	no	—
53	Q₈	—	4	4	no	—
54	D₄	—	2	4	no	—
55	D₄	—	4	4	no	—
56	D₄	—	4	4	no	—
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	4	4	no	—
61	C₂×C₄	—	4	4	no	—
62	C₂×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄	C₂×D₄	1	4	yes	—
65	C₂×C₄	C₂×D₄	1	2	yes	—
66	C₂×C₄	C₂⁴	1	1	yes	characteristic
67	C₈	C₂×D₄	1	8	yes	—
68	C₂²	—	2	2	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	4	no	—
71	C₂²	#127	1	2	yes	—
72	C₂²	C₂²×D₄	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	4	2	no	—
75	C₄	—	4	4	no	—
76	C₄	#127	1	2	yes	—
77	C₄	#137	1	4	yes	complemented
78	C₄	C₂²×D₄	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	—	8	2	no	—
81	C₂	C₂×#127	1	1	yes	characteristic
82	C₂	C₂×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#137	C₂	semidirect
3. C₂×C₄⋊Q₈	C₂	semidirect
4. #497	C₂	semidirect
5. #494	C₂	semidirect
6. #492	C₂	semidirect
7. C₄×M₁₆	C₂	semidirect
40. M₁₆	D₄	semidirect
77. C₄	#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137	2
#532	2
C₂×C₄⋊Q₈	2
#497	2
#494	2
#492	2
C₄×M₁₆	2