Subgroups — #2952

Order 128.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	1	yes	maximal, characteristic
2	#520	C₂	1	2	yes	maximal, complemented
3	#442	C₂	1	2	yes	maximal, complemented
4	#441	C₂	1	2	yes	maximal, complemented
5	C₄×SD₁₆	C₂	1	4	yes	maximal, complemented
6	C₂×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	#417	C₂	1	2	yes	maximal, complemented
8	#405	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41	C₂²	1	2	yes	—
10	#135	C₄	1	8	yes	complemented
11	C₂×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆	C₂²	1	2	yes	—
13	C₂×D₈	C₂²	1	1	yes	characteristic
14	C₂×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₂²	1	1	yes	characteristic
16	C₄×Q₈	C₂²	1	4	yes	—
17	C₄×D₄	—	2	4	no	—
18	C₄×D₄	C₂²	1	4	yes	—
19	#117	—	2	2	no	—
20	#117	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄	—	2	2	no	—
22	C₂×C₄⋊C₄	C₂²	1	2	yes	—
23	#106	C₂²	1	2	yes	—
24	#106	C₂²	1	2	yes	—
25	#103	C₂²	1	4	yes	—
26	#102	C₂²	1	4	yes	—
27	#97	C₂²	1	2	yes	—
28	C₄×C₈	C₂²	1	2	yes	—
29	#41	—	2	4	no	—
30	#41	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈	C₂³	1	2	yes	—
32	C₂×D₄	—	2	2	no	—
33	C₂×D₄	C₂³	1	2	yes	—
34	C₂²×C₄	—	2	2	no	—
35	C₂²×C₄	—	2	2	no	—
36	C₂²×C₄	—	2	4	no	—
37	C₂²×C₄	C₂³	1	1	yes	characteristic
38	Q₁₆	C₂×C₄	1	4	yes	complemented
39	SD₁₆	C₂×C₄	1	8	yes	complemented
40	D₈	C₂×C₄	1	4	yes	complemented
41	M₁₆	—	2	2	no	—
42	C₂×C₈	C₂×C₄	1	4	yes	complemented
43	C₂×C₈	C₂³	1	2	yes	—
44	C₂×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₈	C₂³	1	1	yes	characteristic
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	4	no	—
48	C₄⋊C₄	D₄	1	2	yes	—
49	C₄⋊C₄	C₂³	1	4	yes	—
50	C₂²⋊C₄	—	2	4	no	—
51	C₂²⋊C₄	D₄	1	2	yes	—
52	C₄²	—	2	4	no	—
53	C₄²	C₂³	1	2	yes	—
54	C₂³	—	2	2	no	—
55	C₂³	C₂×D₄	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	C₂²×C₄	1	4	yes	—
58	D₄	—	2	8	no	—
59	D₄	—	2	2	no	—
60	D₄	C₂²×C₄	1	4	yes	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	8	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	8	4	no	—
65	C₂×C₄	—	8	2	no	—
66	C₂×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₄	C₂×D₄	1	4	yes	—
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	#41	1	2	yes	—
70	C₂×C₄	C₂⁴	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	C₂²×C₄	1	2	yes	—
73	C₈	C₂²×C₄	1	2	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	—	4	2	no	—
77	C₂²	—	4	2	no	—
78	C₂²	C₄×D₄	1	2	yes	—
79	C₂²	C₂²×D₄	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	—	2	4	no	—
82	C₄	—	4	4	no	—
83	C₄	C₄×D₄	1	2	yes	—
84	C₄	C₂³×C₄	1	1	yes	characteristic
85	C₄	C₂×#41	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	—	4	4	no	—
88	C₂	C₂×C₄×D₄	1	1	yes	characteristic
89	C₂	#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator	C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×#135	C₂
LCS	G ▷ 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. #520	C₂	semidirect
3. #442	C₂	semidirect
4. #441	C₂	semidirect
5. C₄×SD₁₆	C₂	semidirect
6. C₂×#106	C₂	semidirect
7. #417	C₂	semidirect
8. #405	C₂	semidirect
10. #135	C₄	semidirect
30. #41	C₂×C₄	semidirect
38. Q₁₆	C₂×C₄	semidirect
39. SD₁₆	C₂×C₄	semidirect
40. D₈	C₂×C₄	semidirect
42. C₂×C₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135	2
#520	2
#442	2
#441	2
C₄×SD₁₆	2
C₂×#106	2
#417	2
#405	2