Subgroups — #2275

Order 128.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄	C₂	1	3	yes	maximal
2	C₄×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₄×C₂²⋊C₄	C₂	1	6	yes	maximal, complemented
4	#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#95	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄	C₂²	1	1	yes	characteristic
7	C₂³×C₄	C₂²	1	6	yes	—
8	C₄×D₄	C₄	1	24	yes	complemented
9	C₂×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
11	C₂×C₄²	C₂²	1	6	yes	—
12	C₂×C₄²	C₂²	1	3	yes	—
13	#95	C₂²	1	1	yes	characteristic
14	#95	C₂²	1	6	yes	—
15	#95	C₂²	1	3	yes	—
16	C₂⁴	C₂³	1	2	yes	—
17	C₂×D₄	C₂×C₄	1	12	yes	—
18	C₂²×C₄	—	4	12	no	—
19	C₂²×C₄	C₂×C₄	1	24	yes	complemented
20	C₂²×C₄	C₂³	1	3	yes	—
21	C₂²×C₄	C₂³	1	3	yes	—
22	C₂²×C₄	C₂³	1	6	yes	—
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	C₄⋊C₄	C₂×C₄	1	12	yes	complemented
25	C₂²⋊C₄	C₂×C₄	1	24	yes	complemented
26	C₄²	—	4	6	no	—
27	C₄²	C₂×C₄	1	12	yes	complemented
28	C₂³	—	2	6	no	—
29	C₂³	—	2	2	no	—
30	C₂³	C₂²×C₄	1	12	yes	—
31	C₂³	C₂⁴	1	1	yes	characteristic
32	D₄	C₄²	1	16	yes	complemented
33	C₂×C₄	—	2	3	no	—
34	C₂×C₄	—	2	12	no	—
35	C₂×C₄	—	2	6	no	—
36	C₂×C₄	—	2	3	no	—
37	C₂×C₄	—	4	24	no	—
38	C₂×C₄	C₂²×C₄	1	6	yes	—
39	C₂×C₄	C₂²×C₄	1	6	yes	—
40	C₂×C₄	C₂²×C₄	1	12	yes	—
41	C₂×C₄	C₂²×C₄	1	6	yes	—
42	C₂²	—	2	12	no	—
43	C₂²	—	2	12	no	—
44	C₂²	C₂×C₄²	1	8	yes	—
45	C₂²	C₂³×C₄	1	3	yes	—
46	C₂²	H(5,𝔽₂)	1	3	yes	—
47	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
48	C₄	—	2	6	no	—
49	C₄	—	2	12	no	—
50	C₄	—	2	6	no	—
51	C₄	C₂×C₄²	1	4	yes	—
52	C₂	—	2	8	no	—
53	C₂	C₂²×C₄²	1	1	yes	characteristic
54	C₂	#518	1	3	yes	—
55	C₂	#520	1	3	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁴
commutator	C₂	C₂²×C₄²
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊C₄	C₂	semidirect
3. C₄×C₂²⋊C₄	C₂	semidirect
4. #376	C₂	semidirect
5. C₂×#95	C₂	semidirect
8. C₄×D₄	C₄	semidirect
19. C₂²×C₄	C₂×C₄	semidirect
24. C₄⋊C₄	C₂×C₄	semidirect
25. C₂²⋊C₄	C₂×C₄	semidirect
27. C₄²	C₂×C₄	semidirect
32. D₄	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄	2
C₄×C₄⋊C₄	2
C₄×C₂²⋊C₄	2
#376	2
C₂×#95	2