Finite Group Databasebeta

← #10688705

Subgroups — #10688705

Order 768.

754 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 87 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#74119C211yesmaximal, characteristic
2#69988C212yesmaximal
3C3×#1306C211yesmaximal, characteristic, complemented
4#67686C212yesmaximal, complemented
5#67629C211yesmaximal, characteristic, complemented
6#8752—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×D3×C22⋊C4C412yes—
9C3×#407C2212yes—
10C3×#381C2211yescharacteristic
11#5211C2211yescharacteristic
12#5207C2212yes—
13C2×#823—24no—
14#4955C412yes—
15#4665—24no—
16#2308—31no—
17#1783—32no—
18#1306—32no—
19#1306—31no—
20#1306D311yescharacteristic, complemented
21C24×D3C2×C411yescharacteristic
22C24×D3C2×C411yescharacteristic
23C23×C3×C4C2311yescharacteristic
24C22×D12—22no—
25C22×D12—21no—
26C22×D12—21no—
27C22×D12—44no—
28C22×D12D414yes—
29C22×C4×D3—22no—
30C2×C3×M16—22no—
31C2×C3×C22⋊C4C2×C412yes—
32C2×#217C2×C412yes—
33C2×(C2×D3)⋊C4—22no—
34C2×(C2×D3)⋊C4—24no—
35C2×#208—22no—
36D3×C22⋊C4—44no—
37C3×C22⋊C8—22no—
38C3×C22⋊C8—22no—
39C3×#95—22no—
40C3⋊#95—22no—
41#830—22no—
42#830—22no—
43#823—48no—
44C23×D4—31no—
45C22×C22⋊C4—32no—
46C2×#100—64no—
47#407—32no—
48#407C2×D312yes—
49#386—64no—
50#381—31no—
51#381C2×D311yescharacteristic
52#379—32no—
53C24×C3C22×C411yescharacteristic
54C23×D3—21no—
55C23×D3—21no—
56C23×D3—24no—
57C23×D3—21no—
58C23×D3—21no—
59C23×D3—42no—
60C23×D3—82no—
61C23×D3—82no—
62C23×D3C22⋊C412yes—
63C23×D3C22⋊C412yes—
64C22×C3×C4—22no—
65C22×C3×C4—42no—
66C22×C3×C4C2×D412yes—
67C22×C3×C4#4112yes—
68C22×C3×C4#4111yescharacteristic
69C22×C3×C4#4111yescharacteristic
70C22×Dic3—22no—
71C2×D12—28no—
72C2×D12—24no—
73C2×D12—24no—
74C2×D12—44no—
75C2×D12—48no—
76C2×D12—44no—
77C2×D12—44no—
78C2×C4×D3—44no—
79C2×C4×D3—84no—
80C3×M16—44no—
81C2×C3×C8—42no—
82C3×C22⋊C4—42no—
83#217—42no—
84(C2×D3)⋊C4—44no—
85(C2×D3)⋊C4—84no—
86#208—44no—
87C2×C3⋊C8—42no—
88C25—31no—
89C25—31no—
90C22×D4—34no—
91C22×D4—62no—
92C22×D4—61no—
93C22×D4—61no—
94C22×D4—124no—
95C23×C4—62no—
96C23×C4C22×D311yescharacteristic
97C2×M16—22no—
98C2×M16—62no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—62no—
101C2×C22⋊C4—64no—
102C2×C22⋊C4—124no—
103C2×C22⋊C4C4×D312yes—
104#100—128no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107C22⋊C8—62no—
108C22⋊C8—62no—
109#95—22no—
110#95—62no—
111C23×C3—22no—
112C23×C3—42no—
113C23×C3C2×C22⋊C411yescharacteristic
114C23×C3#11711yescharacteristic
115C23×C3#11711yescharacteristic
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—24no—
119C22×D3—24no—
120C22×D3—22no—
121C22×D3—24no—
122C22×D3—44no—
123C22×D3—42no—
124C22×D3—42no—
125C22×D3—44no—
126C22×D3—82no—
127C22×D3—82no—
128C22×D3—82no—
129C22×D3—82no—
