Subgroups — C₃×Q₈² #5888

Order 192.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊Q₈	C₂	1	9	yes	maximal
2	C₃×C₄×Q₈	C₂	1	6	yes	maximal
3	Q₈²	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂×C₃×Q₈	C₂²	1	6	yes	—
5	C₂×C₃×Q₈	C₂²	1	2	yes	—
6	C₃×C₄⋊C₄	C₂²	1	18	yes	—
7	C₃×C₄²	C₂²	1	9	yes	—
8	C₄⋊Q₈	C₂×C₃	1	9	yes	—
9	C₄×Q₈	C₂×C₃	1	6	yes	—
10	C₃×Q₈	—	4	6	no	—
11	C₃×Q₈	Q₈	1	8	yes	direct product factor
12	C₂×C₃×C₄	C₂³	1	9	yes	—
13	C₂×C₃×C₄	C₂³	1	6	yes	—
14	C₂×Q₈	C₂²×C₃	1	6	yes	—
15	C₂×Q₈	C₂²×C₃	1	2	yes	—
16	C₄⋊C₄	C₂²×C₃	1	18	yes	—
17	C₄²	C₂²×C₃	1	9	yes	—
18	C₂²×C₃	C₂⁴	1	1	yes	characteristic
19	C₃×C₄	—	2	9	no	—
20	C₃×C₄	C₂×Q₈	1	12	yes	—
21	Q₈	—	4	6	no	—
22	Q₈	C₃×Q₈	1	8	yes	direct product factor
23	C₂×C₄	C₂³×C₃	1	9	yes	—
24	C₂×C₄	C₂³×C₃	1	6	yes	—
25	C₂×C₃	C₂²×Q₈	1	2	yes	—
26	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
27	C₂²	C₂⁴×C₃	1	1	yes	characteristic
28	C₄	—	2	9	no	—
29	C₄	C₂×C₃×Q₈	1	12	yes	—
30	C₃	Q₈²	1	1	yes	characteristic, direct product factor, Sylow
31	C₂	C₂²×C₃×Q₈	1	2	yes	—
32	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

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.