Finite Group Databasebeta

← #10698130

Subgroups — #10698130

Order 768.

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

normal subgroups only

This group has 754 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
2#73490C212yesmaximal
3#69990C211yesmaximal, characteristic
4#69445C212yesmaximal
5C3×#1297C211yesmaximal, characteristic
6#10334—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×#924C2212yes—
9C2×#869—22no—
10C3×#407C2211yescharacteristic
11C3×C4×C22⋊C4C2212yes—
12#5207C2211yescharacteristic
13#5137—22no—
14C2×#824—22no—
15#4950—24no—
16#4929—22no—
17#4739—22no—
18#4721—22no—
19#4656—24no—
20#4452—24no—
21#3195—31no—
22#2434—32no—
23#1784—31no—
24#1508—32no—
25#1297D311yescharacteristic
26C23×C3×C4C2311yescharacteristic
27C23×Dic3D412yes—
28C22×Dic6—21no—
29C22×Dic6—22no—
30C22×Dic6—22no—
31C22×Dic6—21no—
32C22×Dic6—44no—
33C22×Dic6D412yes—
34C2×C3×M16D412yes—
35C2×C3×C22⋊C4—22no—
36C2×C3×C42—22no—
37C2×#217—22no—
38C2×C4⋊Dic3—28no—
39C2×C4⋊Dic3D412yes—
40#924—22no—
41#924—22no—
42#924—22no—
43#924—22no—
44C2×Dic3⋊C4—22no—
45C2×#208—21no—
46C2×Dic12—22no—
47#869—48no—
48C3×C22⋊C8—22no—
49C3×#95D414yes—
50C3⋊#95—28no—
51#830—22no—
52#824—44no—
53#817—22no—
54#816—24no—
55C23×Q8—31no—
56C2×C4⋊Q8—62no—
57C2×C22⋊Q8—32no—
58#483—62no—
59#470—62no—
60#451—62no—
61C2×#101—62no—
62#407—31no—
63#407C2×D311yescharacteristic
64#395—64no—
65#391—62no—
66#385—64no—
67C4×C22⋊C4C2×D312yes—
68#328—64no—
69C24×C3C2×D411yescharacteristic
70C22×C3×C4—24no—
71C22×C3×C4—42no—
72C22×C3×C4C2×D411yescharacteristic
73C22×C3×C4C2×D412yes—
74C22×C3×C4C2×D412yes—
75C22×C3×C4#4111yescharacteristic
76C22×Dic3—24no—
77C22×Dic3—22no—
78C22×Dic3—22no—
79C22×Dic3—22no—
80C22×Dic3—44no—
81C2×Dic6—24no—
82C2×Dic6—28no—
83C2×Dic6—22no—
84C2×Dic6—22no—
85C2×Dic6—48no—
86C2×Dic6—44no—
87C2×Dic6—42no—
88C2×Dic6—44no—
89C2×Dic6—41no—
90C2×Dic6—42no—
91C2×Dic6—41no—
92C2×Dic6—44no—
93C3×M16—42no—
94C2×C3×C8—22no—
95C3×C22⋊C4—28no—
96C3×C42—44no—
97#217—42no—
98#217—42no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101C4⋊Dic3—416no—
102Dic3⋊C4—42no—
103Dic3⋊C4—42no—
104#208—42no—
105C2×C3⋊C8—41no—
106Dic12—82no—
107C22×Q8—32no—
108C22×Q8—61no—
109C22×Q8—62no—
110C22×Q8—62no—
111C22×Q8—61no—
112C22×Q8—124no—
113C23×C4—32no—
114C23×C4C22×D311yescharacteristic
115C2×Q16—62no—
116C2×M16—61no—
117C2×M16D1212yes—
118C4⋊Q8—128no—
119C22⋊Q8—62no—
120C22⋊Q8—62no—
121C22⋊Q8—62no—
122C22⋊Q8—62no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—68no—
125C2×C4⋊C4—62no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—62no—
128C2×C42—22no—
129#106—62no—
130#103—64no—
131#101—124no—
132C22⋊C8—22no—
133C22⋊C8—62no—
134#95—68no—
