Finite Group Databasebeta

← C23×#67006 #15660404

Subgroups — C23×#67006 #15660404

Order 2816.

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⋊D11C213yesmaximal, complemented
2C22×#67006C2156yesmaximal, direct product factor
3C23×C22⋊Dic11C213yesmaximal, complemented
4C27×C11C211yesmaximal, characteristic, complemented
5C26×C11—248no—
6C26×C11C2213yes—
7C26×C11C22128yescomplemented
8C25×D11C2211yescharacteristic, complemented
9C23×C22⋊D11—248no—
10C23×C22⋊D11C22184yescomplemented
11C24×Dic11C2213yescomplemented
12C2×#67006C221448yesdirect product factor
13C22×C22⋊Dic11C22184yescomplemented
14C25×C11—2512no—
15C25×C11—248no—
16C25×C11—2672no—
17C25×C11D4148yescomplemented
18C25×C11C23142yes—
19C25×C11C2311yescharacteristic
20C25×C11C231112yescomplemented
21C24×D11—224no—
22C24×D11C23114yescomplemented
23C22×C22⋊D11—2672no—
24C22×C22⋊D11C231336yescomplemented
25C23×Dic11—224no—
26C23×Dic11C23142yescomplemented
27#67006C231512yesdirect product factor
28C2×C22⋊Dic11C231336yescomplemented
29C23×#120—111nomaximal, Sylow
30C24×C11—23584no—
31C24×C11—2384no—
32C24×C11—2336no—
33C24×C11—21344no—
34C24×C11C2×D4124yes—
35C24×C11C2×D41336yescomplemented
36C24×C11C24184yes—
37C24×C11C2417yes—
38C24×C11C24164yescomplemented
39C23×D11—2168no—
40C23×D11—464no—
41C23×D11C24128yescomplemented
42C2×C22⋊D11—21344no—
43C2×C22⋊D11C241192yescomplemented
44C22×Dic11—2168no—
45C22×Dic11C24184yescomplemented
46C22⋊Dic11C241192yescomplemented
47C27D1111yescharacteristic, complemented
48C24×D4—113no—
49C22×#120—1156no—
50C23×C22⋊C4—113no—
51C23×C11—23584no—
52C23×C11—21344no—
53C23×C11—2336no—
54C23×C11—2384no—
55C23×C11#120164yescomplemented
56C23×C11C22×D4184yes—
57C23×C11C22×D41336yescomplemented
58C23×C11C25124yes—
59C23×C11C2517yes—
60C22×D11—2168no—
61C22×D11—4224no—
62C22×D11C2518yescomplemented
63C22⋊D11—2384no—
64C2×Dic11—2168no—
65C2×Dic11C25124yescomplemented
66C26—248no—
67C26—111no—
68C26C2×D1113yes—
69C26C2×D11128yescomplemented
70C23×D4—1184no—
71C23×D4—2248no—
72C24×C4—113no—
73C2×#120—11448no—
74C22×C22⋊C4—1184no—
75C22×C11—2512no—
76C22×C11—248no—
77C22×C11—2672no—
78C22×C11C2×#1201112yescomplemented
79C22×C11C23×D4142yes—
80C22×C11C23×D4148yescomplemented
81C22×C11C2611yescharacteristic
82C2×D11—224no—
83C2×D11—4112no—
84Dic11—224no—
85C25—2512no—
86C25—248no—
87C25—2672no—
88C25—1114no—
89C25—2224no—
90C25C22⋊D11148yescomplemented
91C25C22×D11142yes—
92C25C22×D1111yescharacteristic
93C25C22×D111112yescomplemented
94C22×D4—11336no—
95C22×D4—22672no—
96C23×C4—1142no—
97C23×C4—2224no—
98#120—11512no—
99C2×C22⋊C4—11336no—
100C2×C11—248no—
101C2×C11C22×#120128yescomplemented
102C2×C11C24×D413yes—
103D11—48no—
104C24—23584no—
105C24—2384no—
106C24—2336no—
107C24—21344no—
108C24—1128no—
109C24—22168no—
110C24—4464no—
111C24C2×C22⋊D11124yes—
112C24C2×C22⋊D111336yescomplemented
113C24C23×D11184yes—
114C24C23×D1117yes—
115C24C23×D11164yescomplemented
116C2×D4—11192no—
117C2×D4—221344no—
118C22×C4—1184no—
119C22×C4—22168no—
120C22⋊C4—11192no—
121C11C23×#12011yescharacteristic, complemented, Sylow
122C23—23584no—
123C23—21344no—
124C23—2336no—
125C23—2384no—
126C23—118no—
127C23—22168no—
128C23—44224no—
129C23#67006164yesdirect product factor
130C23C22×C22⋊D11184yes—
131C23C22×C22⋊D111336yescomplemented
132C23C24×D11124yes—
133C23C24×D1117yes—
134D4—22384no—
135C2×C4—1124no—
136C2×C4—22168no—
137C22—2512no—
138C22—248no—
139C22—2672no—
140C22—2224no—
141C22—44112no—
142C22C2×#670061112yesdirect product factor
143C22C23×C22⋊D11142yes—
144C22C23×C22⋊D11148yescomplemented
145C22C25×D1111yescharacteristic
146C4—2224no—
147C2—248no—
148C2—448no—
149C2C22×#67006128yesdirect product factor
150C2C24×C22⋊D1113yes—
subgroupG/H
centerC25C22×D11
commutator, cosocleC22×C11C26
FrattiniC22C25×D11
FittingC27×C11C2
socleC25×C11C23
hypercenterC27D11
hypocenterC11C23×#120
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C25 ◁ C27

Complements

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

NH ≅ G/Ntype
1. C24×C22⋊D11C2semidirect
2. C22×#67006C2direct (N × H)
3. C23×C22⋊Dic11C2semidirect
4. C27×C11C2semidirect
7. C26×C11C22semidirect
8. C25×D11C22semidirect
10. C23×C22⋊D11C22semidirect
11. C24×Dic11C22semidirect
12. C2×#67006C22direct (N × H)
13. C22×C22⋊Dic11C22semidirect
17. C25×C11D4semidirect
20. C25×C11C23semidirect
22. C24×D11C23semidirect
24. C22×C22⋊D11C23semidirect
26. C23×Dic11C23semidirect
27. #67006C23direct (N × H)
28. C2×C22⋊Dic11C23semidirect
35. C24×C11C2×D4semidirect
38. C24×C11C24semidirect
41. C23×D11C24semidirect
43. C2×C22⋊D11C24semidirect
45. C22×Dic11C24semidirect
46. C22⋊Dic11C24semidirect
47. C27D11semidirect
55. C23×C11#120semidirect
57. C23×C11C22×D4semidirect
62. C22×D11C25semidirect
65. C2×Dic11C25semidirect
69. C26C2×D11semidirect
78. C22×C11C2×#120semidirect
80. C22×C11C23×D4semidirect
90. C25C22⋊D11semidirect
93. C25C22×D11semidirect
101. C2×C11C22×#120semidirect
112. C24C2×C22⋊D11semidirect
115. C24C23×D11semidirect
121. C11C23×#120semidirect
129. C23#67006direct (N × H)
131. C23C22×C22⋊D11semidirect
142. C22C2×#67006direct (N × H)
144. C22C23×C22⋊D11semidirect
149. C2C22×#67006direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C22⋊D112
C22×#670062
C23×C22⋊Dic112
C27×C112
C23×#12011