Finite Group Databasebeta

← #10698663

Subgroups — #10698663

Order 768.

518 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#74254C211yesmaximal, characteristic
2#69821C212yesmaximal, complemented
3C2×#4464C212yesmaximal, complemented
4#69548C211yesmaximal, characteristic, complemented
5C3×#1295C211yesmaximal, characteristic, complemented
6#10425—31nomaximal, Sylow
7C2×#875C2212yes—
8C2×C3×#97C2212yes—
9C3×#382C2211yescharacteristic
10C2×#821—22no—
11C2×#816—22no—
12#4991—22no—
13#4935C2211yescharacteristic
14C2×#803C2211yescharacteristic
15#4673—22no—
16#4671—22no—
17#4529—22no—
18#4464—24no—
19#2819—31no—
20#1959—32no—
21#1836—31no—
22C2×#329—32no—
23#1295D311yescharacteristic, complemented
24C22×D12—22no—
25C22×D12D412yes—
26C22×Dic6D412yes—
27C22×C3×C8—22no—
28C2×C3×C4⋊C4D412yes—
29C2×C3×C42C2311yescharacteristic
30C2×(C2×D3)⋊C4—22no—
31C2×(C2×D3)⋊C4—22no—
32C2×(C2×D3)⋊C4—22no—
33C2×(C2×D3)⋊C4—22no—
34C2×C4⋊Dic3—21no—
35C2×C4⋊Dic3—22no—
36C22×C3⋊C8—21no—
37#875—22no—
38#875—22no—
39#875—22no—
40#875—22no—
41C3×#97—22no—
42C3×#97—22no—
43C3×#95—22no—
44C3⋊#95—22no—
45#821—44no—
46#816—44no—
47#803—21no—
48#803D412yescomplemented
49C2×#124—32no—
50C2×#103—62no—
51C2×#102—62no—
52C2×#97—31no—
53C2×#97C2×D312yes—
54#394—62no—
55#393—62no—
56#392—62no—
57#384—31no—
58#382C2×D311yescharacteristic
59#340—62no—
60#329—64no—
61C23×D3—22no—
62C22×C3×C4—22no—
63C22×C3×C4C2×D411yescharacteristic
64C22×C3×C4C2×D412yes—
65C22×Dic3—22no—
66C22×Dic3—41no—
67C2×D12—22no—
68C2×D12—22no—
69C2×D12—42no—
70C2×D12—42no—
71C2×D12—82no—
72C2×D12—82no—
73C2×D12—82no—
74C2×Dic6—22no—
75C2×Dic6—22no—
76C2×Dic6—42no—
77C2×Dic6—42no—
78C2×C3×C8—24no—
79C2×C3×C8—42no—
80C2×C3×C8—42no—
81C3×C4⋊C4—42no—
82C3×C42C2×D411yescharacteristic
83C3×C42#4112yes—
84C3×C42#4111yescharacteristic
85(C2×D3)⋊C4—42no—
86(C2×D3)⋊C4—42no—
87(C2×D3)⋊C4—42no—
88(C2×D3)⋊C4—42no—
89(C2×D3)⋊C4—82no—
90(C2×D3)⋊C4—82no—
91C4⋊Dic3—42no—
92C4⋊Dic3—42no—
93C4⋊Dic3—42no—
94C2×C3⋊C8—22no—
95C2×C3⋊C8—41no—
96C2×C3⋊C8—41no—
97C22×Q8—32no—
98C22×D4—32no—
99C22×D4—62no—
100C22×C8—22no—
101C22×C8—61no—
102#124—62no—
103#124—62no—
104#124—62no—
105#124—62no—
106C2×C4⋊C4—61no—
107C2×C4⋊C4—62no—
108C2×C4⋊C4D1212yes—
109C2×C22⋊C4—62no—
110C2×C22⋊C4—62no—
111C2×C22⋊C4—62no—
112C2×C22⋊C4—62no—
113C2×C42C22×D311yescharacteristic
114#103—124no—
115#102—124no—
116#97—22no—
117#97—22no—
118#97—32no—
119#97—61no—
120#95—22no—
121#95—62no—
122C23×C3#12011yescharacteristic
123C22×D3—42no—
124C22×D3—42no—
125C22×D3—42no—
126C22×D3—42no—
127C22×D3—42no—
128C22×D3—42no—
129C22×D3—42no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4#12112yes—
138C2×C3×C4#12111yescharacteristic
139C2×C3×C4#12312yes—
140C2×C3×C4#12411yescharacteristic
141C2×Dic3—42no—
