Group #13180026 — subgroups

Subgroups — C₂²×C₃³×Q₁₆ #13180026

Order 1728.

3612 conjugacy classes of subgroups; only the 2492 normal subgroups are shown.

normal subgroups only

This group has 3612 conjugacy classes of subgroups — too many to draw. Only its 2492 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	2	yes	maximal
2	C₂×C₃³×Q₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃²×Q₁₆	C₃	1	13	yes	maximal, direct product factor
5	C₂×C₃³×Q₈	—	2	6	no	—
6	C₂×C₃³×Q₈	C₂²	1	12	yes	—
7	C₂²×C₃³×C₄	—	2	2	no	—
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₃³×Q₁₆	C₂²	1	16	yes	direct product factor
10	C₂×C₃³×C₈	C₂²	1	6	yes	—
11	C₂²×C₃²×Q₈	C₂×C₃	1	26	yes	—
12	C₂×C₃²×Q₁₆	C₂×C₃	1	156	yes	direct product factor
13	C₂²×C₃²×C₈	C₂×C₃	1	13	yes	—
14	C₂³×C₃³	D₄	1	1	yes	characteristic
15	C₃³×Q₈	—	2	12	no	—
16	C₃³×Q₈	C₂³	1	8	yes	—
17	C₂×C₃³×C₄	—	2	12	no	—
18	C₂×C₃³×C₄	D₄	1	3	yes	—
19	C₂×C₃³×C₄	C₂³	1	3	yes	—
20	C₃³×C₈	C₂³	1	4	yes	—
21	C₂²×C₃×Q₁₆	C₃²	1	13	yes	direct product factor
22	C₂×C₃²×Q₈	—	2	78	no	—
23	C₂×C₃²×Q₈	C₂²×C₃	1	156	yes	—
24	C₂²×C₃²×C₄	—	2	26	no	—
25	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
26	C₃²×Q₁₆	C₂²×C₃	1	208	yes	direct product factor
27	C₂×C₃²×C₈	C₂²×C₃	1	78	yes	—
28	C₂²×C₃³	Q₁₆	1	4	yes	direct product factor
29	C₂²×C₃³	C₂×D₄	1	3	yes	—
30	C₃³×C₄	—	2	8	no	—
31	C₃³×C₄	C₂×D₄	1	3	yes	—
32	C₃³×C₄	C₂⁴	1	1	yes	characteristic
33	C₂²×C₃×Q₈	C₂×C₃²	1	26	yes	—
34	C₂×C₃×Q₁₆	C₂×C₃²	1	156	yes	direct product factor
35	C₂²×C₃×C₈	C₂×C₃²	1	13	yes	—
36	C₂³×C₃²	C₃×D₄	1	13	yes	—
37	C₃²×Q₈	—	2	156	no	—
38	C₃²×Q₈	C₂³×C₃	1	104	yes	—
39	C₂×C₃²×C₄	—	2	156	no	—
40	C₂×C₃²×C₄	C₃×D₄	1	39	yes	—
41	C₂×C₃²×C₄	C₂³×C₃	1	39	yes	—
42	C₃²×C₈	C₂³×C₃	1	52	yes	—
43	C₂²×Q₁₆	C₃³	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₃³	C₂×Q₁₆	1	6	yes	direct product factor
45	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
46	C₂×C₃×Q₈	—	2	78	no	—
