Finite Group Databasebeta

← #10695856

Subgroups — #10695856

Order 768.

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

normal subgroups only

This group has 878 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, complemented
2C2×#4732C212yesmaximal, complemented
3#69988C211yesmaximal, characteristic, complemented
4#69921C212yesmaximal, complemented
5C3×#1298C211yesmaximal, characteristic, complemented
6#9986—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×#930C2212yes—
9C22×D24—22no—
10C3×#407C2211yescharacteristic
11C2×C3×C22⋊C8C2212yes—
12#5207C2211yescharacteristic
13#5135—22no—
14C2×#823—22no—
15C2×#821—22no—
16#4735—22no—
17#4732—24no—
18#4732—22no—
19#4732—24no—
20#4468—24no—
21#3192—31no—
22C2×#447—32no—
23#1783—31no—
24#1471—32no—
25#1298D311yescharacteristic, complemented
26C24×D3—21no—
27C23×C3×C4C2311yescharacteristic
28C22×#67—22no—
29C22×D12—22no—
30C22×D12—21no—
31C22×D12—22no—
32C22×D12—44no—
33C22×D12D412yes—
34C22×D12D412yes—
35C2×C3×M16D412yes—
36C22×C3×C8—22no—
37#930—22no—
38#930—22no—
39#930—22no—
40#930—22no—
41C2×(C2×D3)⋊C4—22no—
42C2×C4⋊Dic3D412yes—
43C2×#208—21no—
44C2×D24—28no—
45C2×D24—22no—
46C2×D24—48no—
47C3×C22⋊C8—22no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8D414yes—
50#830—22no—
51#823—44no—
52#821—24no—
53#821—28no—
54#817—22no—
55C23×D4—31no—
56C22×D8—62no—
57C2×#121—32no—
58#481—62no—
59#469—62no—
60#447—62no—
61#447—64no—
62#447—64no—
63C2×#102—62no—
64C2×#100—62no—
65#407—31no—
66#407C2×D311yescharacteristic
67C2×C22⋊C8C2×D312yes—
68#323—64no—
69C24×C3#4111yescharacteristic
70C23×D3—22no—
71C23×D3—22no—
72C23×D3—44no—
73C23×D3—42no—
74C23×D3—84no—
75C22×C3×C4—42no—
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D412yes—
78C22×C3×C4C2×D412yes—
79C22×C3×C4C2×D411yescharacteristic
80C2×#67—42no—
81C2×#67—42no—
82C2×#67—84no—
83C22×Dic3—22no—
84C2×D12—28no—
85C2×D12—24no—
86C2×D12—24no—
87C2×D12—22no—
88C2×D12—22no—
89C2×D12—44no—
90C2×D12—48no—
91C2×D12—42no—
92C2×D12—44no—
93C2×D12—41no—
94C2×D12—41no—
95C2×D12—44no—
96C3×M16—42no—
97C2×C3×C8—22no—
98C2×C3×C8—24no—
99C2×C3×C8—44no—
100(C2×D3)⋊C4—42no—
101(C2×D3)⋊C4—42no—
102C4⋊Dic3—22no—
103C4⋊Dic3—22no—
104#208—42no—
105C2×C3⋊C8—41no—
106D24—416no—
107D24—82no—
108C25—61no—
109C22×D4—32no—
110C22×D4—32no—
111C22×D4—62no—
112C22×D4—61no—
113C22×D4—62no—
114C22×D4—62no—
115C22×D4—124no—
116C23×C4C22×D311yescharacteristic
117C2×D8—68no—
118C2×D8—62no—
119C2×D8—128no—
120C2×M16—61no—
121C2×M16D1212yes—
122C22×C8—22no—
123#121—62no—
124#121—62no—
125#121—62no—
126#121—62no—
127C2×C4⋊C4—32no—
128C2×C22⋊C4—62no—
129#106—62no—
130#102—68no—
131#102—64no—
132#100—124no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—62no—
136C22⋊C8D1214yes—
137C23×C3—24no—
138C23×C3—41no—
139C23×C3#12111yescharacteristic
140C23×C3#12112yes—
141C22×D3—24no—
142C22×D3—48no—
143C22×D3—42no—
144C22×D3—42no—
145C22×D3—42no—
146C22×D3—44no—
147C22×D3—84no—
148C22×D3—816no—
149C22×D3—84no—
