Subgroups — #1705⋊C₃ #68017

Order 384.

49 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#1705	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
3	#127⋊C₃	—	4	1	no	maximal
4	#498	—	3	1	no	—
5	#492	—	3	1	no	—
6	C₈²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₄²⋊C₃	—	4	1	no	—
8	C₂×SD₁₆	—	6	2	no	—
9	C₄⋊Q₈	—	3	1	no	—
10	#127	A₄	1	1	yes	characteristic
11	#106	—	6	2	no	—
12	C₄×C₈	—	3	1	no	—
13	C₂×A₄	—	16	1	no	—
14	C₂×Q₈	—	6	1	no	—
15	C₂×D₄	—	6	1	no	—
16	SD₁₆	—	12	4	no	—
17	C₂×C₈	—	3	2	no	—
18	C₄⋊C₄	—	6	2	no	—
19	C₄²	C₂×A₄	1	1	yes	characteristic
20	A₄	—	16	1	no	—
21	C₂³	—	4	1	no	—
22	Q₈	—	12	2	no	—
23	D₄	—	12	2	no	—
24	C₂×C₄	—	3	1	no	—
25	C₂×C₄	—	12	1	no	—
26	C₈	—	3	4	no	—
27	C₂×C₃	—	64	1	no	—
28	C₂²	—	24	1	no	—
29	C₂²	#127⋊C₃	1	1	yes	characteristic
30	C₄	—	3	2	no	—
31	C₄	—	24	1	no	—
32	C₃	—	64	1	no	Sylow
33	C₂	—	3	1	no	—
34	C₂	—	16	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₈²	C₂×C₃
Frattini	C₄²	C₂×A₄
Fitting	#1705	C₃
socle	C₂²	#127⋊C₃
LCS	G ▷ C₈²
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.