47	C₂×C₃×Q₈	C₂²×C₃²	1	156	yes	—
48	C₂²×C₃×C₄	—	2	26	no	—
49	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
50	C₃×Q₁₆	C₂²×C₃²	1	208	yes	direct product factor
51	C₂×C₃×C₈	C₂²×C₃²	1	78	yes	—
52	C₂²×C₃²	C₃×Q₁₆	1	52	yes	direct product factor
53	C₂²×C₃²	C₂×C₃×D₄	1	39	yes	—
54	C₃²×C₄	—	2	104	no	—
55	C₃²×C₄	C₂×C₃×D₄	1	39	yes	—
56	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
57	C₂²×Q₈	C₂×C₃³	1	2	yes	—
58	C₂×Q₁₆	C₂×C₃³	1	12	yes	direct product factor
59	C₂²×C₈	C₂×C₃³	1	1	yes	characteristic
60	C₃³	C₂²×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
61	C₂³×C₃	C₃²×D₄	1	13	yes	—
62	C₃×Q₈	—	2	156	no	—
63	C₃×Q₈	C₂³×C₃²	1	104	yes	—
64	C₂×C₃×C₄	—	2	156	no	—
65	C₂×C₃×C₄	C₃²×D₄	1	39	yes	—
66	C₂×C₃×C₄	C₂³×C₃²	1	39	yes	—
67	C₃×C₈	C₂³×C₃²	1	52	yes	—
68	C₂×C₃²	C₂×C₃×Q₁₆	1	78	yes	direct product factor
69	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
70	C₂×Q₈	—	2	6	no	—
71	C₂×Q₈	C₂²×C₃³	1	12	yes	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
74	Q₁₆	C₂²×C₃³	1	16	yes	direct product factor
75	C₂×C₈	C₂²×C₃³	1	6	yes	—
76	C₂²×C₃	C₃²×Q₁₆	1	52	yes	direct product factor
77	C₂²×C₃	C₂×C₃²×D₄	1	39	yes	—
78	C₃×C₄	—	2	104	no	—
79	C₃×C₄	C₂×C₃²×D₄	1	39	yes	—
80	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
81	C₃²	C₂²×C₃×Q₁₆	1	13	yes	direct product factor
82	C₂³	C₃³×D₄	1	1	yes	characteristic
83	Q₈	—	2	12	no	—
84	Q₈	C₂³×C₃³	1	8	yes	—
85	C₂×C₄	—	2	12	no	—
86	C₂×C₄	C₃³×D₄	1	3	yes	—
87	C₂×C₄	C₂³×C₃³	1	3	yes	—
88	C₈	C₂³×C₃³	1	4	yes	—
89	C₂×C₃	C₂×C₃²×Q₁₆	1	78	yes	direct product factor
90	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
91	C₂²	C₃³×Q₁₆	1	4	yes	direct product factor
92	C₂²	C₂×C₃³×D₄	1	3	yes	—
93	C₄	—	2	8	no	—
94	C₄	C₂×C₃³×D₄	1	3	yes	—
95	C₄	C₂⁴×C₃³	1	1	yes	characteristic
96	C₃	C₂²×C₃²×Q₁₆	1	13	yes	direct product factor
97	C₂	C₂×C₃³×Q₁₆	1	6	yes	direct product factor
98	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃³×Q₈

