Subgroups — C₂×C₃³×C₄⋊Q₈ #13179986

Order 1728.

4060 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	2	yes	maximal
2	C₃³×C₄⋊Q₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃³×C₄⋊C₄	C₂	1	4	yes	maximal
4	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃²×C₄⋊Q₈	C₃	1	13	yes	maximal, direct product factor
6	C₂×C₃³×Q₈	—	2	8	no	—
7	C₂×C₃³×Q₈	C₂²	1	8	yes	—
8	C₂²×C₃³×C₄	C₂²	1	4	yes	—
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃³×C₄	C₂²	1	2	yes	—
11	C₃³×C₄⋊C₄	C₂²	1	16	yes	—
12	C₃³×C₄²	C₂²	1	4	yes	—
13	C₂²×C₃²×Q₈	C₂×C₃	1	26	yes	—
14	C₃²×C₄⋊Q₈	C₂×C₃	1	104	yes	direct product factor
15	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	52	yes	—
16	C₂×C₃²×C₄²	C₂×C₃	1	13	yes	—
17	C₂³×C₃³	C₂³	1	1	yes	characteristic
18	C₃³×Q₈	—	2	16	no	—
19	C₂×C₃³×C₄	—	2	8	no	—
20	C₂×C₃³×C₄	D₄	1	4	yes	—
21	C₂×C₃³×C₄	Q₈	1	8	yes	complemented
22	C₂×C₃³×C₄	C₂³	1	8	yes	—
23	C₂×C₃³×C₄	C₂³	1	2	yes	—
24	C₂×C₃³×C₄	C₂³	1	4	yes	—
25	C₂×C₃×C₄⋊Q₈	C₃²	1	13	yes	direct product factor
26	C₂×C₃²×Q₈	—	2	104	no	—
27	C₂×C₃²×Q₈	C₂²×C₃	1	104	yes	—
28	C₂²×C₃²×C₄	C₂²×C₃	1	52	yes	—
29	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
30	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
31	C₃²×C₄⋊C₄	C₂²×C₃	1	208	yes	—
32	C₃²×C₄²	C₂²×C₃	1	52	yes	—
33	C₂²×C₃³	C₂×D₄	1	2	yes	—
34	C₂²×C₃³	C₂×Q₈	1	4	yes	—
35	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
36	C₃³×C₄	—	2	8	no	—
37	C₃³×C₄	C₂×D₄	1	4	yes	—
38	C₃³×C₄	C₂×Q₈	1	8	yes	complemented
39	C₂²×C₃×Q₈	C₂×C₃²	1	26	yes	—
40	C₃×C₄⋊Q₈	C₂×C₃²	1	104	yes	direct product factor
41	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	52	yes	—
42	C₂×C₃×C₄²	C₂×C₃²	1	13	yes	—
43	C₂³×C₃²	C₂³×C₃	1	13	yes	—
44	C₃²×Q₈	—	2	208	no	—
45	C₂×C₃²×C₄	—	2	104	no	—
46	C₂×C₃²×C₄	C₃×D₄	1	52	yes	—
47	C₂×C₃²×C₄	C₃×Q₈	1	104	yes	complemented
48	C₂×C₃²×C₄	C₂³×C₃	1	104	yes	—
49	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
50	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
51	C₂×C₄⋊Q₈	C₃³	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃³	C₄⋊Q₈	1	4	yes	direct product factor
53	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₃³	C₂²×Q₈	1	2	yes	—
55	C₂×C₃×Q₈	—	2	104	no	—
56	C₂×C₃×Q₈	C₂²×C₃²	1	104	yes	—
57	C₂²×C₃×C₄	C₂²×C₃²	1	52	yes	—
58	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
59	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
60	C₃×C₄⋊C₄	C₂²×C₃²	1	208	yes	—
61	C₃×C₄²	C₂²×C₃²	1	52	yes	—
62	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
63	C₂²×C₃²	C₂×C₃×Q₈	1	52	yes	—
64	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
65	C₃²×C₄	—	2	104	no	—
66	C₃²×C₄	C₂×C₃×D₄	1	52	yes	—
67	C₃²×C₄	C₂×C₃×Q₈	1	104	yes	complemented
68	C₂²×Q₈	C₂×C₃³	1	2	yes	—
69	C₄⋊Q₈	C₂×C₃³	1	8	yes	direct product factor
70	C₂×C₄⋊C₄	C₂×C₃³	1	4	yes	—
71	C₂×C₄²	C₂×C₃³	1	1	yes	characteristic