142C2×Dic3—42no—
143C2×Dic3—42no—
144C2×Dic3—82no—
145C2×Dic3—81no—
146D12—44no—
147D12—82no—
148D12—88no—
149Dic6—44no—
150Dic6—82no—
151C3×C8—44no—
152C3⋊C8—42no—
153C24—62no—
154C2×Q8—62no—
155C2×Q8—62no—
156C2×Q8—122no—
157C2×Q8—122no—
158C2×D4—62no—
159C2×D4—62no—
160C2×D4—122no—
161C2×D4—122no—
162C2×D4—242no—
163C2×D4—242no—
164C2×D4—242no—
165C22×C4—22no—
166C22×C4—62no—
167C22×C4—121no—
168C22×C4C2×D1211yescharacteristic
169C22×C4D3×D412yes—
170C2×C8—24no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—62no—
174C2×C8—121no—
175C2×C8—121no—
176C4⋊C4—42no—
177C4⋊C4—122no—
178C4⋊C4—122no—
179C4⋊C4—122no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C22⋊C4—122no—
183C22⋊C4—122no—
184C22⋊C4—242no—
185C22⋊C4—242no—
186C42#23512yes—
187C42D3×D411yescharacteristic
188C42#23911yescharacteristic
189C22×C3#39411yescharacteristic
190C22×C3#45312yes—
191C22×C3#45411yescharacteristic
192C22×C3#45511yescharacteristic
193C22×C3#45512yes—
194C2×D3—82no—
195C2×D3—82no—
196C2×D3—82no—
197C2×D3—82no—
198C2×D3—82no—
199C2×D3—82no—
200C2×D3—82no—
201C3×C4—22no—
202C3×C4—22no—
203C3×C4—21no—
204C3×C4—21no—
205C3×C4—82no—
206Dic3—82no—
207Dic3—161no—
208C23—122no—
209C23—122no—
210C23—122no—
211C23—122no—
212C23—122no—
213C23—122no—
214C23—122no—
215C23#88211yescharacteristic
216Q8—124no—
217Q8—242no—
218D4—124no—
219D4—242no—
220D4—248no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—122no—
229C2×C4—122no—
230C2×C4—122no—
231C2×C4—241no—
232C2×C4—242no—
233C2×C4#87511yescharacteristic
234C2×C4#88412yes—
235C2×C4#89412yes—
236C2×C4#89511yescharacteristic
237C8—44no—
238C8—122no—
239C2×C3#165611yescharacteristic
240C2×C3#165912yes—
241C2×C3#166211yescharacteristic
242C2×C3#203712yes—
243C2×C3#204011yescharacteristic
244D3—162no—
245C22—242no—
246C22—242no—
247C22—242no—
248C22—242no—
249C22—242no—
250C22—242no—
251C22—242no—
252C22#467211yescharacteristic
253C22#474611yescharacteristic
254C22#475311yescharacteristic
255C22#482212yes—
256C22#482512yes—
257C4—21no—
258C4—21no—
259C4—22no—
260C4—22no—
261C4—82no—
262C4—242no—
263C4—481no—
264C3#1042511yescharacteristic, complemented, Sylow
265C2—482no—
266C2#6835212yes—
267C2#6837011yescharacteristic
268C2#6837411yescharacteristic
269C2#6873312yes—
270C2#6897711yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socleC23×C3#120
prime radical#74254C2
hypercenter#1295D3
hypocenterC3#10425
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. #69821C2semidirect
3. C2×#4464C2semidirect
4. #69548C2semidirect
5. C3×#1295C2semidirect
23. #1295D3semidirect
48. #803D4semidirect
264. C3#10425semidirect

Maximal subgroups

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

subgroupindex
#742542
#698212
C2×#44642
#695482
C3×#12952
#104253