Subgroups — #2875

Order 128.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	2	yes	maximal
2	C₄×#41	C₂	1	1	yes	maximal, characteristic
3	#405	C₂	1	6	yes	maximal, complemented
4	C₄×M₁₆	C₂	1	3	yes	maximal, complemented
5	C₂×#97	C₂	1	3	yes	maximal, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	#131	C₄	1	16	yes	complemented
8	C₂×M₁₆	C₂²	1	6	yes	—
9	C₂²×C₈	C₂²	1	6	yes	—
10	C₄×Q₈	C₄	1	2	yes	—
11	C₄×D₄	C₄	1	6	yes	—
12	#117	C₂²	1	3	yes	—
13	C₂×C₄²	C₂²	1	3	yes	—
14	#97	C₂²	1	1	yes	characteristic
15	#97	C₂²	1	3	yes	—
16	#97	C₂²	1	6	yes	—
17	C₄×C₈	C₂²	1	6	yes	—
18	#41	C₂×C₄	1	8	yes	—
19	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄	C₂×C₄	1	3	yes	—
21	C₂²×C₄	C₂×C₄	1	6	yes	—
22	C₂²×C₄	C₂³	1	3	yes	—
23	M₁₆	C₂×C₄	1	24	yes	complemented
24	C₂×C₈	C₂×C₄	1	24	yes	complemented
25	C₂×C₈	C₂³	1	6	yes	—
26	C₂×C₈	C₂³	1	2	yes	—
27	C₄⋊C₄	C₂×C₄	1	6	yes	—
28	C₂²⋊C₄	C₂×C₄	1	6	yes	—
29	C₄²	C₂×C₄	1	6	yes	—
30	C₄²	C₂³	1	3	yes	—
31	C₄²	C₂³	1	1	yes	characteristic
32	C₂³	C₂²×C₄	1	3	yes	—
33	Q₈	C₄²	1	4	yes	—
34	D₄	C₄²	1	12	yes	—
35	C₂×C₄	—	4	6	no	—
36	C₂×C₄	C₂²×C₄	1	2	yes	—
37	C₂×C₄	C₂²×C₄	1	6	yes	—
38	C₂×C₄	C₂²×C₄	1	3	yes	—
39	C₂×C₄	C₂²×C₄	1	12	yes	—
40	C₂×C₄	C₂⁴	1	1	yes	characteristic
41	C₈	C₂²×C₄	1	4	yes	—
42	C₈	C₂²×C₄	1	12	yes	—
43	C₂²	—	2	6	no	—
44	C₂²	C₂×C₄²	1	6	yes	—
45	C₂²	C₂³×C₄	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	—	2	6	no	—
48	C₄	C₂×C₄²	1	6	yes	—
49	C₄	C₂³×C₄	1	2	yes	—
50	C₂	—	2	6	no	—
51	C₂	C₂²×C₄²	1	1	yes	characteristic
52	C₂	#568	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator	C₂	C₂²×C₄²
socle	C₂²	C₂³×C₄
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #405	C₂	semidirect
4. C₄×M₁₆	C₂	semidirect
5. C₂×#97	C₂	semidirect
7. #131	C₄	semidirect
23. M₁₆	C₂×C₄	semidirect
24. 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×#131	2
C₄×#41	2
#405	2
C₄×M₁₆	2
C₂×#97	2