Finite Group Databasebeta

← C23×#912 #11708971

Subgroups — C23×#912 #11708971

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×#237C212yesmaximal, complemented
2C23×C3×D8C211yesmaximal, characteristic, complemented
3C23×#214C212yesmaximal, complemented
4C22×#912C2156yesmaximal, direct product factor
5C23×Dic12C211yesmaximal, characteristic, complemented
6C23×D3×C8C211yesmaximal, characteristic, complemented
7C23×#135—31nomaximal, Sylow
8C23×C3×D4C2212yes—
9C23×#67—22no—
10C24×Dic3—22no—
11C22×#237—228no—
12C22×#237C22156yescomplemented
13C23×C4×D3C2211yescharacteristic
14C23×Dic6C2212yes—
15C22×C3×D8C22128yescomplemented
16C23×C3×C8C2211yescharacteristic
17C22×#214C22156yescomplemented
18C23×C3⋊C8C2211yescharacteristic
19C2×#912C221448yesdirect product factor
20C22×Dic12C22128yescomplemented
21C22×D3×C8C22128yescomplemented
22C23×#41—32no—
23C22×#135—356no—
24C23×Q16—31no—
25C23×SD16—32no—
26C23×D8D311yescharacteristic, complemented
27C24×C8—31no—
28C25×C3—22no—
29C24×D3D411yescharacteristic
30C22×C3×D4—214no—
31C22×C3×D4C23128yes—
32C23×C3×C4C2311yescharacteristic
33C22×#67—256no—
34C23×Dic3—22no—
35C23×Dic3—256no—
36C23×Dic3D411yescharacteristic
37C2×#237—2336no—
38C2×#237C231224yescomplemented
39C22×C4×D3D4114yes—
40C22×C4×D3C23114yes—
41C22×Dic6—214no—
42C22×Dic6C23128yes—
43C2×C3×D8C231112yescomplemented
44C22×C3×C8C23114yes—
45C2×#214C231224yescomplemented
46C22×C3⋊C8C23114yes—
47#912C231512yesdirect product factor
48C2×Dic12C231112yescomplemented
49C2×D3×C8C231112yescomplemented
50C22×#41—356no—
51C22×#41—628no—
52C23×Q8—32no—
53C23×D4—62no—
54C23×D4C2×D312yes—
55C24×C4—31no—
56C24×C4—62no—
57C2×#135—3448no—
58C22×Q16—328no—
59C22×SD16—356no—
60C22×D8C2×D3128yescomplemented
61C23×C8—31no—
62C23×C8—328no—
63C23×C8C2×D311yescharacteristic
64C24×C3—228no—
65C24×C3—416no—
66C24×C3C2×D411yescharacteristic
67C23×D3—28no—
68C23×D3C2×D4114yes—
69C2×C3×D4—284no—
70C2×C3×D4C24156yes—
71C22×C3×C4C2×D417yes—
72C22×C3×C4C2417yes—
73C2×#67—2224no—
74C22×Dic3—228no—
75C22×Dic3—2224no—
76C22×Dic3C2×D4114yes—
77#237—2448no—
78#237C241128yescomplemented
79C2×C4×D3C2×D4184yes—
80C2×C4×D3C24128yes—
81C2×Dic6—284no—
82C2×Dic6C24156yes—
83C3×D8C24164yescomplemented
84C2×C3×C8C24128yes—
85#214C241128yescomplemented
86C2×C3⋊C8C24128yes—
87Dic12C24164yescomplemented
88D3×C8C24164yescomplemented
89C25—22no—
90C25—31no—
91C2×#41—3224no—
92C2×#41—6336no—
93C22×Q8—328no—
94C22×Q8—614no—
95C22×D4—214no—
96C22×D4—656no—
97C22×D4C22×D3128yes—
98C23×C4—31no—
99C23×C4—314no—
100C23×C4—314no—
101C23×C4—656no—
102C23×C4—62no—
103C23×C4C22×D311yescharacteristic
104#135—3512no—
105C2×Q16—3112no—
106C2×SD16—3224no—
107C2×D8C22×D31112yescomplemented
108C22×C8—314no—
109C22×C8—3112no—
110C22×C8C22×D3114yes—
111C23×C3—256no—
112C23×C3—4112no—
113C23×C3#13518yescomplemented
114C23×C3C22×D417yes—
115C22×D3—256no—
116C22×D3C22×D4128yes—
117C3×D4—256no—
118C3×D4C25116yes—
119C2×C3×C4C22×D4121yes—
120C2×C3×C4C2517yes—
121#67—2128no—
122C2×Dic3—256no—
123C2×Dic3—2128no—
124C2×Dic3C22×D4128yes—
125C4×D3C22×D4156yes—
126C4×D3C2518yes—
127Dic6—256no—
128Dic6C25116yes—
129C3×C8C2518yes—
130C3⋊C8C2518yes—
131C24—228no—
132C24—314no—
133C24—416no—
134C24—68no—
135C24D3×D411yescharacteristic
136#41—3128no—
137#41—6448no—
138C2×Q8—356no—
139C2×Q8—684no—
140C2×D4—284no—
141C2×D4—6224no—
142C2×D4C23×D3156yes—
143C22×C4—314no—
144C22×C4—384no—
145C22×C4—328no—
146C22×C4—6224no—
147C22×C4—628no—
148C22×C4D3×D417yes—
149C22×C4C23×D317yes—
150Q16—364no—
151SD16—3128no—
152D8C23×D3164yescomplemented
153C2×C8—328no—
154C2×C8—364no—
155C2×C8C23×D3128yes—
156C22×C3—216no—
157C22×C3—4112no—
158C22×C3C2×#135128yescomplemented
159C22×C3C23×D417yes—
160C2×D3—256no—
161C2×D3C23×D418yes—
162C3×C4C23×D417yes—
163C3×C4C2611yescharacteristic
164Dic3—216no—
165Dic3C23×D418yes—
166C23—256no—
167C23—328no—
168C23—4112no—
169C23—656no—
170C23#91218yesdirect product factor
171C23C2×D3×D417yes—
172Q8—316no—
173Q8—656no—
174D4—256no—
175D4—6128no—
176D4C24×D3116yes—
177C2×C4—328no—
178C2×C4—356no—
179C2×C4—38no—
180C2×C4—6128no—
181C2×C4—656no—
182C2×C4C2×D3×D4121yes—
183C2×C4C24×D317yes—
184C8—38no—
185C8C24×D318yes—
186C2×C3—416no—
187C2×C3C22×#135114yescomplemented
188C2×C3C24×D411yescharacteristic
189D3—28no—
190C22—216no—
191C22—38no—
192C22—4112no—
193C22—656no—
194C22C2×#912128yesdirect product factor
195C22C22×D3×D417yes—
196C4—38no—
197C4—616no—
198C4C22×D3×D417yes—
199C4C25×D311yescharacteristic
200C3C23×#13511yescharacteristic, complemented, Sylow
201C2—416no—
202C2—68no—
203C2C22×#912114yesdirect product factor
204C2C23×D3×D411yescharacteristic
subgroupG/H
centerC24D3×D4
commutator, cosocleC3×C4C26
FrattiniC4C25×D3
FittingC23×C3×D8C2
socleC24×C3C2×D4
hypercenterC23×D8D3
hypocenterC3C23×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×D8

