Finite Group Databasebeta

← C23×#14519746 #30586644

Subgroups — C23×#14519746 #30586644

Order 15616.

42862 conjugacy classes of subgroups; only the 5451 normal subgroups are shown.

normal subgroups only

This group has 42862 conjugacy classes of subgroups — too many to draw. Only its 5451 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C22⋊D61C213yesmaximal, complemented
2C22×#14519746C2156yesmaximal, direct product factor
3C23×C22⋊Dic61C213yesmaximal, complemented
4C27×C61C211yesmaximal, characteristic, complemented
5C25×D61C2211yescharacteristic, complemented
6C23×C22⋊D61—248no—
7C23×C22⋊D61C22184yescomplemented
8C24×Dic61C2213yescomplemented
9C2×#14519746C221448yesdirect product factor
10C22×C22⋊Dic61C22184yescomplemented
11C26×C61—248no—
12C26×C61C2213yes—
13C26×C61C22128yescomplemented
14C24×D61—224no—
15C24×D61C23114yescomplemented
16C22×C22⋊D61—2672no—
17C22×C22⋊D61C231336yescomplemented
18C23×Dic61—224no—
19C23×Dic61C23142yescomplemented
20#14519746C231512yesdirect product factor
21C2×C22⋊Dic61C231336yescomplemented
22C25×C61—2512no—
23C25×C61—248no—
24C25×C61—2672no—
25C25×C61D4148yescomplemented
26C25×C61C23142yes—
27C25×C61C2311yescharacteristic
28C25×C61C231112yescomplemented
29C24×C61—23584no—
30C24×C61—2384no—
31C24×C61—2336no—
32C24×C61—21344no—
33C24×C61C2×D4124yes—
34C24×C61C2×D41336yescomplemented
35C24×C61C24184yes—
36C24×C61C2417yes—
37C24×C61C24164yescomplemented
38C23×D61—2168no—
39C23×D61—464no—
40C23×D61C24128yescomplemented
41C2×C22⋊D61—21344no—
42C2×C22⋊D61C241192yescomplemented
43C22×Dic61—2168no—
44C22×Dic61C24184yescomplemented
45C22⋊Dic61C241192yescomplemented
46C23×C61—23584no—
47C23×C61—21344no—
48C23×C61—2336no—
49C23×C61—2384no—
50C23×C61#120164yescomplemented
51C23×C61C22×D4184yes—
52C23×C61C22×D41336yescomplemented
53C23×C61C25124yes—
54C23×C61C2517yes—
55C22×D61—2168no—
56C22×D61—4224no—
57C22×D61C2518yescomplemented
58C22⋊D61—2384no—
59C2×Dic61—2168no—
60C2×Dic61C25124yescomplemented
61C23×#120—611nomaximal, Sylow
62C22×C61—2512no—
63C22×C61—248no—
64C22×C61—2672no—
65C22×C61C2×#1201112yescomplemented
66C22×C61C23×D4142yes—
67C22×C61C23×D4148yescomplemented
68C22×C61C2611yescharacteristic
69C2×D61—224no—
70C2×D61—4112no—
71Dic61—224no—
72C27D6111yescharacteristic, complemented
73C24×D4—613no—
74C22×#120—6156no—
75C23×C22⋊C4—613no—
76C2×C61—248no—
77C2×C61C22×#120128yescomplemented
78C2×C61C24×D413yes—
79D61—48no—
80C26—248no—
81C26—611no—
82C26C2×D6113yes—
83C26C2×D61128yescomplemented
84C23×D4—6184no—
85C23×D4—12248no—
86C24×C4—613no—
87C2×#120—61448no—
88C22×C22⋊C4—6184no—
89C61C23×#12011yescharacteristic, complemented, Sylow
90C25—2512no—
91C25—248no—
92C25—2672no—
93C25—6114no—
94C25—12224no—
95C25C22⋊D61148yescomplemented
96C25C22×D61142yes—
97C25C22×D6111yescharacteristic
98C25C22×D611112yescomplemented
99C22×D4—61336no—
100C22×D4—122672no—
101C23×C4—6142no—
102C23×C4—12224no—
103#120—61512no—
104C2×C22⋊C4—61336no—
105C24—23584no—
106C24—2336no—
107C24—21344no—
108C24—2384no—
109C24—6128no—
110C24—122168no—
111C24—24464no—
112C24C2×C22⋊D61124yes—
113C24C2×C22⋊D611336yescomplemented
114C24C23×D61184yes—
115C24C23×D6117yes—
116C24C23×D61164yescomplemented
117C2×D4—61192no—
118C2×D4—1221344no—
119C22×C4—6184no—
120C22×C4—122168no—
121C22⋊C4—61192no—
122C23—23584no—
123C23—21344no—
124C23—2336no—
125C23—2384no—
126C23—618no—
127C23—122168no—
128C23—244224no—
129C23#14519746164yesdirect product factor
130C23C22×C22⋊D61184yes—
131C23C22×C22⋊D611336yescomplemented
132C23C24×D61124yes—
133C23C24×D6117yes—
134D4—122384no—
135C2×C4—6124no—
136C2×C4—122168no—
137C22—2512no—
138C22—2672no—
139C22—248no—
140C22—12224no—
141C22—244112no—
142C22C2×#145197461112yesdirect product factor
143C22C23×C22⋊D61142yes—
144C22C23×C22⋊D61148yescomplemented
145C22C25×D6111yescharacteristic
146C4—12224no—
147C2—248no—
148C2—2448no—
149C2C22×#14519746128yesdirect product factor
150C2C24×C22⋊D6113yes—
subgroupG/H
centerC25C22×D61
commutator, cosocleC22×C61C26
FrattiniC22C25×D61
FittingC27×C61C2
socleC25×C61C23
hypercenterC27D61
hypocenterC61C23×#120
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C25 ◁ C27

