Subgroups — #3142

Order 128.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532	C₂	1	1	yes	maximal, characteristic
2	C₂×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	#491	C₂	1	2	yes	maximal, complemented
4	#490	C₂	1	2	yes	maximal, complemented
5	#488	C₂	1	4	yes	maximal, complemented
6	#417	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	2	yes	—
9	C₂×M₁₆	C₂²	1	2	yes	—
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	#127	C₂²	1	1	yes	characteristic
12	#125	—	2	2	no	—
13	#125	C₂²	1	2	yes	—
14	#125	C₂²	1	2	yes	—
15	#124	C₂²	1	2	yes	—
16	#121	—	2	2	no	—
17	C₄×D₄	—	2	2	no	—
18	C₂×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄²	C₂²	1	1	yes	characteristic
21	#103	C₂²	1	8	yes	—
22	#102	C₂²	1	8	yes	—
23	#97	C₂²	1	2	yes	—
24	C₄×C₈	C₂²	1	2	yes	—
25	#41	—	2	8	no	—
26	C₂×Q₈	C₂³	1	2	yes	—
27	C₂×D₄	—	2	2	no	—
28	C₂×D₄	—	2	2	no	—
29	C₂×D₄	C₂³	1	2	yes	—
30	C₂²×C₄	—	2	2	no	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	D₄	1	2	yes	—
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	M₁₆	—	2	4	no	—
35	C₂×C₈	C₂³	1	4	yes	—
36	C₄⋊C₄	—	2	4	no	—
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	—	2	2	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	C₂³	1	4	yes	—
41	C₂²⋊C₄	—	2	4	no	—
42	C₄²	D₄	1	2	yes	—
43	C₄²	C₂³	1	1	yes	characteristic
44	C₄²	C₂³	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	C₂×D₄	1	1	yes	characteristic
47	Q₈	—	2	4	no	—
48	D₄	—	2	4	no	—
49	D₄	—	4	4	no	—
50	D₄	—	4	4	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	—	4	4	no	—
55	C₂×C₄	—	4	4	no	—
56	C₂×C₄	C₂×D₄	1	2	yes	—
57	C₂×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	#41	1	4	yes	—
60	C₂×C₄	C₂⁴	1	1	yes	characteristic
61	C₈	—	2	4	no	—
62	C₂²	—	2	2	no	—
63	C₂²	—	4	2	no	—
64	C₂²	—	4	4	no	—
65	C₂²	#124	1	2	yes	—
66	C₂²	C₂²×D₄	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	2	no	—
69	C₄	—	4	2	no	—
70	C₄	—	4	4	no	—
71	C₄	#124	1	2	yes	—
72	C₄	C₂×#41	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	—	8	2	no	—
75	C₂	C₂×#124	1	1	yes	characteristic
76	C₂	#575	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
2. C₂×#125	C₂	semidirect
3. #491	C₂	semidirect
4. #490	C₂	semidirect
5. #488	C₂	semidirect
6. #417	C₂	semidirect
7. C₄×M₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#532	2
C₂×#125	2
#491	2
#490	2
#488	2
#417	2
C₄×M₁₆	2