Finite Group Databasebeta

← #10698085

Subgroups — #10698085

Order 768.

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

normal subgroups only

This group has 806 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#78641C211yesmaximal, characteristic
2#73491C212yesmaximal, complemented
3#69990C211yesmaximal, characteristic, complemented
4#69444C212yesmaximal, complemented
5C3×#1297C211yesmaximal, characteristic, complemented
6#10327—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×#930C2212yes—
9C2×#875—22no—
10C3×#407C2211yescharacteristic
11C3×C4×C22⋊C4C2212yes—
12#5207C2211yescharacteristic
13C2×#824—22no—
14#5135—22no—
15#4958—24no—
16#4929—22no—
17#4735—22no—
18#4732—22no—
19#4673—24no—
20#4451—24no—
21#3192—31no—
22#2436—32no—
23#1784—31no—
24#1507—32no—
25#1297D311yescharacteristic, complemented
26C23×C3×C4C2311yescharacteristic
27C22×#67—22no—
28C23×Dic3D412yes—
29C22×D12D412yes—
30C22×Dic6—21no—
31C22×Dic6—22no—
32C22×Dic6—22no—
33C22×Dic6—21no—
34C22×Dic6—44no—
35C2×C3×M16D412yes—
36C2×C3×C22⋊C4—22no—
37C2×C3×C42—22no—
38#930—22no—
39#930—22no—
40#930—22no—
41#930—22no—
42C2×(C2×D3)⋊C4—28no—
43C2×(C2×D3)⋊C4—22no—
44C2×C4⋊Dic3D412yes—
45C2×#208—21no—
46C2×D24—22no—
47#875—48no—
48C3×C22⋊C8—22no—
49C3×#95D414yes—
50C3⋊#95—28no—
51#830—22no—
52#824—44no—
53#821—24no—
54#817—22no—
55C23×Q8—31no—
56C2×#124—62no—
57C2×#121—32no—
58#481—62no—
59#469—62no—
60#447—62no—
61C2×#101—62no—
62#407—31no—
63#407C2×D311yescharacteristic
64#393—64no—
65#391—62no—
66#386—64no—
67C4×C22⋊C4C2×D312yes—
68#327—64no—
69C24×C3C2×D411yescharacteristic
70C23×D3—22no—
71C22×C3×C4—24no—
72C22×C3×C4—42no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D412yes—
76C22×C3×C4#4111yescharacteristic
77C2×#67—48no—
78C2×#67—42no—
79C2×#67—42no—
80C22×Dic3—24no—
81C22×Dic3—22no—
82C22×Dic3—22no—
83C22×Dic3—44no—
84C2×D12—22no—
85C2×D12—22no—
86C2×D12—44no—
87C2×Dic6—24no—
88C2×Dic6—28no—
89C2×Dic6—48no—
90C2×Dic6—44no—
91C2×Dic6—42no—
92C2×Dic6—44no—
93C2×Dic6—41no—
94C2×Dic6—42no—
95C2×Dic6—41no—
96C3×M16—42no—
97C2×C3×C8—22no—
98C3×C22⋊C4—44no—
99C3×C42—44no—
100(C2×D3)⋊C4—416no—
101(C2×D3)⋊C4—42no—
102(C2×D3)⋊C4—42no—
103C4⋊Dic3—22no—
104C4⋊Dic3—22no—
105#208—42no—
106C2×C3⋊C8—41no—
107D24—82no—
108C22×Q8—61no—
109C22×Q8—62no—
110C22×Q8—62no—
111C22×Q8—61no—
112C22×Q8—124no—
113C22×D4—32no—
114C22×D4—62no—
115C23×C4—32no—
116C23×C4C22×D311yescharacteristic
117C2×D8—62no—
118C2×M16—61no—
119C2×M16D1212yes—
120#124—128no—
121#121—62no—
122#121—62no—
123#121—62no—
124#121—62no—
125C2×C4⋊C4—32no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—68no—
128C2×C22⋊C4—62no—
129C2×C42—22no—
130#106—62no—
131#102—64no—
132#101—124no—
133C22⋊C8—22no—
134C22⋊C8—62no—
135#95—68no—
136#95D1214yes—
137C23×C3—24no—
138C23×C3—41no—
139C23×C3#12012yes—
140C23×C3#12111yescharacteristic
141C22×D3—44no—
142C22×D3—44no—
