Subgroups — #3237

Order 128.

161 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	#533	C₂	1	1	yes	maximal, characteristic
3	#501	C₂	1	2	yes	maximal, complemented
4	#491	C₂	1	2	yes	maximal, complemented
5	#485	C₂	1	4	yes	maximal
6	#477	C₂	1	2	yes	maximal, complemented
7	#475	C₂	1	2	yes	maximal, complemented
8	#432	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	#125	—	2	1	no	—
14	#125	C₂²	1	2	yes	—
15	C₂²⋊Q₈	—	2	1	no	—
16	C₂²⋊Q₈	—	2	1	no	—
17	C₂²⋊Q₈	—	2	1	no	—
18	C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₄×Q₈	—	2	1	no	—
20	C₄×Q₈	—	2	1	no	—
21	C₄×Q₈	—	2	1	no	—
22	C₄×Q₈	C₂²	1	2	yes	—
23	#117	—	2	1	no	—
24	#117	C₂²	1	2	yes	—
25	C₂×C₄²	C₂²	1	1	yes	characteristic
26	#107	C₂²	1	4	yes	—
27	#106	C₂²	1	4	yes	—
28	C₄⋊C₈	C₂²	1	2	yes	—
29	#103	C₂²	1	4	yes	—
30	#103	C₂²	1	4	yes	—
31	C₂²⋊C₈	C₂²	1	2	yes	—
32	#97	C₂²	1	2	yes	—
33	C₂×Q₈	—	2	1	no	—
34	C₂×Q₈	—	2	1	no	—
35	C₂×Q₈	—	2	1	no	—
36	C₂×Q₈	C₂³	1	2	yes	—
37	C₂²×C₄	D₄	1	1	yes	characteristic
38	C₂²×C₄	D₄	1	1	yes	characteristic
39	C₂²×C₄	C₂³	1	1	yes	characteristic
40	C₂×C₈	C₂³	1	4	yes	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	2	no	—
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	2	1	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	C₂³	1	2	yes	—
51	C₄⋊C₄	C₂³	1	2	yes	—
52	C₄⋊C₄	C₂³	1	2	yes	—
53	C₂²⋊C₄	—	2	2	no	—
54	C₂²⋊C₄	—	2	2	no	—
55	C₄²	—	2	2	no	—
56	C₄²	—	2	2	no	—
57	C₄²	D₄	1	1	yes	characteristic
58	C₄²	D₄	1	1	yes	characteristic
59	C₄²	C₂³	1	1	yes	characteristic
60	C₄²	C₂³	1	1	yes	characteristic
61	C₂³	C₂×D₄	1	1	yes	characteristic
62	Q₈	—	2	4	no	—
63	Q₈	—	4	2	no	—
64	Q₈	—	4	2	no	—
65	Q₈	—	4	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄	C₂⁴	1	1	yes	characteristic
79	C₈	—	2	4	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂²×D₄	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	4	2	no	—
89	C₄	—	4	2	no	—
90	C₄	—	4	2	no	—
91	C₄	—	4	2	no	—
92	C₄	#137	1	2	yes	—
93	C₄	2₋¹⁺⁴	1	2	yes	—
94	C₂	—	4	1	no	—
95	C₂	#536	1	1	yes	characteristic
96	C₂	C₂×#137	1	1	yes	characteristic
97	C₂	#575	1	1	yes	characteristic

	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 ◁ 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
3. #501	C₂	semidirect
4. #491	C₂	semidirect
6. #477	C₂	semidirect
7. #475	C₂	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#501	2
#491	2
#485	2
#477	2
#475	2
#432	2