Subgroups — #2956

Order 128.

229 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	#568	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	C₂×#104	C₂	1	2	yes	maximal, complemented
7	C₄×M₁₆	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	4	no	—
13	#131	C₂²	1	8	yes	complemented
14	C₂×M₁₆	—	2	4	no	—
15	C₂×M₁₆	C₂²	1	2	yes	—
16	C₂×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	#108	C₂²	1	4	yes	complemented
20	#104	C₂²	1	4	yes	complemented
21	#104	C₂²	1	4	yes	complemented
22	#97	C₂²	1	2	yes	complemented
23	C₄×C₈	C₂²	1	2	yes	complemented
24	#41	—	2	2	no	—
25	#41	—	2	2	no	—
26	#41	C₂×C₄	1	4	yes	—
27	#41	C₂³	1	4	yes	—
28	C₂×Q₈	C₂×C₄	1	2	yes	—
29	C₂×D₄	—	2	2	no	—
30	C₂×D₄	C₂×C₄	1	2	yes	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	Q₁₆	C₂×C₄	1	4	yes	complemented
35	SD₁₆	C₂×C₄	1	8	yes	complemented
36	D₈	C₂×C₄	1	4	yes	complemented
37	M₁₆	—	2	2	no	—
38	M₁₆	—	2	8	no	—
39	M₁₆	C₂×C₄	1	4	yes	—
40	M₁₆	D₄	1	4	yes	—
41	M₁₆	C₂³	1	4	yes	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	2	8	no	—
44	C₂×C₈	C₂³	1	2	yes	—
45	C₂×C₈	C₂³	1	2	yes	—
46	C₄²	—	2	1	no	—
47	C₄²	C₂³	1	1	yes	characteristic
48	C₄²	C₂³	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	C₂³	#41	1	1	yes	characteristic
51	Q₈	—	2	2	no	—
52	Q₈	C₂²×C₄	1	4	yes	—
53	D₄	—	2	4	no	—
54	D₄	—	2	4	no	—
55	D₄	—	2	2	no	—
56	D₄	C₂²×C₄	1	4	yes	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	C₂²×C₄	1	2	yes	—
64	C₂×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₄	#41	1	1	yes	characteristic
66	C₂×C₄	C₂⁴	1	1	yes	characteristic
67	C₈	—	2	4	no	—
68	C₈	C₂²×C₄	1	4	yes	—
69	C₈	C₂×D₄	1	4	yes	—
70	C₂²	—	2	4	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	8	2	no	—
73	C₂²	C₄×D₄	1	2	yes	—
74	C₂²	C₂×#41	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	4	no	—
77	C₄	—	4	1	no	—
78	C₄	C₄×D₄	1	2	yes	—
79	C₄	C₂³×C₄	1	1	yes	characteristic
80	C₄	C₂²×D₄	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	2	2	no	—
83	C₂	—	4	4	no	—
84	C₂	C₂×C₄×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×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 ◁ C₄ ◁ C₂×M₁₆ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #568	C₂	semidirect
3. #444	C₂	semidirect
4. #443	C₂	semidirect
5. #430	C₂	semidirect
6. C₂×#104	C₂	semidirect
7. C₄×M₁₆	C₂	semidirect
9. #137	C₄	semidirect
10. AGL(1,ℤ/8)	C₄	semidirect
13. #131	C₂²	semidirect
19. #108	C₂²	semidirect
20. #104	C₂²	semidirect
21. #104	C₂²	semidirect
22. #97	C₂²	semidirect
23. C₄×C₈	C₂²	semidirect
34. Q₁₆	C₂×C₄	semidirect
35. SD₁₆	C₂×C₄	semidirect
36. D₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#575	2
#568	2
#444	2
#443	2
#430	2
C₂×#104	2
C₄×M₁₆	2