Subgroups — #2955

Order 128.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#575	C₂	1	1	yes	maximal, characteristic
2	C₂×#131	C₂	1	2	yes	maximal, complemented
3	#444	C₂	1	4	yes	maximal, complemented
4	#443	C₂	1	4	yes	maximal, complemented
5	#430	C₂	1	1	yes	maximal, characteristic, complemented
6	#421	C₂	1	2	yes	maximal, complemented
7	#405	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	2	yes	—
9	#137	C₄	1	4	yes	complemented
10	AGL(1,ℤ/8)	C₄	1	4	yes	complemented
11	#135	C₂²	1	4	yes	—
12	#131	—	2	2	no	—
13	#131	—	2	2	no	—
14	#131	C₂²	1	4	yes	complemented
15	#131	C₂²	1	4	yes	complemented
16	C₂×M₁₆	—	2	2	no	—
17	C₂×M₁₆	C₂²	1	2	yes	—
18	C₂×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₈	—	2	2	no	—
20	C₂²×C₈	C₂²	1	1	yes	characteristic
21	#117	C₂²	1	1	yes	characteristic
22	#108	C₂²	1	4	yes	complemented
23	#104	C₂²	1	8	yes	complemented
24	#97	C₂²	1	2	yes	complemented
25	C₄×C₈	C₂²	1	2	yes	complemented
26	#41	—	2	2	no	—
27	#41	—	2	2	no	—
28	#41	C₂×C₄	1	4	yes	—
29	#41	C₂³	1	4	yes	—
30	C₂×Q₈	C₂×C₄	1	2	yes	—
31	C₂×D₄	—	2	2	no	—
32	C₂×D₄	C₂×C₄	1	2	yes	—
33	C₂²×C₄	—	2	2	no	—
34	C₂²×C₄	C₂³	1	1	yes	characteristic
35	Q₁₆	C₂×C₄	1	4	yes	—
36	SD₁₆	C₂×C₄	1	8	yes	—
37	D₈	C₂×C₄	1	4	yes	—
38	M₁₆	—	2	2	no	—
39	M₁₆	—	2	4	no	—
40	M₁₆	—	2	4	no	—
41	M₁₆	C₂×C₄	1	4	yes	—
42	M₁₆	C₂³	1	4	yes	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	—	2	4	no	—
46	C₂×C₈	D₄	1	2	yes	—
47	C₂×C₈	D₄	1	2	yes	—
48	C₂×C₈	C₂³	1	2	yes	—
49	C₂×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊C₄	—	2	1	no	—
52	C₂²⋊C₄	—	2	1	no	—
53	C₄²	C₂³	1	2	yes	—
54	C₂³	—	2	2	no	—
55	C₂³	#41	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	C₂²×C₄	1	4	yes	—
58	D₄	—	2	4	no	—
59	D₄	—	2	4	no	—
60	D₄	—	2	2	no	—
61	D₄	C₂²×C₄	1	4	yes	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₄	C₂×D₄	1	2	yes	—
68	C₂×C₄	#41	1	1	yes	characteristic
69	C₂×C₄	C₂⁴	1	1	yes	characteristic
70	C₈	—	2	4	no	—
71	C₈	C₂²×C₄	1	4	yes	—
72	C₈	C₂×D₄	1	2	yes	—
73	C₈	C₂×D₄	1	1	yes	characteristic
74	C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²	—	2	4	no	—
76	C₂²	—	4	1	no	—
77	C₂²	—	4	4	no	—
78	C₂²	C₄×D₄	1	2	yes	—
79	C₂²	C₂×#41	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	—	4	2	no	—
82	C₄	C₄×D₄	1	2	yes	—
83	C₄	C₂³×C₄	1	1	yes	characteristic
84	C₄	C₂²×D₄	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	—	2	2	no	—
87	C₂	—	4	4	no	—
88	C₂	C₂×C₄×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄
commutator	C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂	C₂×C₄×D₄
prime radical	#575	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ 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×#131	C₂	semidirect
3. #444	C₂	semidirect
4. #443	C₂	semidirect
5. #430	C₂	semidirect
6. #421	C₂	semidirect
7. #405	C₂	semidirect
9. #137	C₄	semidirect
10. AGL(1,ℤ/8)	C₄	semidirect
14. #131	C₂²	semidirect
15. #131	C₂²	semidirect
22. #108	C₂²	semidirect
23. #104	C₂²	semidirect
24. #97	C₂²	semidirect
25. C₄×C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#575	2
C₂×#131	2
#444	2
#443	2
#430	2
#421	2
#405	2