Subgroups — #3177

Order 128.

169 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#541	C₂	1	2	yes	maximal
2	#499	C₂	1	1	yes	maximal, characteristic, complemented
3	#494	C₂	1	1	yes	maximal, characteristic, complemented
4	#491	C₂	1	2	yes	maximal, complemented
5	#470	C₂	1	2	yes	maximal
6	#467	C₂	1	2	yes	maximal
7	#441	C₂	1	2	yes	maximal, complemented
8	C₄×Q₁₆	C₂	1	2	yes	maximal, complemented
9	#405	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₁₆	C₂²	1	2	yes	—
11	C₂×Q₁₆	C₂²	1	2	yes	—
12	C₂×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈	C₂²	1	2	yes	—
15	#126	—	2	4	no	—
16	#125	—	2	4	no	—
17	#125	C₂²	1	2	yes	—
18	C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₄×Q₈	C₂²	1	4	yes	—
20	#117	C₂²	1	1	yes	characteristic
21	#107	C₂²	1	2	yes	—
22	#106	C₂²	1	2	yes	—
23	#103	C₂²	1	4	yes	—
24	#103	C₂²	1	4	yes	—
25	#97	C₂²	1	1	yes	characteristic
26	#97	C₂²	1	1	yes	characteristic
27	C₄×C₈	C₂²	1	1	yes	characteristic
28	C₄×C₈	C₂²	1	1	yes	characteristic
29	C₂×Q₈	C₂³	1	4	yes	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	Q₁₆	—	2	4	no	—
32	Q₁₆	—	2	4	no	—
33	M₁₆	—	2	2	no	—
34	C₂×C₈	—	2	2	no	—
35	C₂×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈	C₂³	1	1	yes	characteristic
37	C₂×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊C₄	—	2	4	no	—
40	C₄⋊C₄	—	2	4	no	—
41	C₄⋊C₄	—	2	4	no	—
42	C₄⋊C₄	—	2	4	no	—
43	C₄⋊C₄	D₄	1	2	yes	—
44	C₄⋊C₄	C₂³	1	4	yes	—
45	C₂²⋊C₄	—	2	4	no	—
46	C₂²⋊C₄	D₄	1	2	yes	—
47	C₄²	—	2	4	no	—
48	C₄²	C₂³	1	1	yes	characteristic
49	C₄²	C₂³	1	1	yes	characteristic
50	C₂³	C₂×D₄	1	1	yes	characteristic
51	Q₈	—	2	4	no	—
52	Q₈	—	2	4	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₄	C₂×D₄	1	2	yes	—
59	C₂×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄	C₂⁴	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	2	1	no	—
63	C₈	#41	1	2	yes	—
64	C₈	#41	1	2	yes	—
65	C₂²	—	2	1	no	—
66	C₂²	—	2	2	no	—
67	C₂²	C₂²×D₄	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	2	no	—
71	C₄	—	4	4	no	—
72	C₄	—	4	4	no	—
73	C₄	C₂×#41	1	1	yes	characteristic
74	C₄	C₂×#41	1	1	yes	characteristic
75	C₂	—	4	1	no	—
76	C₂	#532	1	1	yes	characteristic
77	C₂	#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #499	C₂	semidirect
3. #494	C₂	semidirect
4. #491	C₂	semidirect
7. #441	C₂	semidirect
8. 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
#541	2
#499	2
#494	2
#491	2
#470	2
#467	2
#441	2
C₄×Q₁₆	2
#405	2