C₂

yes

maximal

C₂×C₃³×Q₁₆

C₂

yes

maximal, direct product factor

C₂²×C₃³×C₈

C₂

yes

maximal, characteristic

C₂²×C₃²×Q₁₆

C₃

yes

maximal, direct product factor

C₂×C₃³×Q₈

—

C₂×C₃³×Q₈

C₂²

yes

—

C₂²×C₃³×C₄

—

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₃³×Q₁₆

C₂²

yes

direct product factor

C₂×C₃³×C₈

C₂²

yes

—

C₂²×C₃²×Q₈

C₂×C₃

yes

—

C₂×C₃²×Q₁₆

C₂×C₃

156

yes

direct product factor

C₂²×C₃²×C₈

C₂×C₃

yes

—

C₂³×C₃³

D₄

yes

characteristic

C₃³×Q₈

—

C₃³×Q₈

C₂³

yes

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

D₄

yes

—

C₂×C₃³×C₄

C₂³

yes

—

C₃³×C₈

C₂³

yes

—

C₂²×C₃×Q₁₆

C₃²

yes

direct product factor

C₂×C₃²×Q₈

—

C₂×C₃²×Q₈

C₂²×C₃

156

yes

—

C₂²×C₃²×C₄

—

C₂²×C₃²×C₄

C₂²×C₃

yes

—

C₃²×Q₁₆

C₂²×C₃

208

yes

direct product factor

C₂×C₃²×C₈

C₂²×C₃

yes

—

C₂²×C₃³

Q₁₆

yes

direct product factor

C₂²×C₃³

C₂×D₄

yes

—

C₃³×C₄

—

C₃³×C₄

C₂×D₄

yes

—

C₃³×C₄

C₂⁴

yes

characteristic

C₂²×C₃×Q₈

C₂×C₃²

yes

—

C₂×C₃×Q₁₆

C₂×C₃²

156

yes

direct product factor

C₂²×C₃×C₈

C₂×C₃²

yes

—

C₂³×C₃²

C₃×D₄

yes

—

C₃²×Q₈

—

156

—

C₃²×Q₈

C₂³×C₃

104

yes

—

C₂×C₃²×C₄

—

156

—

C₂×C₃²×C₄

C₃×D₄

yes

—

C₂×C₃²×C₄

C₂³×C₃

yes

—

C₃²×C₈

C₂³×C₃

yes

—

C₂²×Q₁₆

C₃³

yes

characteristic, direct product factor, Sylow

C₂×C₃³

C₂×Q₁₆

yes

direct product factor

C₂×C₃³

C₂²×D₄

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂²×C₃²

156

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²×C₃²

yes

—

C₃×Q₁₆

C₂²×C₃²

208

yes

direct product factor

C₂×C₃×C₈

C₂²×C₃²

yes

—

C₂²×C₃²

C₃×Q₁₆

yes

direct product factor

C₂²×C₃²

C₂×C₃×D₄

yes

—

C₃²×C₄

—

104

—

C₃²×C₄

C₂×C₃×D₄

yes

—

C₃²×C₄

C₂⁴×C₃

yes

—

C₂²×Q₈

C₂×C₃³

yes

—

C₂×Q₁₆

C₂×C₃³

yes

direct product factor

C₂²×C₈

C₂×C₃³

yes

characteristic

C₃³

C₂²×Q₁₆

yes

characteristic, direct product factor, Sylow

C₂³×C₃

C₃²×D₄

yes

—

C₃×Q₈

—

156

—

C₃×Q₈

C₂³×C₃²

104

yes

—

C₂×C₃×C₄

—

156

—

C₂×C₃×C₄

C₃²×D₄

yes

—

C₂×C₃×C₄

C₂³×C₃²

yes

—

C₃×C₈

C₂³×C₃²

yes

—

C₂×C₃²

C₂×C₃×Q₁₆

yes

direct product factor

C₂×C₃²

C₂²×C₃×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

C₂²×C₃³

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₃³

yes

characteristic

Q₁₆

C₂²×C₃³

yes

direct product factor

C₂×C₈

C₂²×C₃³

yes

—

C₂²×C₃

C₃²×Q₁₆

yes

direct product factor

C₂²×C₃

C₂×C₃²×D₄

yes

—

C₃×C₄

—

104

—

C₃×C₄

C₂×C₃²×D₄

yes

—

C₃×C₄

C₂⁴×C₃²

yes

—

C₃²

C₂²×C₃×Q₁₆

yes

direct product factor

C₂³

C₃³×D₄

yes

characteristic

Q₈

—

Q₈

C₂³×C₃³

yes

—

C₂×C₄

—

C₂×C₄

C₃³×D₄

yes

—

C₂×C₄

C₂³×C₃³

yes

—

C₈

C₂³×C₃³

yes

—

C₂×C₃

C₂×C₃²×Q₁₆

yes

direct product factor

C₂×C₃

C₂²×C₃²×D₄

yes

—

C₂²

C₃³×Q₁₆

yes

direct product factor

C₂²

C₂×C₃³×D₄

yes

—

C₄

—

C₄

C₂×C₃³×D₄

yes

—

C₄

C₂⁴×C₃³

yes

characteristic

C₃

C₂²×C₃²×Q₁₆

yes

direct product factor

C₂

C₂×C₃³×Q₁₆

yes

direct product factor

C₂

C₂²×C₃³×D₄

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃³	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ C₂²×C₃³×C₄ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃³