Complements

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

NH ≅ G/Ntype
1. C23×#237C2semidirect
2. C23×C3×D8C2semidirect
3. C23×#214C2semidirect
4. C22×#912C2direct (N × H)
5. C23×Dic12C2semidirect
6. C23×D3×C8C2semidirect
12. C22×#237C22semidirect
15. C22×C3×D8C22semidirect
17. C22×#214C22semidirect
19. C2×#912C22direct (N × H)
20. C22×Dic12C22semidirect
21. C22×D3×C8C22semidirect
26. C23×D8D3semidirect
38. C2×#237C23semidirect
43. C2×C3×D8C23semidirect
45. C2×#214C23semidirect
47. #912C23direct (N × H)
48. C2×Dic12C23semidirect
49. C2×D3×C8C23semidirect
60. C22×D8C2×D3semidirect
78. #237C24semidirect
83. C3×D8C24semidirect
85. #214C24semidirect
87. Dic12C24semidirect
88. D3×C8C24semidirect
107. C2×D8C22×D3semidirect
113. C23×C3#135semidirect
152. D8C23×D3semidirect
158. C22×C3C2×#135semidirect
170. C23#912direct (N × H)
187. C2×C3C22×#135semidirect
194. C22C2×#912direct (N × H)
200. C3C23×#135semidirect
203. C2C22×#912direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#2372
C23×C3×D82
C23×#2142
C22×#9122
C23×Dic122
C23×D3×C82
C23×#1353