Subgroups — #3393

Order 128.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556	C₂	1	2	yes	maximal, complemented
2	#495	C₂	1	2	yes	maximal
3	#494	C₂	1	1	yes	maximal, characteristic, complemented
4	#464	C₂	1	4	yes	maximal, complemented
5	#461	C₂	1	2	yes	maximal, complemented
6	C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₁₆	C₂	1	2	yes	maximal, complemented
8	C₄×D₈	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₁₆	C₂²	1	2	yes	—
10	C₂×Q₁₆	C₂²	1	2	yes	—
11	C₂×SD₁₆	C₂²	1	4	yes	—
12	C₂×D₈	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈	C₂²	1	2	yes	—
14	#126	—	2	4	no	—
15	#124	C₂²	1	4	yes	—
16	C₂²⋊Q₈	—	2	2	no	—
17	C₄×Q₈	C₂²	1	4	yes	—
18	C₄×Q₈	C₂²	1	1	yes	characteristic
19	C₄×D₄	C₂²	1	2	yes	—
20	#117	—	2	2	no	—
21	#107	C₂²	1	1	yes	characteristic
22	C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₄⋊C₈	C₂²	1	2	yes	—
24	#103	C₂²	1	4	yes	—
25	#102	C₂²	1	2	yes	—
26	C₄×C₈	C₂²	1	1	yes	characteristic
27	C₄×C₈	C₂²	1	2	yes	—
28	C₂×Q₈	D₄	1	1	yes	characteristic
29	C₂×Q₈	C₂³	1	4	yes	—
30	C₂×D₄	C₂³	1	2	yes	—
31	C₂²×C₄	—	2	2	no	—
32	Q₁₆	—	2	4	no	—
33	Q₁₆	—	4	2	no	—
34	SD₁₆	—	4	4	no	—
35	D₈	—	2	2	no	—
36	C₂×C₈	C₂³	1	1	yes	characteristic
37	C₂×C₈	C₂³	1	2	yes	—
38	C₂×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	—	2	4	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₄⋊C₄	D₄	1	2	yes	—
44	C₄⋊C₄	C₂³	1	2	yes	—
45	C₂²⋊C₄	—	2	4	no	—
46	C₂²⋊C₄	—	2	2	no	—
47	C₂²⋊C₄	—	2	4	no	—
48	C₄²	—	2	4	no	—
49	C₄²	C₂³	1	1	yes	characteristic
50	C₄²	C₂³	1	2	yes	—
51	C₂³	—	2	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	4	no	—
54	Q₈	—	2	4	no	—
55	D₄	—	2	2	no	—
56	D₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	—	4	4	no	—
60	C₂×C₄	C₂×D₄	1	2	yes	—
61	C₂×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄	C₂⁴	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₈	—	2	1	no	—
67	C₈	#41	1	2	yes	—
68	C₂²	—	4	2	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	C₂²×D₄	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	2	2	no	—
74	C₄	—	2	1	no	—
75	C₄	—	4	4	no	—
76	C₄	—	4	2	no	—
77	C₄	#135	1	2	yes	—
78	C₄	C₂×#41	1	1	yes	characteristic
79	C₄	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂	—	8	2	no	—
81	C₂	#550	1	1	yes	characteristic
82	C₂	C₂×#135	1	1	yes	characteristic
83	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #556	C₂	semidirect
3. #494	C₂	semidirect
4. #464	C₂	semidirect
5. #461	C₂	semidirect
6. C₈×Q₈	C₂	semidirect
7. C₄×Q₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#556	2
#495	2
#494	2
#464	2
#461	2
C₈×Q₈	2
C₄×Q₁₆	2
C₄×D₈	2