72	C₃³	C₂×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
73	C₂³×C₃	C₂³×C₃²	1	13	yes	—
74	C₃×Q₈	—	2	208	no	—
75	C₂×C₃×C₄	—	2	104	no	—
76	C₂×C₃×C₄	C₃²×D₄	1	52	yes	—
77	C₂×C₃×C₄	C₃²×Q₈	1	104	yes	complemented
78	C₂×C₃×C₄	C₂³×C₃²	1	104	yes	—
79	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
80	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
81	C₂×C₃²	C₃×C₄⋊Q₈	1	52	yes	direct product factor
82	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
83	C₂×C₃²	C₂²×C₃×Q₈	1	26	yes	—
84	C₂×Q₈	—	2	8	no	—
85	C₂×Q₈	C₂²×C₃³	1	8	yes	—
86	C₂²×C₄	C₂²×C₃³	1	4	yes	—
87	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
88	C₂²×C₄	C₂²×C₃³	1	2	yes	—
89	C₄⋊C₄	C₂²×C₃³	1	16	yes	—
90	C₄²	C₂²×C₃³	1	4	yes	—
91	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
92	C₂²×C₃	C₂×C₃²×Q₈	1	52	yes	—
93	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
94	C₃×C₄	—	2	104	no	—
95	C₃×C₄	C₂×C₃²×D₄	1	52	yes	—
96	C₃×C₄	C₂×C₃²×Q₈	1	104	yes	complemented
97	C₃²	C₂×C₃×C₄⋊Q₈	1	13	yes	direct product factor
98	C₂³	C₂³×C₃³	1	1	yes	characteristic
99	Q₈	—	2	16	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	C₃³×D₄	1	4	yes	—
102	C₂×C₄	C₃³×Q₈	1	8	yes	complemented
103	C₂×C₄	C₂³×C₃³	1	8	yes	—
104	C₂×C₄	C₂³×C₃³	1	2	yes	—
105	C₂×C₄	C₂³×C₃³	1	4	yes	—
106	C₂×C₃	C₃²×C₄⋊Q₈	1	52	yes	direct product factor
107	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
108	C₂×C₃	C₂²×C₃²×Q₈	1	26	yes	—
109	C₂²	C₂×C₃³×D₄	1	2	yes	—
110	C₂²	C₂×C₃³×Q₈	1	4	yes	—
111	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
112	C₄	—	2	8	no	—
113	C₄	C₂×C₃³×D₄	1	4	yes	—
114	C₄	C₂×C₃³×Q₈	1	8	yes	complemented
115	C₃	C₂×C₃²×C₄⋊Q₈	1	13	yes	direct product factor
116	C₂	C₃³×C₄⋊Q₈	1	4	yes	direct product factor
117	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
118	C₂	C₂²×C₃³×Q₈	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄⋊Q₈	C₂	direct (N × H)
5. C₂×C₃²×C₄⋊Q₈	C₃	direct (N × H)
14. C₃²×C₄⋊Q₈	C₂×C₃	direct (N × H)
21. C₂×C₃³×C₄	Q₈	semidirect
25. C₂×C₃×C₄⋊Q₈	C₃²	direct (N × H)
38. C₃³×C₄	C₂×Q₈	semidirect
40. C₃×C₄⋊Q₈	C₂×C₃²	direct (N × H)
47. C₂×C₃²×C₄	C₃×Q₈	semidirect
51. C₂×C₄⋊Q₈	C₃³	direct (N × H)
52. C₂×C₃³	C₄⋊Q₈	direct (N × H)
67. C₃²×C₄	C₂×C₃×Q₈	semidirect
69. C₄⋊Q₈	C₂×C₃³	direct (N × H)
72. C₃³	C₂×C₄⋊Q₈	direct (N × H)
77. C₂×C₃×C₄	C₃²×Q₈	semidirect
81. C₂×C₃²	C₃×C₄⋊Q₈	direct (N × H)
96. C₃×C₄	C₂×C₃²×Q₈	semidirect
97. C₃²	C₂×C₃×C₄⋊Q₈	direct (N × H)
102. C₂×C₄	C₃³×Q₈	semidirect
106. C₂×C₃	C₃²×C₄⋊Q₈	direct (N × H)
114. C₄	C₂×C₃³×Q₈	semidirect
115. C₃	C₂×C₃²×C₄⋊Q₈	direct (N × H)
116. C₂	C₃³×C₄⋊Q₈	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×C₃³×Q₈	2
C₃³×C₄⋊Q₈	2
C₂×C₃³×C₄⋊C₄	2
C₂×C₃³×C₄²	2
C₂×C₃²×C₄⋊Q₈	3

Subgroups — C2×C33×C4⋊Q8 #13179986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×C₄⋊Q₈ #13179986