Subgroups — C₂×#336 #1727

Order 128.

264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈	C₂	1	2	yes	maximal
2	C₂²×C₄²	C₂	1	1	yes	maximal, characteristic
3	#336	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄	C₄	1	2	yes	—
5	C₂³×C₄	C₂²	1	1	yes	characteristic
6	C₂²×C₈	—	2	4	no	—
7	C₂²×C₈	—	2	4	no	—
8	C₂²×C₈	C₄	1	8	yes	complemented
9	C₂²×C₈	C₂²	1	4	yes	—
10	C₂×C₄²	—	2	4	no	—
11	C₂×C₄²	C₄	1	2	yes	—
12	C₂×C₄²	C₂²	1	2	yes	—
13	C₂⁴	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄	—	2	4	no	—
15	C₂²×C₄	—	2	4	no	—
16	C₂²×C₄	C₈	1	8	yes	complemented
17	C₂²×C₄	C₂×C₄	1	4	yes	—
18	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄	C₂×C₄	1	4	yes	—
20	C₂²×C₄	D₄	1	2	yes	—
21	C₂²×C₄	D₄	1	4	yes	—
22	C₂²×C₄	Q₈	1	2	yes	—
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₈	—	2	16	no	—
25	C₂×C₈	—	2	4	no	—
26	C₂×C₈	—	2	4	no	—
27	C₂×C₈	C₂×C₄	1	8	yes	complemented
28	C₄²	—	2	4	no	—
29	C₄²	—	2	4	no	—
30	C₂³	C₄²	1	2	yes	—
31	C₂³	C₂²⋊C₄	1	2	yes	—
32	C₂³	C₄⋊C₄	1	2	yes	—
33	C₂³	C₂×C₈	1	4	yes	—
34	C₂³	M₁₆	1	4	yes	—
35	C₂³	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄	—	2	16	no	—
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	C₄²	1	2	yes	—
40	C₂×C₄	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄	C₂²⋊C₄	1	8	yes	—
42	C₂×C₄	C₄⋊C₄	1	2	yes	—
43	C₂×C₄	C₄⋊C₄	1	8	yes	—
44	C₂×C₄	C₂×C₈	1	8	yes	complemented
45	C₂×C₄	C₂²×C₄	1	2	yes	—
46	C₂×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄	C₂×D₄	1	2	yes	—
48	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₈	—	2	8	no	—
50	C₂²	#95	1	4	yes	—
51	C₂²	C₄×C₈	1	4	yes	—
52	C₂²	#97	1	4	yes	—
53	C₂²	C₂²⋊C₈	1	8	yes	—
54	C₂²	C₄⋊C₈	1	8	yes	—
55	C₂²	C₂×C₄²	1	1	yes	characteristic
56	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²	C₂²×C₈	1	2	yes	—
59	C₂²	C₂×M₁₆	1	2	yes	—
60	C₄	—	2	8	no	—
61	C₄	#95	1	4	yes	—
62	C₄	C₂×C₂²⋊C₄	1	2	yes	—
63	C₄	C₂×C₄⋊C₄	1	2	yes	—
64	C₂	#336	1	8	yes	direct product factor
65	C₂	C₂×#95	1	1	yes	characteristic
66	C₂	C₂×C₄×C₈	1	1	yes	characteristic
67	C₂	C₂×#97	1	1	yes	characteristic
68	C₂	C₂×C₂²⋊C₈	1	2	yes	—
69	C₂	C₂×C₄⋊C₈	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²
commutator	C₂	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄	C₂³
socle, prime radical	C₂⁴	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #336	C₂	direct (N × H)
8. C₂²×C₈	C₄	semidirect
16. C₂²×C₄	C₈	semidirect
27. C₂×C₈	C₂×C₄	semidirect
44. C₂×C₄	C₂×C₈	semidirect
64. C₂	#336	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈	2
C₂²×C₄²	2
#336	2

Subgroups — C2×#336 #1727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#336 #1727