Finite Group Databasebeta

← #10696042

Subgroups — #10696042

Order 768.

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

normal subgroups only

This group has 566 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#79309C212yesmaximal
2#73397C211yesmaximal, characteristic, complemented
3#70770C212yesmaximal, complemented
4#69440C211yesmaximal, characteristic
5C3×#1274C211yesmaximal, characteristic, complemented
6#10014—31nomaximal, Sylow
7#5477C2211yescharacteristic
8C2×C4×Dic6C2212yes—
9C3×#431C2212yes—
10C3×#376C2211yescharacteristic
11C4×C4⋊Dic3—22no—
12C2×#805—22no—
13#4924C2211yescharacteristic
14#4851—22no—
15#4848—22no—
16#4738—22no—
17#4735—22no—
18C4×#204—24no—
19#4661—22no—
20#4656—22no—
21#4456—24no—
22#3103—32no—
23#2399—31no—
24#1963—31no—
25#1487—32no—
26#1274D311yescharacteristic, complemented
27C2×#235—21no—
28C22×Dic6D412yes—
29C2×C3×C42D412yes—
30C2×C3×C42C2311yescharacteristic
31#930—22no—
32C2×C4⋊Dic3—24no—
33C2×C4⋊Dic3D412yes—
34C2×Dic3⋊C4—22no—
35C2×C4×Dic3—22no—
36C2×#208—21no—
37C2×#204—24no—
38#875—21no—
39#874D411yescharacteristic
40C4×D12—22no—
41#869D411yescharacteristic
42C4×Dic6—22no—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C4×Dic6—22no—
46C3×C4⋊C8—22no—
47C3×C22⋊C8—22no—
48C3×C4×C8D414yes—
49C3×#95—22no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52#821—24no—
53#817—24no—
54#816—24no—
55#805—44no—
56C4⋊(C3⋊C8)—22no—
57#532—31no—
58C2×C4×Q8—32no—
59#481—62no—
60#475—62no—
61#460—62no—
62#448—62no—
63C4×SD16—64no—
64#431C2×D312yes—
65#423—31no—
66C2×#104—62no—
67#398—62no—
68#395—62no—
69C4×C4⋊C4—62no—
70#376C2×D311yescharacteristic
71Q8⋊C8—64no—
72C22×C3×C4—22no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D412yes—
75C2×#67—41no—
76C22×Dic3—22no—
77C22×Dic3—22no—
78#235—42no—
79#235—42no—
80C2×D12—22no—
81C2×D12—21no—
82C2×C4×D3—41no—
83C2×Dic6—22no—
84C2×Dic6—22no—
85C2×Dic6—21no—
86C2×Dic6—44no—
87C2×C3×C8—24no—
88C3×C42—22no—
89C3×C42SD1614yescomplemented
90C3×C42C2×D412yes—
91C3×C42C2×D411yescharacteristic
92C3×C42#4111yescharacteristic
93(C2×D3)⋊C4—42no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97C4⋊Dic3—48no—
98Dic3⋊C4—42no—
99Dic3⋊C4—42no—
100C4×Dic3—42no—
101C4×Dic3—42no—
102#208—42no—
103C2×C3⋊C8—41no—
104#204—48no—
105C2×#41—61no—
106C22×Q8—32no—
107C2×SD16—64no—
108C2×M16—61no—
109C4⋊Q8—31no—
110#127—31no—
111#124—61no—
112#121—62no—
113C4×Q8—62no—
114C4×Q8—62no—
115C4×Q8—62no—
116C4×Q8—62no—
117C4×D4—62no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—64no—
120C2×C4⋊C4—62no—
121C2×C42—62no—
122C2×C42D1212yes—
123C2×C42C22×D311yescharacteristic
124#106—64no—
125C4⋊C8—22no—
126C4⋊C8—62no—
127#104—124no—
128#103—64no—
129#102—64no—
130C22⋊C8—22no—
131C4×C8D1214yes—
132#95—22no—
133#95—62no—
134#95—62no—
135C23×C3#12011yescharacteristic
136C22×D3—41no—
137C2×C3×C4—24no—
138C2×C3×C4—42no—
139C2×C3×C4—44no—
140C2×C3×C4#12012yes—
141C2×C3×C4#12112yes—
142C2×C3×C4#12111yescharacteristic
143C2×C3×C4#12711yescharacteristic
144C2×C3×C4C2×SD1612yes—
145C2×C3×C4#13712yes—
146#67—82no—
147C2×Dic3—24no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—44no—
151C2×Dic3—42no—
152C2×Dic3—41no—
153D12—44no—
154D12—42no—
155C4×D3—81no—
156C4×D3—81no—
157Dic6—24no—
158Dic6—24no—
159Dic6—42no—
160Dic6—42no—
161Dic6—42no—
162C3×C8—44no—
163C3⋊C8—81no—
164#41—122no—
165#41—122no—
166C2×Q8—62no—
167C2×Q8—62no—
168C2×Q8—61no—
169C2×Q8—124no—
170C2×D4—62no—
171C2×D4—61no—
172C2×D4—121no—
173C22×C4—22no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4—121no—
177C22×C4C2×D1211yescharacteristic
178C22×C4D3×D412yes—
179SD16—128no—
180M16—122no—
181C2×C8—24no—
182C2×C8—121no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—128no—
187C4⋊C4—122no—
188C4⋊C4—122no—
189C22⋊C4—122no—
190C42—22no—
191C42—122no—
192C42—122no—
193C42#20414yescomplemented
194C42C2×D1212yes—
195C42D3×D411yescharacteristic
196C42#23911yescharacteristic
197C22×C3—22no—
198C22×C3#39211yescharacteristic
199C22×C3#44812yes—
200C2×D3—82no—
201C2×D3—81no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—44no—
205C3×C4#46112yes—
206C3×C4#49211yescharacteristic
207C3×C4#49711yescharacteristic
208Dic3—44no—
209Dic3—82no—
210Dic3—81no—
211C23—121no—
212C23#88211yescharacteristic
213Q8—64no—
214Q8—64no—
215Q8—122no—
216Q8—122no—
217Q8—122no—
218D4—124no—
219D4—122no—
220D4—242no—
221C2×C4—24no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—64no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—124no—
228C2×C4—122no—
229C2×C4—121no—
230C2×C4—241no—
231C2×C4—241no—
232C2×C4#87411yescharacteristic
233C2×C4#88212yes—
234C2×C4#89512yes—
235C2×C4#89511yescharacteristic
236C2×C4C2×#20412yes—
237C2×C4#90912yes—
238C8—44no—
239C8—241no—
240C2×C3—22no—
241C2×C3#201012yes—
242C2×C3#201311yescharacteristic
243D3—161no—
244C22—22no—
245C22—241no—
246C22—242no—
247C22#467111yescharacteristic
248C22#473812yes—
249C4—22no—
250C4—22no—
251C4—44no—
252C4—124no—
253C4—241no—
254C4—242no—
255C4#469311yescharacteristic
256C4#471511yescharacteristic
257C4#484112yes—
258C3#1001411yescharacteristic, complemented, Sylow
259C2—22no—
260C2—481no—
261C2#6868711yescharacteristic
262C2#6870312yes—
subgroupG/H
centerC2×C4#874
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1274D3
hypocenterC3#10014
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
2. #73397C2semidirect
3. #70770C2semidirect
5. C3×#1274C2semidirect
26. #1274D3semidirect
89. C3×C42SD16semidirect
193. C42#204semidirect
258. C3#10014semidirect

Maximal subgroups

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

subgroupindex
#793092
#733972
#707702
#694402
C3×#12742
#100143