D₄

commutator, Frattini, cosocle

C₄

C₂⁴×C₃³

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃³ ◁ C₂²×C₃³×C₄ ◁ G

N	H ≅ G/N	type
2. C₂×C₃³×Q₁₆	C₂	direct (N × H)
4. C₂²×C₃²×Q₁₆	C₃	direct (N × H)
9. C₃³×Q₁₆	C₂²	direct (N × H)
12. C₂×C₃²×Q₁₆	C₂×C₃	direct (N × H)
21. C₂²×C₃×Q₁₆	C₃²	direct (N × H)
26. C₃²×Q₁₆	C₂²×C₃	direct (N × H)
28. C₂²×C₃³	Q₁₆	direct (N × H)
34. C₂×C₃×Q₁₆	C₂×C₃²	direct (N × H)
43. C₂²×Q₁₆	C₃³	direct (N × H)
44. C₂×C₃³	C₂×Q₁₆	direct (N × H)
50. C₃×Q₁₆	C₂²×C₃²	direct (N × H)
52. C₂²×C₃²	C₃×Q₁₆	direct (N × H)
58. C₂×Q₁₆	C₂×C₃³	direct (N × H)
60. C₃³	C₂²×Q₁₆	direct (N × H)
68. C₂×C₃²	C₂×C₃×Q₁₆	direct (N × H)
74. Q₁₆	C₂²×C₃³	direct (N × H)
76. C₂²×C₃	C₃²×Q₁₆	direct (N × H)
81. C₃²	C₂²×C₃×Q₁₆	direct (N × H)
89. C₂×C₃	C₂×C₃²×Q₁₆	direct (N × H)
91. C₂²	C₃³×Q₁₆	direct (N × H)
96. C₃	C₂²×C₃²×Q₁₆	direct (N × H)
97. C₂	C₂×C₃³×Q₁₆	direct (N × H)

H ≅ G/N

type

2. C₂×C₃³×Q₁₆

C₂

direct (N × H)

4. C₂²×C₃²×Q₁₆

C₃

direct (N × H)

9. C₃³×Q₁₆

C₂²

direct (N × H)

12. C₂×C₃²×Q₁₆

C₂×C₃

direct (N × H)

21. C₂²×C₃×Q₁₆

C₃²

direct (N × H)

26. C₃²×Q₁₆

C₂²×C₃

direct (N × H)

28. C₂²×C₃³

Q₁₆

direct (N × H)

34. C₂×C₃×Q₁₆

C₂×C₃²

direct (N × H)

43. C₂²×Q₁₆

C₃³

direct (N × H)

44. C₂×C₃³

C₂×Q₁₆

direct (N × H)

50. C₃×Q₁₆

C₂²×C₃²

direct (N × H)

52. C₂²×C₃²

C₃×Q₁₆

direct (N × H)

58. C₂×Q₁₆

C₂×C₃³

direct (N × H)

60. C₃³

C₂²×Q₁₆

direct (N × H)

68. C₂×C₃²

C₂×C₃×Q₁₆

direct (N × H)

74. Q₁₆

C₂²×C₃³

direct (N × H)

76. C₂²×C₃

C₃²×Q₁₆

direct (N × H)

81. C₃²

C₂²×C₃×Q₁₆

direct (N × H)

89. C₂×C₃

C₂×C₃²×Q₁₆

direct (N × H)

91. C₂²

C₃³×Q₁₆

direct (N × H)

96. C₃

C₂²×C₃²×Q₁₆

direct (N × H)

97. C₂

C₂×C₃³×Q₁₆

direct (N × H)

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

subgroup	index
C₂²×C₃³×Q₈	2
C₂×C₃³×Q₁₆	2
C₂²×C₃³×C₈	2
C₂²×C₃²×Q₁₆	3

subgroup

index

C₂²×C₃³×Q₈

C₂×C₃³×Q₁₆

C₂²×C₃³×C₈

C₂²×C₃²×Q₁₆

Subgroups — C₂²×C₃³×Q₁₆ #13180026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C33×Q16 #13180026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃³×Q₁₆ #13180026