Finite Group Databasebeta

← C₂⁴×C₃×SD₁₆ #10681685

Subgroups — C₂⁴×C₃×SD₁₆ #10681685

Order 768.

13310 conjugacy classes of subgroups; only the 6878 normal subgroups are shown.

normal subgroups only

This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 6878 normal subgroups are shown.

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

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₃×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃×SD₁₆	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×SD₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂⁶×C₃	—	2	1	no	—
7	C₂³×C₃×Q₈	—	2	15	no	—
8	C₂³×C₃×Q₈	C₂²	1	30	yes	complemented
9	C₂³×C₃×D₄	—	2	15	no	—
10	C₂³×C₃×D₄	C₂²	1	30	yes	—
11	C₂⁴×C₃×C₄	—	2	1	no	—
12	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×SD₁₆	C₂²	1	560	yes	direct product factor
14	C₂³×C₃×C₈	C₂²	1	30	yes	complemented
15	C₂⁴×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
16	C₂⁴×D₄	C₂×C₃	1	1	yes	characteristic
17	C₂³×SD₁₆	C₂×C₃	1	60	yes	direct product factor
18	C₂⁴×C₈	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂⁵×C₃	—	2	30	no	—
20	C₂⁵×C₃	—	4	16	no	—
21	C₂⁵×C₃	D₄	1	1	yes	characteristic
22	C₂²×C₃×Q₈	—	2	210	no	—
23	C₂²×C₃×Q₈	C₂³	1	140	yes	complemented
24	C₂²×C₃×D₄	—	2	210	no	—
25	C₂²×C₃×D₄	C₂³	1	140	yes	—
26	C₂³×C₃×C₄	—	2	30	no	—
27	C₂³×C₃×C₄	D₄	1	15	yes	—
28	C₂³×C₃×C₄	C₂³	1	15	yes	—
29	C₂×C₃×SD₁₆	C₂³	1	960	yes	direct product factor
30	C₂²×C₃×C₈	C₂³	1	140	yes	complemented
31	C₂⁶	—	2	1	no	—
32	C₂³×Q₈	—	2	15	no	—
33	C₂³×Q₈	C₂²×C₃	1	30	yes	complemented
34	C₂³×D₄	—	2	15	no	—
35	C₂³×D₄	C₂²×C₃	1	30	yes	—
36	C₂⁴×C₄	—	2	1	no	—
37	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
38	C₂²×SD₁₆	C₂²×C₃	1	560	yes	direct product factor
39	C₂³×C₈	C₂²×C₃	1	30	yes	complemented
40	C₂⁴×C₃	—	2	140	no	—
41	C₂⁴×C₃	—	4	240	no	—
42	C₂⁴×C₃	SD₁₆	1	16	yes	direct product factor
43	C₂⁴×C₃	C₂×D₄	1	15	yes	—
44	C₂×C₃×Q₈	—	2	420	no	—
45	C₂×C₃×Q₈	C₂⁴	1	120	yes	complemented
46	C₂×C₃×D₄	—	2	420	no	—
47	C₂×C₃×D₄	C₂⁴	1	120	yes	—
48	C₂²×C₃×C₄	—	2	140	no	—
49	C₂²×C₃×C₄	C₂×D₄	1	105	yes	—
50	C₂²×C₃×C₄	C₂⁴	1	35	yes	—
51	C₃×SD₁₆	C₂⁴	1	256	yes	direct product factor
52	C₂×C₃×C₈	C₂⁴	1	120	yes	complemented
53	C₂⁵	—	2	30	no	—
54	C₂⁵	—	4	16	no	—
55	C₂⁵	C₃×D₄	1	1	yes	characteristic
56	C₂²×Q₈	—	2	210	no	—
57	C₂²×Q₈	C₂³×C₃	1	140	yes	complemented
58	C₂²×D₄	—	2	210	no	—
59	C₂²×D₄	C₂³×C₃	1	140	yes	—
60	C₂³×C₄	—	2	30	no	—
61	C₂³×C₄	C₃×D₄	1	15	yes	—
62	C₂³×C₄	C₂³×C₃	1	15	yes	—
63	C₂×SD₁₆	C₂³×C₃	1	960	yes	direct product factor
64	C₂²×C₈	C₂³×C₃	1	140	yes	complemented
65	C₂³×C₃	—	2	120	no	—
66	C₂³×C₃	—	4	560	no	—
67	C₂³×C₃	C₂×SD₁₆	1	120	yes	direct product factor
68	C₂³×C₃	C₂²×D₄	1	35	yes	—
69	C₃×Q₈	—	2	120	no	—
70	C₃×Q₈	C₂⁵	1	16	yes	complemented
71	C₃×D₄	—	2	120	no	—
72	C₃×D₄	C₂⁵	1	16	yes	—
73	C₂×C₃×C₄	—	2	120	no	—
74	C₂×C₃×C₄	C₂²×D₄	1	105	yes	—
75	C₂×C₃×C₄	C₂⁵	1	15	yes	—
76	C₃×C₈	C₂⁵	1	16	yes	complemented
