Subgroups — #1658⋊C₃ #68016

Order 384.

47 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#564⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#1658	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
3	#127⋊C₃	—	4	1	no	maximal
4	#564	C₂×C₃	1	1	yes	characteristic, complemented
5	#490	—	3	1	no	—
6	#330	—	3	1	no	—
7	C₄²⋊C₃	—	4	1	no	—
8	C₄²⋊C₃	—	8	2	no	—
9	#127	A₄	1	1	yes	characteristic
10	#125	—	3	1	no	—
11	#125	—	6	2	no	—
12	#102	—	6	2	no	—
13	#97	—	3	1	no	—
14	C₂×A₄	—	16	1	no	—
15	C₂×D₄	—	6	1	no	—
16	C₂×C₈	—	6	1	no	—
17	C₄⋊C₄	—	3	2	no	—
18	C₄⋊C₄	—	6	2	no	—
19	C₄⋊C₄	—	6	2	no	—
20	C₄²	—	2	2	no	—
21	C₄²	C₂×A₄	1	1	yes	characteristic
22	A₄	—	16	1	no	—
23	C₂³	—	4	1	no	—
24	D₄	—	12	2	no	—
25	C₂×C₄	—	3	1	no	—
26	C₂×C₄	—	6	2	no	—
27	C₈	—	12	1	no	—
28	C₂×C₃	—	64	1	no	—
29	C₂²	—	24	1	no	—
30	C₂²	#120⋊C₃	1	1	yes	characteristic
31	C₄	—	6	1	no	—
32	C₄	—	12	2	no	—
33	C₃	—	64	1	no	Sylow
34	C₂	—	3	1	no	—
35	C₂	—	16	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#564	C₂×C₃
Frattini	C₄²	C₂×A₄
Fitting	#1658	C₃
socle	C₂²	#120⋊C₃
LCS	G ▷ #564
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.