143C22×D3—42no—
144C22×D3—44no—
145C2×C3×C4—22no—
146C2×C3×C4—24no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—44no—
150C2×C3×C4—44no—
151C2×C3×C4—44no—
152C2×C3×C4#12011yescharacteristic
153C2×C3×C4#12112yes—
154C2×C3×C4#12711yescharacteristic
155#67—84no—
156#67—84no—
157C2×Dic3—28no—
158C2×Dic3—24no—
159C2×Dic3—44no—
160C2×Dic3—42no—
161C2×Dic3—42no—
162C2×Dic3—44no—
163C2×Dic3—42no—
164D12—44no—
165D12—82no—
166Dic6—44no—
167Dic6—48no—
168Dic6—44no—
169C3×C8—42no—
170C3⋊C8—81no—
171C24—62no—
172C24D3×D411yescharacteristic
173C2×Q8—64no—
174C2×Q8—68no—
175C2×Q8—124no—
176C2×Q8—122no—
177C2×Q8—128no—
178C2×Q8—124no—
179C2×Q8—121no—
180C2×Q8—121no—
181C2×Q8—122no—
182C2×D4—62no—
183C2×D4—62no—
184C2×D4—124no—
185C2×D4—128no—
186C2×D4—122no—
187C2×D4—122no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—64no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—124no—
194C22×C4C2×D1211yescharacteristic
195C22×C4C2×D1212yes—
196C22×C4D3×D412yes—
197C22×C4#23911yescharacteristic
198D8—242no—
199M16—42no—
200M16—122no—
201C2×C8—22no—
202C2×C8—121no—
203C4⋊C4—62no—
204C4⋊C4—62no—
205C22⋊C4—44no—
206C22⋊C4—1216no—
207C22⋊C4—122no—
208C22⋊C4—122no—
209C42—44no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—44no—
215C22×C3#39211yescharacteristic
216C22×C3#45414yes—
217C2×D3—88no—
218C2×D3—82no—
219C2×D3—82no—
220C2×D3—82no—
221C3×C4—21no—
222C3×C4—22no—
223C3×C4—84no—
224C3×C4B(4,𝔽2)12yes—
225Dic3—44no—
226Dic3—82no—
227C23—24no—
228C23—41no—
229C23—124no—
230C23—124no—
231C23—122no—
232C23—124no—
233C23#88212yes—
234C23#89511yescharacteristic
235Q8—124no—
236Q8—128no—
237Q8—124no—
238D4—124no—
239D4—242no—
240D4—244no—
241D4—244no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—68no—
250C2×C4—64no—
251C2×C4—124no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—124no—
255C2×C4—122no—
256C2×C4#87411yescharacteristic
257C2×C4#88211yescharacteristic
258C2×C4#89512yes—
259C8—42no—
260C8—241no—
261C2×C3—24no—
262C2×C3—41no—
263C2×C3#200312yes—
264C2×C3#202111yescharacteristic
265D3—162no—
266C22—22no—
267C22—22no—
268C22—21no—
269C22—22no—
270C22—44no—
271C22—248no—
272C22—242no—
273C22—242no—
274C22—242no—
275C22#467111yescharacteristic
276C22#482714yes—
277C4—21no—
278C4—22no—
279C4—84no—
280C4—124no—
281C4—242no—
282C4#474112yes—
283C3#1032711yescharacteristic, complemented, Sylow
284C2—24no—
285C2—41no—
286C2—482no—
287C2#6876712yes—
288C2#6896511yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#392
prime radical#78641C2
hypercenter#1297D3
hypocenterC3#10327
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
2. #73491C2semidirect
3. #69990C2semidirect
4. #69444C2semidirect
5. C3×#1297C2semidirect
25. #1297D3semidirect
283. C3#10327semidirect

Maximal subgroups

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

subgroupindex
#786412
#734912
#699902
#694442
C3×#12972
#103273