Subgroups — C₂×C₃²×Q₁₆ #64183

Order 288.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	2	yes	maximal
2	C₃²×Q₁₆	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×Q₁₆	C₃	1	4	yes	maximal, direct product factor
5	C₃²×Q₈	—	2	2	no	—
6	C₃²×Q₈	C₂²	1	4	yes	—
7	C₂×C₃²×C₄	—	2	2	no	—
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₃²×C₈	C₂²	1	2	yes	—
10	C₂×C₃×Q₈	C₂×C₃	1	8	yes	—
11	C₃×Q₁₆	C₂×C₃	1	16	yes	direct product factor
12	C₂×C₃×C₈	C₂×C₃	1	4	yes	—
13	C₂²×C₃²	D₄	1	1	yes	characteristic
14	C₃²×C₄	—	2	4	no	—
15	C₃²×C₄	D₄	1	1	yes	characteristic
16	C₃²×C₄	C₂³	1	1	yes	characteristic
17	C₂×Q₁₆	C₃²	1	1	yes	characteristic, direct product factor, Sylow
18	C₃×Q₈	—	2	8	no	—
19	C₃×Q₈	C₂²×C₃	1	16	yes	—
20	C₂×C₃×C₄	—	2	8	no	—
21	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
22	C₃×C₈	C₂²×C₃	1	8	yes	—
23	C₂×C₃²	Q₁₆	1	2	yes	direct product factor
24	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
25	C₂×Q₈	C₂×C₃²	1	2	yes	—
26	Q₁₆	C₂×C₃²	1	4	yes	direct product factor
27	C₂×C₈	C₂×C₃²	1	1	yes	characteristic
28	C₂²×C₃	C₃×D₄	1	4	yes	—
29	C₃×C₄	—	2	16	no	—
30	C₃×C₄	C₃×D₄	1	4	yes	—
31	C₃×C₄	C₂³×C₃	1	4	yes	—
32	C₃²	C₂×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
33	Q₈	—	2	2	no	—
34	Q₈	C₂²×C₃²	1	4	yes	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
37	C₈	C₂²×C₃²	1	2	yes	—
38	C₂×C₃	C₃×Q₁₆	1	8	yes	direct product factor
39	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
40	C₂²	C₃²×D₄	1	1	yes	characteristic
41	C₄	—	2	4	no	—
42	C₄	C₃²×D₄	1	1	yes	characteristic
43	C₄	C₂³×C₃²	1	1	yes	characteristic
44	C₃	C₂×C₃×Q₁₆	1	4	yes	direct product factor
45	C₂	C₃²×Q₁₆	1	2	yes	direct product factor
46	C₂	C₂×C₃²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×Q₁₆	C₂	direct (N × H)
4. C₂×C₃×Q₁₆	C₃	direct (N × H)
11. C₃×Q₁₆	C₂×C₃	direct (N × H)
17. C₂×Q₁₆	C₃²	direct (N × H)
23. C₂×C₃²	Q₁₆	direct (N × H)
26. Q₁₆	C₂×C₃²	direct (N × H)
32. C₃²	C₂×Q₁₆	direct (N × H)
38. C₂×C₃	C₃×Q₁₆	direct (N × H)
44. C₃	C₂×C₃×Q₁₆	direct (N × H)
45. C₂	C₃²×Q₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×Q₈	2
C₃²×Q₁₆	2
C₂×C₃²×C₈	2
C₂×C₃×Q₁₆	3

Subgroups — C2×C32×Q16 #64183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×Q₁₆ #64183