Finite Group Databasebeta

← C23×#919 #11708975

Subgroups — C23×#919 #11708975

Order 768.

11422 conjugacy classes of subgroups; only the 3659 normal subgroups are shown.

normal subgroups only

This group has 11422 conjugacy classes of subgroups — too many to draw. Only its 3659 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
1C23×#239C212yesmaximal, complemented
2C23×C3×Q16C211yesmaximal, characteristic, complemented
3C23×#215C212yesmaximal, complemented
4C22×#919C2156yesmaximal, direct product factor
5C23×D24C211yesmaximal, characteristic, complemented
6C23×D3×C8C211yesmaximal, characteristic, complemented
7C23×#135—31nomaximal, Sylow
8C23×C3×Q8C2212yes—
9C22×#239—228no—
10C22×#239C22156yescomplemented
11C23×D12—22no—
12C23×D12C2212yes—
13C23×C4×D3—22no—
14C23×C4×D3C2211yescharacteristic
15C22×C3×Q16C22128yescomplemented
16C23×C3×C8C2211yescharacteristic
17C22×#215C22156yescomplemented
18C23×C3⋊C8C2211yescharacteristic
19C2×#919C221448yesdirect product factor
20C22×D24C22128yescomplemented
21C22×D3×C8C22128yescomplemented
22C23×#41—32no—
23C22×#135—356no—
24C23×Q16D311yescharacteristic, complemented
25C23×SD16—32no—
26C23×D8—31no—
27C24×C8—31no—
28C24×D3—22no—
29C24×D3D411yescharacteristic
30C22×C3×Q8—214no—
31C22×C3×Q8C23128yes—
32C23×C3×C4—22no—
33C23×C3×C4C2311yescharacteristic
34C23×Dic3D411yescharacteristic
35C2×#239—2336no—
36C2×#239C231224yescomplemented
37C22×D12—256no—
38C22×D12—214no—
39C22×D12C23128yes—
40C22×C4×D3—256no—
41C22×C4×D3D4114yes—
42C22×C4×D3C23114yes—
43C2×C3×Q16C231112yescomplemented
44C22×C3×C8C23114yes—
45C2×#215C231224yescomplemented
46C22×C3⋊C8C23114yes—
47#919C231512yesdirect product factor
48C2×D24C231112yescomplemented
49C2×D3×C8C231112yescomplemented
50C22×#41—356no—
51C22×#41—628no—
52C23×Q8C2×D312yes—
53C23×D4—32no—
54C23×D4—62no—
55C24×C4—31no—
56C24×C4—62no—
57C2×#135—3448no—
58C22×Q16C2×D3128yescomplemented
59C22×SD16—356no—
60C22×D8—328no—
61C23×C8—31no—
62C23×C8—328no—
63C23×C8C2×D311yescharacteristic
64C24×C3C2×D411yescharacteristic
65C23×D3—28no—
66C23×D3—228no—
67C23×D3—416no—
68C23×D3C2×D4114yes—
69C2×C3×Q8—284no—
70C2×C3×Q8C24156yes—
71C22×C3×C4—228no—
72C22×C3×C4C2×D417yes—
73C22×C3×C4C2417yes—
74C22×Dic3C2×D4114yes—
75#239—2448no—
76#239C241128yescomplemented
77C2×D12—2224no—
78C2×D12—284no—
79C2×D12C24156yes—
80C2×C4×D3—2224no—
81C2×C4×D3C2×D4184yes—
82C2×C4×D3C24128yes—
83C3×Q16C24164yescomplemented
84C2×C3×C8C24128yes—
85#215C241128yescomplemented
86C2×C3⋊C8C24128yes—
87D24C24164yescomplemented
88D3×C8C24164yescomplemented
89C25—31no—
90C25—62no—
91C2×#41—3224no—
92C2×#41—6336no—
93C22×Q8—214no—
94C22×Q8C22×D3128yes—
95C22×D4—328no—
96C22×D4—656no—
97C22×D4—614no—
98C23×C4—22no—
99C23×C4—31no—
100C23×C4—314no—
101C23×C4—314no—
102C23×C4—656no—
103C23×C4C22×D311yescharacteristic
104#135—3512no—
105C2×Q16C22×D31112yescomplemented
106C2×SD16—3224no—
107C2×D8—3112no—
108C22×C8—314no—
109C22×C8—3112no—
110C22×C8C22×D3114yes—
111C23×C3#13518yescomplemented
112C23×C3C22×D417yes—
113C22×D3—256no—
114C22×D3—256no—
115C22×D3—4112no—
116C22×D3C22×D4128yes—
117C3×Q8—256no—
118C3×Q8C25116yes—
119C2×C3×C4—256no—
120C2×C3×C4C22×D4121yes—
121C2×C3×C4C2517yes—
122C2×Dic3C22×D4128yes—
123D12—2128no—
124D12—256no—
125D12C25116yes—
126C4×D3—2128no—
127C4×D3C22×D4156yes—
128C4×D3C2518yes—
129C3×C8C2518yes—
130C3⋊C8C2518yes—
131C24—314no—
132C24—68no—
133C24—628no—
134C24—1216no—
135C24D3×D411yescharacteristic
136#41—3128no—
137#41—6448no—
138C2×Q8—284no—
139C2×Q8C23×D3156yes—
140C2×D4—356no—
141C2×D4—6224no—
142C2×D4—684no—
143C22×C4—228no—
144C22×C4—314no—
145C22×C4—384no—
146C22×C4—328no—
147C22×C4—6224no—
148C22×C4D3×D417yes—
149C22×C4C23×D317yes—
150Q16C23×D3164yescomplemented
151SD16—3128no—
152D8—364no—
153C2×C8—328no—
154C2×C8—364no—
155C2×C8C23×D3128yes—
156C22×C3C2×#135128yescomplemented
157C22×C3C23×D417yes—
158C2×D3—256no—
159C2×D3—216no—
160C2×D3—4112no—
161C2×D3C23×D418yes—
162C3×C4—216no—
163C3×C4C23×D417yes—
164C3×C4C2611yescharacteristic
165Dic3C23×D418yes—
166C23—328no—
167C23—656no—
168C23—656no—
169C23—12112no—
170C23#91918yesdirect product factor
171C23C2×D3×D417yes—
172Q8—256no—
173Q8C24×D3116yes—
174D4—316no—
175D4—6128no—
176D4—656no—
177C2×C4—256no—
178C2×C4—328no—
179C2×C4—356no—
180C2×C4—38no—
181C2×C4—6128no—
182C2×C4C2×D3×D4121yes—
183C2×C4C24×D317yes—
184C8—38no—
185C8C24×D318yes—
186C2×C3C22×#135114yescomplemented
187C2×C3C24×D411yescharacteristic
188D3—28no—
189D3—416no—
190C22—38no—
191C22—656no—
192C22—616no—
193C22—12112no—
194C22C2×#919128yesdirect product factor
195C22C22×D3×D417yes—
196C4—216no—
197C4—38no—
198C4C22×D3×D417yes—
199C4C25×D311yescharacteristic
200C3C23×#13511yescharacteristic, complemented, Sylow
201C2—68no—
202C2—1216no—
203C2C22×#919114yesdirect product factor
204C2C23×D3×D411yescharacteristic
subgroupG/H
centerC24D3×D4
commutator, cosocleC3×C4C26
FrattiniC4C25×D3
FittingC23×C3×Q16C2
socleC24×C3C2×D4
hypercenterC23×Q16D3
hypocenterC3C23×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×Q16