150C22×D3—84no—
151C22×D3—82no—
152C22×D3—82no—
153C2×C3×C4—22no—
154C2×C3×C4—24no—
155C2×C3×C4—42no—
156C2×C3×C4—41no—
157C2×C3×C4#12012yes—
158C2×C3×C4#12011yescharacteristic
159C2×C3×C4#12711yescharacteristic
160#67—84no—
161#67—84no—
162C2×Dic3—42no—
163C2×Dic3—82no—
164D12—28no—
165D12—44no—
166D12—44no—
167D12—44no—
168D12—44no—
169D12—44no—
170C3×C8—44no—
171C3×C8—42no—
172C3⋊C8—81no—
173C24—62no—
174C24—62no—
175C24—124no—
176C24—122no—
177C24—244no—
178C24#23911yescharacteristic
179C2×D4—68no—
180C2×D4—64no—
181C2×D4—64no—
182C2×D4—62no—
183C2×D4—62no—
184C2×D4—124no—
185C2×D4—122no—
186C2×D4—124no—
187C2×D4—124no—
188C2×D4—128no—
189C2×D4—122no—
190C2×D4—122no—
191C2×D4—121no—
192C2×D4—121no—
193C2×D4—244no—
194C22×C4—42no—
195C22×C4—62no—
196C22×C4C2×D1211yescharacteristic
197C22×C4C2×D1212yes—
198C22×C4D3×D412yes—
199C22×C4D3×D411yescharacteristic
200D8—1216no—
201D8—242no—
202M16—42no—
203M16—122no—
204C2×C8—22no—
205C2×C8—24no—
206C2×C8—44no—
207C2×C8—121no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C22⋊C4—122no—
211C22⋊C4—122no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—44no—
217C22×C3#39211yescharacteristic
218C22×C3#47214yes—
219C2×D3—44no—
220C2×D3—816no—
221C2×D3—84no—
222C2×D3—84no—
223C2×D3—84no—
224C2×D3—82no—
225C2×D3—82no—
226C2×D3—82no—
227C2×D3—164no—
228C3×C4—21no—
229C3×C4—22no—
230C3×C4B(4,𝔽2)12yes—
231Dic3—82no—
232C23—24no—
233C23—41no—
234C23—64no—
235C23—128no—
236C23—122no—
237C23—122no—
238C23—122no—
239C23—124no—
240C23—244no—
241C23—244no—
242C23—242no—
243C23—242no—
244C23—2416no—
245C23—244no—
246C23#89511yescharacteristic
247C23#89512yes—
248D4—68no—
249D4—124no—
250D4—124no—
251D4—124no—
252D4—124no—
253D4—124no—
254D4—244no—
255D4—244no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—122no—
261C2×C4—242no—
262C2×C4#87411yescharacteristic
263C2×C4#88212yes—
264C2×C4#88211yescharacteristic
265C8—44no—
266C8—42no—
267C8—241no—
268C2×C3—24no—
269C2×C3—41no—
270C2×C3#201812yes—
271C2×C3#202111yescharacteristic
272D3—84no—
273D3—162no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—21no—
278C22—44no—
279C22—124no—
280C22—2416no—
281C22—244no—
282C22—244no—
283C22—244no—
284C22—242no—
285C22—242no—
286C22—242no—
287C22—484no—
288C22#467111yescharacteristic
289C22#489714yes—
290C4—22no—
291C4—21no—
292C4—242no—
293C4#474112yes—
294C3#998611yescharacteristic, complemented, Sylow
295C2—24no—
296C2—41no—
297C2—244no—
298C2—482no—
299C2#6894212yes—
300C2#6896511yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1298C2
socleC22×C3#392
hypercenter#1298D3
hypocenterC3#9986
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
1. #78641C2semidirect
2. C2×#4732C2semidirect
3. #69988C2semidirect
4. #69921C2semidirect
5. C3×#1298C2semidirect
25. #1298D3semidirect
294. C3#9986semidirect

Maximal subgroups

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

subgroupindex
#786412
C2×#47322
#699882
#699212
C3×#12982
#99863