Finite Group Databasebeta

← #10692821

Subgroups — #10692821

Order 768.

456 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2371C211yesmaximal, characteristic, complemented
2#83012C211yesmaximal, characteristic
3#71736C211yesmaximal, characteristic, complemented
4#69432C212yesmaximal
5#67615C212yesmaximal, complemented
6#9442—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8#5477C2211yescharacteristic
9C3×#387—22no—
10C3×#387—22no—
11C3×C4×C4⋊C4—22no—
12C3×#376C2212yes—
13#5069C412yescomplemented
14#5068—22no—
15#5067C412yescomplemented
16#4966—22no—
17#4926C2212yes—
18C4×#208C2211yescharacteristic
19#4543—22no—
20#4451—22no—
21#4451—22no—
22#3142—31no—
23#2371D311yescharacteristic, complemented
24#1534—31no—
25#1505—32no—
26#1300—32no—
27C2×#235—21no—
28C3×#125—22no—
29C3×#125C2×C412yes—
30C3×#125D412yes—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4D412yes—
34C2×C3×C42—22no—
35C2×C3×C42—22no—
36C2×C3×C42C2311yescharacteristic
37#930—22no—
38C2×#208—21no—
39#875—21no—
40#874C2×C411yescharacteristic
41C4×D12—22no—
42#869C2×C411yescharacteristic
43C3×#95—22no—
44C3×#95—22no—
45C3×#95—22no—
46C3×#95D412yes—
47C3×#95D412yes—
48#830—22no—
49#810—24no—
50#809—24no—
51C4⋊(C3⋊C8)—22no—
52#803—22no—
53#803C2×C412yes—
54C4×C3⋊C8—21no—
55#532—31no—
56C2×#125C2×D311yescharacteristic
57#491—32no—
58#490—32no—
59#488—62no—
60#432—32no—
61#417—62no—
62C4×M16—31no—
63#387—22no—
64#387—22no—
65C4×C4⋊C4—22no—
66#376C2×D312yes—
67#330—62no—
68#327—62no—
69#327—62no—
70C22×C3×C4—22no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D412yes—
75C2×#67—41no—
76#235—44no—
77C2×D12—21no—
78C2×D12—22no—
79C2×C4×D3—41no—
80C2×Dic6—21no—
81C3×C4⋊C4—24no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—44no—
85C3×C4⋊C4C22⋊C414yes—
86C3×C42—44no—
87C3×C42—44no—
88C3×C42C22×C411yescharacteristic
89C3×C42C2×D411yescharacteristic
90C3×C42#4112yes—
91(C2×D3)⋊C4—42no—
92C4⋊Dic3—22no—
93#208—42no—
94C2×C3⋊C8—22no—
95C2×C3⋊C8—42no—
96C2×#41—61no—
97C2×M16—61no—
98C4⋊Q8—31no—
99#127—31no—
100#125—22no—
101#125C4×D312yes—
102#125D1212yes—
103#124—61no—
104#121—62no—
105C4×D4—62no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4#6712yes—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42C22×D311yescharacteristic
112C4⋊C8—62no—
113#103—64no—
114#102—64no—
115C22⋊C8—62no—
116#97—32no—
117#97—62no—
118C4×C8—61no—
119#95—22no—
120#95—22no—
121#95—22no—
122#95#6712yes—
123#95#6712yes—
124C23×C3#12011yescharacteristic
125C22×D3—41no—
126C2×C3×C4—22no—
127C2×C3×C4—24no—
128C2×C3×C4—24no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4C2×C22⋊C411yescharacteristic
138C2×C3×C4C4×D412yes—
139C2×C3×C4#12112yes—
140C2×C3×C4#12311yescharacteristic
141#67—82no—
142C2×Dic3—41no—
143D12—42no—
144D12—82no—
145C4×D3—82no—
146Dic6—42no—
147C3⋊C8—42no—
148C3⋊C8—82no—
149#41—124no—
150C2×Q8—61no—
151C2×D4—61no—
152C2×D4—62no—
153C2×D4—121no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—121no—
158C22×C4C2×#6711yescharacteristic
159C22×C4C2×#6712yes—
160M16—122no—
161C2×C8—62no—
162C2×C8—122no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—24no—
166C4⋊C4—44no—
167C4⋊C4—62no—
168C4⋊C4(C2×D3)⋊C414yes—
169C22⋊C4—122no—
170C42—44no—
171C42—44no—
172C42C2×C4×D311yescharacteristic
173C42C2×D1211yescharacteristic
174C42#23512yes—
175C22×C3—22no—
176C22×C3#38611yescharacteristic
177C22×C3#45412yes—
178C2×D3—81no—
179C2×D3—82no—
180C3×C4—21no—
181C3×C4—21no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—44no—
185C3×C4—44no—
186C3×C4—82no—
187Dic3—81no—
188C23—121no—
189C23#95311yescharacteristic
190Q8—122no—
191D4—122no—
192D4—242no—
193D4—242no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—121no—
206C2×C4—242no—
207C2×C4C2×(C2×D3)⋊C411yescharacteristic
208C2×C4C4×#6712yes—
209C2×C4#92911yescharacteristic
210C2×C4#93012yes—
211C8—122no—
212C8—242no—
213C2×C3—22no—
214C2×C3#188811yescharacteristic
215C2×C3#190612yes—
216D3—161no—
217C22—22no—
218C22—242no—
219C22—241no—
220C22#521311yescharacteristic
221C22#524512yes—
222C4—22no—
223C4—22no—
224C4—21no—
225C4—21no—
226C4—44no—
227C4—44no—
228C4—82no—
229C4—241no—
230C3#944211yescharacteristic, complemented, Sylow
231C2—22no—
232C2—481no—
233C2#7164512yes—
234C2#7166111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2371C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2371D3
hypocenterC3#9442
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
1. C3×#2371C2semidirect
3. #71736C2semidirect
5. #67615C2semidirect
13. #5069C4semidirect
15. #5067C4semidirect
23. #2371D3semidirect
230. C3#9442semidirect

Maximal subgroups

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

subgroupindex
C3×#23712
#830122
#717362
#694322
#676152
#94423