Subgroups — C₂²×C₃×Q₁₆ #5901

Order 192.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	2	yes	maximal
2	C₂×C₃×Q₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂×C₃×Q₈	—	2	6	no	—
6	C₂×C₃×Q₈	C₂²	1	12	yes	—
7	C₂²×C₃×C₄	—	2	2	no	—
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₃×Q₁₆	C₂²	1	16	yes	direct product factor
10	C₂×C₃×C₈	C₂²	1	6	yes	—
11	C₂²×Q₈	C₂×C₃	1	2	yes	—
12	C₂×Q₁₆	C₂×C₃	1	12	yes	direct product factor
13	C₂²×C₈	C₂×C₃	1	1	yes	characteristic
14	C₂³×C₃	D₄	1	1	yes	characteristic
15	C₃×Q₈	—	2	12	no	—
16	C₃×Q₈	C₂³	1	8	yes	—
17	C₂×C₃×C₄	—	2	12	no	—
18	C₂×C₃×C₄	D₄	1	3	yes	—
19	C₂×C₃×C₄	C₂³	1	3	yes	—
20	C₃×C₈	C₂³	1	4	yes	—
21	C₂×Q₈	—	2	6	no	—
22	C₂×Q₈	C₂²×C₃	1	12	yes	—
23	C₂²×C₄	—	2	2	no	—
24	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
25	Q₁₆	C₂²×C₃	1	16	yes	direct product factor
26	C₂×C₈	C₂²×C₃	1	6	yes	—
27	C₂²×C₃	Q₁₆	1	4	yes	direct product factor
28	C₂²×C₃	C₂×D₄	1	3	yes	—
29	C₃×C₄	—	2	8	no	—
30	C₃×C₄	C₂×D₄	1	3	yes	—
31	C₃×C₄	C₂⁴	1	1	yes	characteristic
32	C₂³	C₃×D₄	1	1	yes	characteristic
33	Q₈	—	2	12	no	—
34	Q₈	C₂³×C₃	1	8	yes	—
35	C₂×C₄	—	2	12	no	—
36	C₂×C₄	C₃×D₄	1	3	yes	—
37	C₂×C₄	C₂³×C₃	1	3	yes	—
38	C₈	C₂³×C₃	1	4	yes	—
39	C₂×C₃	C₂×Q₁₆	1	6	yes	direct product factor
40	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
41	C₂²	C₃×Q₁₆	1	4	yes	direct product factor
42	C₂²	C₂×C₃×D₄	1	3	yes	—
43	C₄	—	2	8	no	—
44	C₄	C₂×C₃×D₄	1	3	yes	—
45	C₄	C₂⁴×C₃	1	1	yes	characteristic
46	C₃	C₂²×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
47	C₂	C₂×C₃×Q₁₆	1	6	yes	direct product factor
48	C₂	C₂²×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×Q₁₆	C₂	direct (N × H)
4. C₂²×Q₁₆	C₃	direct (N × H)
9. C₃×Q₁₆	C₂²	direct (N × H)
12. C₂×Q₁₆	C₂×C₃	direct (N × H)
25. Q₁₆	C₂²×C₃	direct (N × H)
27. C₂²×C₃	Q₁₆	direct (N × H)
39. C₂×C₃	C₂×Q₁₆	direct (N × H)
41. C₂²	C₃×Q₁₆	direct (N × H)
46. C₃	C₂²×Q₁₆	direct (N × H)
47. C₂	C₂×C₃×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
C₂²×C₃×Q₈	2
C₂×C₃×Q₁₆	2
C₂²×C₃×C₈	2
C₂²×Q₁₆	3

Subgroups — C22×C3×Q16 #5901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×Q₁₆ #5901