Subgroups — Q₈×#16812367 #42122926

Order 27712.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16812367	C₂	1	3	yes	maximal
2	C₄×#16812367	C₂	1	3	yes	maximal
3	Q₈×#13181884	C₂	1	1	yes	maximal, characteristic
4	C₂×#16812367	C₂²	1	3	yes	—
5	C₂×#16812367	C₂²	1	1	yes	characteristic
6	C₄⋊#13181884	C₄	1	3	yes	—
7	C₄×#13181884	C₂²	1	3	yes	—
8	Q₈×D₄₃₃	C₄	1	1	yes	characteristic
9	#16812367	—	2	3	no	—
10	#16812367	Q₈	1	2	yes	direct product factor
11	C₂×#13181884	C₂×C₄	1	3	yes	—
12	C₂×#13181884	C₂³	1	1	yes	characteristic
13	Dic₈₆₆	C₈	1	3	yes	complemented
14	C₄×D₄₃₃	C₂×C₄	1	3	yes	—
15	Q₈×C₄₃₃	C₈	1	1	yes	characteristic, complemented
16	C₂×D₄₃₃	C₂²×C₄	1	1	yes	characteristic
17	#13181884	—	2	3	no	—
18	#13181884	C₂×Q₈	1	1	yes	characteristic
19	#13181884	#41	1	1	yes	characteristic
20	C₄×C₄₃₃	C₂×C₈	1	3	yes	—
21	Dic₄₃₃	C₂×C₈	1	3	yes	—
22	C₂×C₄₃₃	C₂²×C₈	1	1	yes	characteristic
23	D₄₃₃	C₄×Q₈	1	1	yes	characteristic
24	D₄₃₃	#131	1	1	yes	characteristic
25	C₄₃₃	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
26	C₈×Q₈	—	433	1	no	maximal, Sylow
27	C₄×Q₈	—	433	1	no	—
28	C₄⋊C₈	—	433	3	no	—
29	C₄×C₈	—	433	3	no	—
30	C₂×Q₈	—	433	1	no	—
31	C₂×C₈	—	433	3	no	—
32	C₂×C₈	—	433	1	no	—
33	C₄⋊C₄	—	433	3	no	—
34	C₄²	—	433	3	no	—
35	Q₈	—	433	3	no	—
36	Q₈	#16812367	1	1	yes	characteristic, direct product factor
37	C₂×C₄	—	433	3	no	—
38	C₂×C₄	—	433	1	no	—
39	C₂×C₄	—	433	3	no	—
40	C₈	—	866	3	no	—
41	C₈	—	433	2	no	—
42	C₂²	—	433	1	no	—
43	C₄	—	866	3	no	—
44	C₄	—	433	1	no	—
45	C₄	—	433	1	no	—
46	C₄	—	433	3	no	—
47	C₄	C₂×#16812367	1	3	yes	—
48	C₂	—	433	1	no	—
49	C₂	—	433	1	no	—
50	C₂	C₂²×#16812367	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#16812367
commutator, socle	C₂×C₄₃₃	C₂²×C₈
Fitting	Q₈×C₄₃₃	C₈
prime radical	C₂×D₄₃₃	C₂²×C₄
cosocle	C₂×#13181884	C₂³
hypercenter	Q₈	#16812367
hypocenter	C₄₃₃	C₈×Q₈
LCS	G ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
10. #16812367	Q₈	direct (N × H)
13. Dic₈₆₆	C₈	semidirect
15. Q₈×C₄₃₃	C₈	semidirect
25. C₄₃₃	C₈×Q₈	semidirect
36. Q₈	#16812367	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄⋊#16812367	2
C₄×#16812367	2
Q₈×#13181884	2
C₈×Q₈	433

Subgroups — Q8×#16812367 #42122926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#16812367 #42122926