Finite Group Databasebeta

← #10696047

Subgroups — #10696047

Order 768.

696 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 105 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#79298C212yesmaximal, complemented
2#73418C211yesmaximal, characteristic
3#70762C212yesmaximal, complemented
4#69441C211yesmaximal, characteristic, complemented
5C3×#1274C211yesmaximal, characteristic, complemented
6#10015—31nomaximal, Sylow
7C2×C4×D12—22no—
8C2×C4×D12C2212yes—
9#5470C2211yescharacteristic
10C3×#431C2212yes—
11C3×#376C2211yescharacteristic
12#4924C2211yescharacteristic
13#4842—22no—
14#4841—22no—
15#4731—22no—
16#4721—22no—
17C4×#204—24no—
18#4677—22no—
19#4671—22no—
20#4523—22no—
21#4459—24no—
22#3101—32no—
23#2397—31no—
24#1995—31no—
25#1486—32no—
26#1274D311yescharacteristic, complemented
27C22×D12—22no—
28C22×D12D412yes—
29C22×C4×D3—22no—
30C2×C3×C42D412yes—
31C2×C3×C42C2311yescharacteristic
32C2×(C2×D3)⋊C4—24no—
33C2×(C2×D3)⋊C4—22no—
34#926—21no—
35C2×C4⋊Dic3—22no—
36C2×C4⋊Dic3D412yes—
37#924—22no—
38C2×#208—21no—
39C2×#204—24no—
40C4×D12—22no—
41C4×D12—22no—
42C4×D12—22no—
43C4×D12—22no—
44C4×D12—44no—
45C4×D12—44no—
46#870—21no—
47#869D412yescomplemented
48C4×Dic6—22no—
49C3×C4⋊C8—22no—
50C3×C22⋊C8—22no—
51C3×C4×C8D414yescomplemented
52C3×#95—22no—
53#821—24no—
54#817—24no—
55#816—24no—
56C4⋊(C3⋊C8)—22no—
57#533—31no—
58C2×C4×D4—32no—
59C2×C4×D4—62no—
60#483—62no—
61#476—62no—
62#461—62no—
63#450—62no—
64C4×SD16—64no—
65#431C2×D312yes—
66#423—31no—
67#399—62no—
68#392—62no—
69#376C2×D311yescharacteristic
70#339—62no—
71D4⋊C8—64no—
72C23×D3—22no—
73C22×C3×C4—22no—
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4C2×D412yes—
76C22×Dic3—22no—
77C2×D12—22no—
78C2×D12—22no—
79C2×D12—44no—
80C2×D12—44no—
81C2×D12—48no—
82C2×C4×D3—42no—
83C2×C4×D3—42no—
84C2×C4×D3—44no—
85C2×C4×D3—44no—
86C2×Dic6—22no—
87C2×C3×C8—24no—
88C3×C42—22no—
89C3×C42SD1614yescomplemented
90C3×C42C2×D412yes—
91C3×C42C2×D411yescharacteristic
92C3×C42#4111yescharacteristic
93#217—41no—
94(C2×D3)⋊C4—48no—
95(C2×D3)⋊C4—42no—
96(C2×D3)⋊C4—42no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101C4⋊Dic3—44no—
102Dic3⋊C4—42no—
103C4×Dic3—41no—
104#208—42no—
105C2×C3⋊C8—41no—
106#204—48no—
107C22×D4—32no—
108C22×D4—62no—
109C23×C4—62no—
110C2×SD16—64no—
111C2×M16—61no—
112C4⋊Q8—32no—
113#125—61no—
114C22⋊Q8—62no—
115C4×Q8—62no—
116C4×D4—62no—
117C4×D4—62no—
118C4×D4—62no—
119C4×D4—62no—
120C4×D4—124no—
121C4×D4—124no—
122#117—61no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—62no—
125C2×C22⋊C4—64no—
126C2×C22⋊C4—62no—
127C2×C42D1212yes—
128C2×C42C22×D311yescharacteristic
129#106—64no—
130C4⋊C8—22no—
131C4⋊C8—62no—
132#103—64no—
133#102—64no—
134C22⋊C8—22no—
135C4×C8D1214yescomplemented
136#95—22no—
137C23×C3#12011yescharacteristic
138C22×D3—24no—
139C22×D3—44no—
140C22×D3—42no—
141C22×D3—42no—
142C22×D3—44no—
143C2×C3×C4—24no—
144C2×C3×C4—42no—
145C2×C3×C4—44no—