77	C₂⁴	—	2	140	no	—
78	C₂⁴	—	4	240	no	—
79	C₂⁴	C₃×SD₁₆	1	16	yes	direct product factor
80	C₂⁴	C₂×C₃×D₄	1	15	yes	—
81	C₂×Q₈	—	2	420	no	—
82	C₂×Q₈	C₂⁴×C₃	1	120	yes	complemented
83	C₂×D₄	—	2	420	no	—
84	C₂×D₄	C₂⁴×C₃	1	120	yes	—
85	C₂²×C₄	—	2	140	no	—
86	C₂²×C₄	C₂×C₃×D₄	1	105	yes	—
87	C₂²×C₄	C₂⁴×C₃	1	35	yes	—
88	SD₁₆	C₂⁴×C₃	1	256	yes	direct product factor
89	C₂×C₈	C₂⁴×C₃	1	120	yes	complemented
90	C₂²×C₃	—	2	16	no	—
91	C₂²×C₃	—	4	240	no	—
92	C₂²×C₃	C₂²×SD₁₆	1	140	yes	direct product factor
93	C₂²×C₃	C₂³×D₄	1	15	yes	—
94	C₃×C₄	—	2	16	no	—
95	C₃×C₄	C₂³×D₄	1	15	yes	—
96	C₃×C₄	C₂⁶	1	1	yes	characteristic
97	C₂³	—	2	120	no	—
98	C₂³	—	4	560	no	—
99	C₂³	C₂×C₃×SD₁₆	1	120	yes	direct product factor
100	C₂³	C₂²×C₃×D₄	1	35	yes	—
101	Q₈	—	2	120	no	—
102	Q₈	C₂⁵×C₃	1	16	yes	complemented
103	D₄	—	2	120	no	—
104	D₄	C₂⁵×C₃	1	16	yes	—
105	C₂×C₄	—	2	120	no	—
106	C₂×C₄	C₂²×C₃×D₄	1	105	yes	—
107	C₂×C₄	C₂⁵×C₃	1	15	yes	—
108	C₈	C₂⁵×C₃	1	16	yes	complemented
109	C₂×C₃	—	4	16	no	—
110	C₂×C₃	C₂³×SD₁₆	1	30	yes	direct product factor
111	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
112	C₂²	—	2	16	no	—
113	C₂²	—	4	240	no	—
114	C₂²	C₂²×C₃×SD₁₆	1	140	yes	direct product factor
115	C₂²	C₂³×C₃×D₄	1	15	yes	—
116	C₄	—	2	16	no	—
117	C₄	C₂³×C₃×D₄	1	15	yes	—
118	C₄	C₂⁶×C₃	1	1	yes	characteristic
119	C₃	C₂⁴×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
120	C₂	—	4	16	no	—
121	C₂	C₂³×C₃×SD₁₆	1	30	yes	direct product factor
122	C₂	C₂⁴×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₃
prime radical	C₂⁴×C₃×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ C₂⁴×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×Q₈	C₂	semidirect
3. C₂³×C₃×SD₁₆	C₂	direct (N × H)
4. C₂⁴×C₃×C₈	C₂	semidirect
5. C₂⁴×SD₁₆	C₃	direct (N × H)
8. C₂³×C₃×Q₈	C₂²	semidirect
13. C₂²×C₃×SD₁₆	C₂²	direct (N × H)
14. C₂³×C₃×C₈	C₂²	semidirect
15. C₂⁴×Q₈	C₂×C₃	semidirect
17. C₂³×SD₁₆	C₂×C₃	direct (N × H)
18. C₂⁴×C₈	C₂×C₃	semidirect
23. C₂²×C₃×Q₈	C₂³	semidirect
29. C₂×C₃×SD₁₆	C₂³	direct (N × H)
30. C₂²×C₃×C₈	C₂³	semidirect
33. C₂³×Q₈	C₂²×C₃	semidirect
38. C₂²×SD₁₆	C₂²×C₃	direct (N × H)
39. C₂³×C₈	C₂²×C₃	semidirect
42. C₂⁴×C₃	SD₁₆	direct (N × H)
45. C₂×C₃×Q₈	C₂⁴	semidirect
51. C₃×SD₁₆	C₂⁴	direct (N × H)
52. C₂×C₃×C₈	C₂⁴	semidirect
57. C₂²×Q₈	C₂³×C₃	semidirect
63. C₂×SD₁₆	C₂³×C₃	direct (N × H)
64. C₂²×C₈	C₂³×C₃	semidirect
67. C₂³×C₃	C₂×SD₁₆	direct (N × H)
70. C₃×Q₈	C₂⁵	semidirect
76. C₃×C₈	C₂⁵	semidirect
79. C₂⁴	C₃×SD₁₆	direct (N × H)
82. C₂×Q₈	C₂⁴×C₃	semidirect
88. SD₁₆	C₂⁴×C₃	direct (N × H)
89. C₂×C₈	C₂⁴×C₃	semidirect
92. C₂²×C₃	C₂²×SD₁₆	direct (N × H)
99. C₂³	C₂×C₃×SD₁₆	direct (N × H)
102. Q₈	C₂⁵×C₃	semidirect
108. C₈	C₂⁵×C₃	semidirect
110. C₂×C₃	C₂³×SD₁₆	direct (N × H)
114. C₂²	C₂²×C₃×SD₁₆	direct (N × H)
119. C₃	C₂⁴×SD₁₆	direct (N × H)
121. C₂	C₂³×C₃×SD₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×Q₈	2
C₂⁴×C₃×D₄	2
C₂³×C₃×SD₁₆	2
C₂⁴×C₃×C₈	2
C₂⁴×SD₁₆	3