Subgroups — (C₃⋊D₃)⋊ΓL(1,𝔽₈) #67364

Order 378.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
3	D₃×ΓL(1,𝔽₈)	—	3	1	no	maximal
4	C₃²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
5	C₃×ΓL(1,𝔽₈)	—	2	1	no	—
6	C₃×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
7	(C₃⋊D₃)⋊C₃	—	7	1	no	maximal, Hall
8	D₃×C₇	—	3	1	no	—
9	D₃×C₇	—	9	1	no	—
10	C₂×ΓL(1,𝔽₈)	—	9	1	no	—
11	H(3,𝔽₃)	—	7	1	no	Sylow
12	C₃×C₇	—	3	1	no	—
13	C₃×C₇	C₃×D₃	1	1	yes	characteristic
14	ΓL(1,𝔽₈)	—	3	1	no	—
15	ΓL(1,𝔽₈)	—	6	1	no	—
16	C₃⋊D₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
17	C₃×D₃	—	21	1	no	—
18	C₂×C₇	—	9	1	no	Hall
19	C₃²	—	7	1	no	—
20	C₃²	—	14	1	no	—
21	C₃²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₇	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₃	—	63	1	no	—
24	D₃	—	3	1	no	—
25	D₃	—	9	1	no	—
26	C₃	—	3	1	no	—
27	C₃	—	21	1	no	—
28	C₃	—	42	1	no	—
29	C₃	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
30	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃²×C₇	C₂×C₃
Frattini	C₃	D₃×ΓL(1,𝔽₈)
socle	C₃×C₇	C₃×D₃
LCS	G ▷ C₃²×C₇
UCS	C₁

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

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