146C2×C3×C4#12012yes—
147C2×C3×C4#12112yes—
148C2×C3×C4#12111yescharacteristic
149C2×C3×C4#12711yescharacteristic
150C2×C3×C4C2×SD1612yes—
151C2×C3×C4AGL(1,ℤ/8)12yes—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—44no—
155D12—24no—
156D12—24no—
157D12—42no—
158D12—42no—
159D12—416no—
160C4×D3—84no—
161C4×D3—82no—
162C4×D3—82no—
163Dic6—44no—
164C3×C8—44no—
165C3⋊C8—81no—
166C24—62no—
167C2×Q8—62no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—124no—
171C2×D4—124no—
172C2×D4—128no—
173C22×C4—22no—
174C22×C4—62no—
175C22×C4—122no—
176C22×C4—122no—
177C22×C4—124no—
178C22×C4—124no—
179C22×C4C2×D1211yescharacteristic
180C22×C4D3×D412yes—
181SD16—128no—
182M16—122no—
183C2×C8—24no—
184C2×C8—121no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4—62no—
188C4⋊C4—62no—
189C4⋊C4—124no—
190C4⋊C4—122no—
191C22⋊C4—128no—
192C22⋊C4—121no—
193C22⋊C4—122no—
194C22⋊C4—122no—
195C42—22no—
196C42—121no—
197C42#20414yescomplemented
198C42C2×D1212yes—
199C42D3×D411yescharacteristic
200C42#23911yescharacteristic
201C22×C3—22no—
202C22×C3#39211yescharacteristic
203C22×C3#45012yes—
204C2×D3—48no—
205C2×D3—44no—
206C2×D3—84no—
207C2×D3—82no—
208C2×D3—82no—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—44no—
212C3×C4#46012yes—
213C3×C4#49211yescharacteristic
214C3×C4#49611yescharacteristic
215Dic3—82no—
216Dic3—82no—
217C23—64no—
218C23—124no—
219C23—122no—
220C23—122no—
221C23—124no—
222C23#88211yescharacteristic
223Q8—124no—
224D4—64no—
225D4—64no—
226D4—122no—
227D4—122no—
228D4—1216no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—122no—
233C2×C4—122no—
234C2×C4—124no—
235C2×C4—242no—
236C2×C4—242no—
237C2×C4—244no—
238C2×C4#87411yescharacteristic
239C2×C4#88212yes—
240C2×C4#89512yes—
241C2×C4#89511yescharacteristic
242C2×C4C2×#20412yes—
243C2×C4#90812yes—
244C8—44no—
245C8—241no—
246C2×C3—22no—
247C2×C3#201012yes—
248C2×C3#201411yescharacteristic
249D3—84no—
250C22—22no—
251C22—128no—
252C22—124no—
253C22—244no—
254C22—242no—
255C22—242no—
256C22#467111yescharacteristic
257C22#473112yes—
258C4—22no—
259C4—22no—
260C4—44no—
261C4—242no—
262C4—242no—
263C4#469311yescharacteristic
264C4#471211yescharacteristic
265C4#484812yes—
266C3#1001511yescharacteristic, complemented, Sylow
267C2—22no—
268C2—244no—
269C2#6867611yescharacteristic
270C2#6870312yes—
subgroupG/H
centerC2×C4#874
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#392
prime radical#73418C2
hypercenter#1274D3
hypocenterC3#10015
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
1. #79298C2semidirect
3. #70762C2semidirect
4. #69441C2semidirect
5. C3×#1274C2semidirect
26. #1274D3semidirect
47. #869D4semidirect
51. C3×C4×C8D4semidirect
89. C3×C42SD16semidirect
135. C4×C8D12semidirect
197. C42#204semidirect
266. C3#10015semidirect

Maximal subgroups

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

subgroupindex
#792982
#734182
#707622
#694412
C3×#12742
#100153