Finite Group Databasebeta

← #10695885

Subgroups — #10695885

Order 768.

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

normal subgroups only

This group has 698 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#78646C211yesmaximal, characteristic
2C2×#4739C212yesmaximal
3#69990C211yesmaximal, characteristic
4#69923C212yesmaximal
5C3×#1298C211yesmaximal, characteristic
6#9991—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×#924C2212yes—
9C22×Dic12—22no—
10C3×#407C2211yescharacteristic
11C2×C3×C22⋊C8C2212yes—
12#5207C2211yescharacteristic
13#5137—22no—
14C2×#824—22no—
15C2×#816—22no—
16#4739—24no—
17#4739—22no—
18#4739—24no—
19#4721—22no—
20#4469—24no—
21#3195—31no—
22C2×#451—32no—
23#1784—31no—
24#1472—32no—
25#1298D311yescharacteristic
26C23×C3×C4C2311yescharacteristic
27C23×Dic3—21no—
28C22×Dic6—22no—
29C22×Dic6—21no—
30C22×Dic6—22no—
31C22×Dic6—44no—
32C22×Dic6D412yes—
33C22×Dic6D412yes—
34C2×C3×M16D412yes—
35C22×C3×C8—22no—
36C2×#217—22no—
37C2×C4⋊Dic3D412yes—
38#924—22no—
39#924—22no—
40#924—22no—
41#924—22no—
42C2×Dic3⋊C4—22no—
43C2×#208—21no—
44C2×Dic12—28no—
45C2×Dic12—22no—
46C2×Dic12—48no—
47C3×C22⋊C8—22no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8D414yes—
50#830—22no—
51#824—44no—
52#817—22no—
53#816—24no—
54#816—28no—
55C23×Q8—31no—
56C22×Q16—62no—
57C2×C22⋊Q8—32no—
58#483—62no—
59#470—62no—
60#451—62no—
61#451—64no—
62#451—64no—
63C2×#103—62no—
64C2×#101—62no—
65#407—31no—
66#407C2×D311yescharacteristic
67C2×C22⋊C8C2×D312yes—
68#324—64no—
69C24×C3#4111yescharacteristic
70C22×C3×C4—42no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D412yes—
74C22×C3×C4C2×D411yescharacteristic
75C22×Dic3—22no—
76C22×Dic3—22no—
77C22×Dic3—22no—
78C22×Dic3—44no—
79C22×Dic3—42no—
80C2×Dic6—28no—
81C2×Dic6—24no—
82C2×Dic6—24no—
83C2×Dic6—22no—
84C2×Dic6—22no—
85C2×Dic6—44no—
86C2×Dic6—48no—
87C2×Dic6—44no—
88C2×Dic6—42no—
89C2×Dic6—41no—
90C2×Dic6—41no—
91C2×Dic6—44no—
92C3×M16—42no—
93C2×C3×C8—22no—
94C2×C3×C8—24no—
95C2×C3×C8—44no—
96#217—42no—
97#217—42no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100Dic3⋊C4—42no—
101Dic3⋊C4—42no—
102#208—42no—
103C2×C3⋊C8—41no—
104Dic12—416no—
105Dic12—82no—
106C22×Q8—32no—
107C22×Q8—32no—
108C22×Q8—61no—
109C22×Q8—62no—
110C22×Q8—62no—
111C22×Q8—124no—
112C23×C4—61no—
113C23×C4C22×D311yescharacteristic
114C2×Q16—68no—
115C2×Q16—62no—
116C2×Q16—128no—
117C2×M16—61no—
118C2×M16D1212yes—
119C22×C8—22no—
120C22⋊Q8—62no—
121C22⋊Q8—62no—
122C22⋊Q8—62no—
123C22⋊Q8—62no—
124C2×C4⋊C4—32no—
125C2×C4⋊C4—62no—
126C2×C22⋊C4—62no—
127#106—62no—
128#103—68no—
129#103—64no—
130#101—124no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—62no—
134C22⋊C8D1214yes—
135C23×C3—24no—
136C23×C3—41no—
137C23×C3#12111yescharacteristic
138C23×C3#12112yes—
139C2×C3×C4—22no—
140C2×C3×C4—24no—
141C2×C3×C4—42no—
142C2×C3×C4—41no—
143C2×C3×C4#12012yes—
144C2×C3×C4#12011yescharacteristic
145C2×C3×C4#12711yescharacteristic
146C2×Dic3—24no—
147C2×Dic3—48no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151C2×Dic3—42no—
152C2×Dic3—44no—
153C2×Dic3—82no—
154Dic6—28no—
155Dic6—44no—
156Dic6—44no—
157Dic6—44no—
158Dic6—44no—
159Dic6—44no—
160C3×C8—44no—
161C3×C8—42no—
162C3⋊C8—81no—
163C24#23911yescharacteristic
164C2×Q8—64no—
165C2×Q8—68no—
166C2×Q8—64no—
167C2×Q8—62no—
168C2×Q8—62no—
169C2×Q8—124no—
170C2×Q8—122no—
171C2×Q8—124no—
172C2×Q8—124no—
173C2×Q8—128no—
174C2×Q8—121no—
175C2×Q8—121no—
176C22×C4—42no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—124no—
181C22×C4—122no—
182C22×C4C2×D1211yescharacteristic
183C22×C4C2×D1212yes—
184C22×C4D3×D412yes—
185C22×C4D3×D411yescharacteristic
186Q16—1216no—
187Q16—242no—
188M16—42no—
189M16—122no—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—44no—
193C2×C8—121no—
194C4⋊C4—62no—
195C4⋊C4—62no—
196C4⋊C4—122no—
197C4⋊C4—122no—
198C22⋊C4—122no—
199C22⋊C4—122no—
200C22×C3—22no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—21no—
204C22×C3—44no—
205C22×C3#39211yescharacteristic
206C22×C3#47314yes—
207C3×C4—21no—
208C3×C4—22no—
209C3×C4B(4,𝔽2)12yes—
210Dic3—44no—
211Dic3—82no—
212Dic3—82no—
213C23—24no—
214C23—41no—
215C23#89511yescharacteristic
216C23#89512yes—
217Q8—68no—
218Q8—124no—
219Q8—124no—
220Q8—124no—
221Q8—124no—
222Q8—124no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—64no—
228C2×C4—122no—
229C2×C4—128no—
230C2×C4—122no—
231C2×C4—124no—
232C2×C4—122no—
233C2×C4—122no—
234C2×C4—242no—
235C2×C4#87411yescharacteristic
236C2×C4#88212yes—
237C2×C4#88211yescharacteristic
238C8—44no—
239C8—42no—
240C8—241no—
241C2×C3—24no—
242C2×C3—41no—
243C2×C3#202012yes—
244C2×C3#202111yescharacteristic
245C22—22no—
246C22—22no—
247C22—22no—
248C22—21no—
249C22—44no—
250C22#467111yescharacteristic
251C22#489614yes—
252C4—22no—
253C4—21no—
254C4—124no—
255C4—242no—
256C4—242no—
257C4#474112yes—
258C3#999111yescharacteristic, complemented, Sylow
259C2—24no—
260C2—41no—
261C2#6894412yes—
262C2#6896511yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1298C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenter#1298D3
hypocenterC3#9991
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
258. C3#9991semidirect

Maximal subgroups

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

subgroupindex
#786462
C2×#47392
#699902
#699232
C3×#12982
#99913