Complements

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

NH ≅ G/Ntype
1. C23×#239C2semidirect
2. C23×C3×Q16C2semidirect
3. C23×#215C2semidirect
4. C22×#919C2direct (N × H)
5. C23×D24C2semidirect
6. C23×D3×C8C2semidirect
10. C22×#239C22semidirect
15. C22×C3×Q16C22semidirect
17. C22×#215C22semidirect
19. C2×#919C22direct (N × H)
20. C22×D24C22semidirect
21. C22×D3×C8C22semidirect
24. C23×Q16D3semidirect
36. C2×#239C23semidirect
43. C2×C3×Q16C23semidirect
45. C2×#215C23semidirect
47. #919C23direct (N × H)
48. C2×D24C23semidirect
49. C2×D3×C8C23semidirect
58. C22×Q16C2×D3semidirect
76. #239C24semidirect
83. C3×Q16C24semidirect
85. #215C24semidirect
87. D24C24semidirect
88. D3×C8C24semidirect
105. C2×Q16C22×D3semidirect
111. C23×C3#135semidirect
150. Q16C23×D3semidirect
156. C22×C3C2×#135semidirect
170. C23#919direct (N × H)
186. C2×C3C22×#135semidirect
194. C22C2×#919direct (N × H)
200. C3C23×#135semidirect
203. C2C22×#919direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#2392
C23×C3×Q162
C23×#2152
C22×#9192
C23×D242
C23×D3×C82
C23×#1353