135#95D1214yes—
136C23×C3—24no—
137C23×C3—41no—
138C23×C3#12012yes—
139C23×C3#12111yescharacteristic
140C2×C3×C4—22no—
141C2×C3×C4—24no—
142C2×C3×C4—28no—
143C2×C3×C4—42no—
144C2×C3×C4—41no—
145C2×C3×C4—44no—
146C2×C3×C4—44no—
147C2×C3×C4#12011yescharacteristic
148C2×C3×C4#12112yes—
149C2×C3×C4#12711yescharacteristic
150C2×Dic3—28no—
151C2×Dic3—24no—
152C2×Dic3—44no—
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—44no—
156C2×Dic3—42no—
157C2×Dic3—42no—
158C2×Dic3—44no—
159Dic6—44no—
160Dic6—48no—
161Dic6—44no—
162Dic6—44no—
163Dic6—82no—
164C3×C8—42no—
165C3⋊C8—81no—
166C24D3×D411yescharacteristic
167C2×Q8—64no—
168C2×Q8—68no—
169C2×Q8—62no—
170C2×Q8—62no—
171C2×Q8—124no—
172C2×Q8—122no—
173C2×Q8—124no—
174C2×Q8—128no—
175C2×Q8—124no—
176C2×Q8—121no—
177C2×Q8—121no—
178C2×Q8—122no—
179C22×C4—24no—
180C22×C4—42no—
181C22×C4—64no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—124no—
186C22×C4C2×D1211yescharacteristic
187C22×C4C2×D1212yes—
188C22×C4D3×D412yes—
189C22×C4#23911yescharacteristic
190Q16—242no—
191M16—42no—
192M16—122no—
193C2×C8—22no—
194C2×C8—121no—
195C4⋊C4—62no—
196C4⋊C4—62no—
197C4⋊C4—1216no—
198C4⋊C4—122no—
199C4⋊C4—122no—
200C22⋊C4—28no—
201C22⋊C4—122no—
202C22⋊C4—122no—
203C42—44no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—21no—
208C22×C3—44no—
209C22×C3#39211yescharacteristic
210C22×C3#45614yes—
211C3×C4—21no—
212C3×C4—22no—
213C3×C4—48no—
214C3×C4B(4,𝔽2)12yes—
215Dic3—44no—
216Dic3—82no—
217Dic3—82no—
218C23—24no—
219C23—41no—
220C23#88212yes—
221C23#89511yescharacteristic
222Q8—124no—
223Q8—128no—
224Q8—124no—
225Q8—124no—
226Q8—242no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—28no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—68no—
235C2×C4—64no—
236C2×C4—124no—
237C2×C4—122no—
238C2×C4—122no—
239C2×C4—124no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4—124no—
243C2×C4#87411yescharacteristic
244C2×C4#88211yescharacteristic
245C2×C4#89512yes—
246C8—42no—
247C8—241no—
248C2×C3—24no—
249C2×C3—41no—
250C2×C3#200512yes—
251C2×C3#202111yescharacteristic
252C22—22no—
253C22—22no—
254C22—21no—
255C22—22no—
256C22—44no—
257C22#467111yescharacteristic
258C22#482614yes—
259C4—21no—
260C4—22no—
261C4—48no—
262C4—124no—
263C4—242no—
264C4—242no—
265C4#474112yes—
266C3#1033411yescharacteristic, complemented, Sylow
267C2—24no—
268C2—41no—
269C2#6876812yes—
270C2#6896511yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1297D3
hypocenterC3#10334
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
266. C3#10334semidirect

Maximal subgroups

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

subgroupindex
#786462
#734902
#699902
#694452
C3×#12972
#103343