Subgroups — D₈×#13132997 #41003876

Order 26944.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#13132997	C₂	1	2	yes	maximal, complemented
2	#27212435	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212428	C₂	1	2	yes	maximal, complemented
4	C₈×#13132997	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13132997	—	2	2	no	—
7	D₄₂₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊#13132997	C₂²	1	2	yes	—
9	C₂²⋊#13132997	—	2	2	no	—
10	C₄×#13132997	C₂²	1	1	yes	characteristic
11	D₄×D₄₂₁	C₂²	1	2	yes	—
12	D₃₃₆₈	C₄	1	1	yes	characteristic, complemented
13	#20269756	C₄	1	2	yes	complemented
14	C₈×D₄₂₁	C₂²	1	1	yes	characteristic
15	D₈×C₄₂₁	C₄	1	1	yes	characteristic, complemented
16	C₂×#13132997	—	2	2	no	—
17	C₂×#13132997	—	4	4	no	—
18	C₂×#13132997	D₄	1	1	yes	characteristic
19	C₂×#13132997	D₄	1	1	yes	characteristic
20	#16809897	—	2	1	no	—
21	C₂²×D₄₂₁	—	2	2	no	—
22	D₁₆₈₄	C₂×C₄	1	2	yes	complemented
23	C₂²⋊D₄₂₁	—	2	2	no	—
24	C₄×D₄₂₁	C₂³	1	1	yes	characteristic
25	#16809890	C₂×C₄	1	1	yes	characteristic, complemented
26	D₄×C₄₂₁	C₂×C₄	1	2	yes	complemented
27	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂²×C₄₂₁	—	2	2	no	—
29	C₂×D₄₂₁	—	2	2	no	—
30	C₂×D₄₂₁	—	4	2	no	—
31	C₂×D₄₂₁	—	4	2	no	—
32	C₂×D₄₂₁	C₂×D₄	1	1	yes	characteristic
33	#13132997	—	2	1	no	—
34	#13132997	—	4	2	no	—
35	#13132997	D₈	1	2	yes	direct product factor
36	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
37	Dic₄₂₁	#41	1	1	yes	characteristic
38	C₂×C₄₂₁	—	4	2	no	—
39	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
40	D₄₂₁	—	4	2	no	—
41	D₄₂₁	C₂×D₈	1	1	yes	characteristic
42	D₄₂₁	#135	1	1	yes	characteristic
43	C₄₂₁	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
44	C₄×D₈	—	421	1	no	maximal, Sylow
45	C₂×D₈	—	421	1	no	—
46	C₄×D₄	—	421	2	no	—
47	#107	—	421	1	no	—
48	#102	—	421	2	no	—
49	C₄×C₈	—	421	1	no	—
50	C₂×D₄	—	421	2	no	—
51	C₂²×C₄	—	842	2	no	—
52	D₈	—	421	2	no	—
53	D₈	—	421	1	no	—
54	D₈	#13132997	1	1	yes	characteristic, direct product factor
55	C₂×C₈	—	421	1	no	—
56	C₂×C₈	—	421	1	no	—
57	C₄⋊C₄	—	421	2	no	—
58	C₂²⋊C₄	—	842	2	no	—
59	C₄²	—	421	1	no	—
60	C₂³	—	842	2	no	—
61	D₄	—	842	2	no	—
62	D₄	—	421	2	no	—
63	D₄	C₂×#13132997	1	2	yes	complemented
64	C₂×C₄	—	842	2	no	—
65	C₂×C₄	—	1684	4	no	—
66	C₂×C₄	—	421	1	no	—
67	C₂×C₄	—	421	1	no	—
68	C₂×C₄	—	421	1	no	—
69	C₈	—	842	1	no	—
70	C₈	—	421	1	no	—
71	C₈	C₂×#13132997	1	1	yes	characteristic, complemented
72	C₂²	—	2	2	no	—
73	C₂²	—	842	2	no	—
74	C₂²	—	1684	2	no	—
75	C₂²	—	1684	2	no	—
76	C₂²	—	421	1	no	—
77	C₄	—	842	1	no	—
78	C₄	—	1684	2	no	—
79	C₄	—	421	2	no	—
80	C₄	—	421	1	no	—
81	C₄	C₂²×#13132997	1	1	yes	characteristic
82	C₂	—	4	2	no	—
83	C₂	—	1684	2	no	—
84	C₂	—	421	1	no	—
85	C₂	—	421	1	no	—
86	C₂	D₄×#13132997	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13132997
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₄	C₂²×#13132997
Fitting	D₈×C₄₂₁	C₄
socle	C₂×C₄₂₁	C₄×D₄
prime radical	D₈×D₄₂₁	C₂
cosocle	C₄×D₄₂₁	C₂³
hypercenter	D₈	#13132997
hypocenter	C₄₂₁	C₄×D₈
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#13132997	C₂	semidirect
2. #27212435	C₂	semidirect
3. #27212428	C₂	semidirect
4. C₈×#13132997	C₂	semidirect
12. D₃₃₆₈	C₄	semidirect
13. #20269756	C₄	semidirect
15. D₈×C₄₂₁	C₄	semidirect
22. D₁₆₈₄	C₂×C₄	semidirect
25. #16809890	C₂×C₄	semidirect
26. D₄×C₄₂₁	C₂×C₄	semidirect
27. C₈×C₄₂₁	C₂×C₄	semidirect
35. #13132997	D₈	direct (N × H)
43. C₄₂₁	C₄×D₈	semidirect
54. D₈	#13132997	direct (N × H)
63. D₄	C₂×#13132997	semidirect
71. C₈	C₂×#13132997	semidirect

Maximal subgroups

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

subgroup	index
D₄×#13132997	2
#27212435	2
#27212428	2
C₈×#13132997	2
D₈×D₄₂₁	2
C₄×D₈	421

Subgroups — D8×#13132997 #41003876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×#13132997 #41003876