Complements

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

NH ≅ G/Ntype
1. C24×C22⋊D61C2semidirect
2. C22×#14519746C2direct (N × H)
3. C23×C22⋊Dic61C2semidirect
4. C27×C61C2semidirect
5. C25×D61C22semidirect
7. C23×C22⋊D61C22semidirect
8. C24×Dic61C22semidirect
9. C2×#14519746C22direct (N × H)
10. C22×C22⋊Dic61C22semidirect
13. C26×C61C22semidirect
15. C24×D61C23semidirect
17. C22×C22⋊D61C23semidirect
19. C23×Dic61C23semidirect
20. #14519746C23direct (N × H)
21. C2×C22⋊Dic61C23semidirect
25. C25×C61D4semidirect
28. C25×C61C23semidirect
34. C24×C61C2×D4semidirect
37. C24×C61C24semidirect
40. C23×D61C24semidirect
42. C2×C22⋊D61C24semidirect
44. C22×Dic61C24semidirect
45. C22⋊Dic61C24semidirect
50. C23×C61#120semidirect
52. C23×C61C22×D4semidirect
57. C22×D61C25semidirect
60. C2×Dic61C25semidirect
65. C22×C61C2×#120semidirect
67. C22×C61C23×D4semidirect
72. C27D61semidirect
77. C2×C61C22×#120semidirect
83. C26C2×D61semidirect
89. C61C23×#120semidirect
95. C25C22⋊D61semidirect
98. C25C22×D61semidirect
113. C24C2×C22⋊D61semidirect
116. C24C23×D61semidirect
129. C23#14519746direct (N × H)
131. C23C22×C22⋊D61semidirect
142. C22C2×#14519746direct (N × H)
144. C22C23×C22⋊D61semidirect
149. C2C22×#14519746direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C22⋊D612
C22×#145197462
C23×C22⋊Dic612
C27×C612
C23×#12061