130C22×D3—88no—
131C22×D3—88no—
132C22×D3#9914yes—
133C2×C3×C4—24no—
134C2×C3×C4—42no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—44no—
138C2×C3×C4—42no—
139C2×C3×C4#12312yes—
140C2×C3×C4#12312yes—
141C2×Dic3—44no—
142C2×Dic3—42no—
143D12—48no—
144D12—48no—
145C4×D3—88no—
146C3×C8—82no—
147C3⋊C8—82no—
148C24—32no—
149C24—32no—
150C24—64no—
151C24—61no—
152C24—61no—
153C24—61no—
154C24—61no—
155C24—122no—
156C24—242no—
157C24—242no—
158C24C2×C4×D311yescharacteristic
159C2×D4—68no—
160C2×D4—64no—
161C2×D4—64no—
162C2×D4—124no—
163C2×D4—128no—
164C2×D4—124no—
165C2×D4—124no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—62no—
169C22×C4—124no—
170C22×C4—244no—
171C22×C4#23512yes—
172C22×C4D3×D412yes—
173C22×C4#23711yescharacteristic
174C22×C4#23911yescharacteristic
175M16—44no—
176M16—124no—
177C2×C8—42no—
178C2×C8—122no—
179C22⋊C4—42no—
180C22⋊C4—124no—
181C22⋊C4—122no—
182C22⋊C4—244no—
183C22×C3—22no—
184C22×C3—21no—
185C22×C3—21no—
186C22×C3—22no—
187C22×C3—22no—
188C22×C3—44no—
189C22×C3#38111yescharacteristic
190C22×C3C2×#9912yes—
191C2×D3—24no—
192C2×D3—44no—
193C2×D3—44no—
194C2×D3—44no—
195C2×D3—42no—
196C2×D3—44no—
197C2×D3—84no—
198C2×D3—84no—
199C2×D3—84no—
200C2×D3—84no—
201C2×D3—84no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—82no—
205Dic3—82no—
206C23—22no—
207C23—34no—
208C23—42no—
209C23—64no—
210C23—64no—
211C23—62no—
212C23—62no—
213C23—62no—
214C23—64no—
215C23—124no—
216C23—122no—
217C23—122no—
218C23—124no—
219C23—242no—
220C23—242no—
221C23—242no—
222C23—242no—
223C23—248no—
224C23—248no—
225C23#87211yescharacteristic
226C23D3×C22⋊C411yescharacteristic
227C23#89211yescharacteristic
228D4—128no—
229D4—128no—
230C2×C4—24no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—124no—
237C2×C4—122no—
238C2×C4—248no—
239C2×C4#88312yes—
240C2×C4#89412yes—
241C8—82no—
242C8—242no—
243C2×C3—22no—
244C2×C3—41no—
245C2×C3—41no—
246C2×C3#181911yescharacteristic
247C2×C3#182912yes—
248D3—44no—
249D3—82no—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—22no—
255C22—44no—
256C22—64no—
257C22—124no—
258C22—124no—
259C22—124no—
260C22—122no—
261C22—124no—
262C22—244no—
263C22—244no—
264C22—244no—
265C22—244no—
266C22—244no—
267C22#466411yescharacteristic
268C22D3×#9912yes—
269C4—22no—
270C4—22no—
271C4—82no—
272C4—242no—
273C3#875211yescharacteristic, complemented, Sylow
274C2—22no—
275C2—41no—
276C2—41no—
277C2—124no—
278C2—242no—
279C2#6896911yescharacteristic
280C2#6902612yes—
subgroupG/H
centerC22#4664
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3#381
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenter#1306D3
hypocenterC3#8752
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
3. C3×#1306C2semidirect
4. #67686C2semidirect
5. #67629C2semidirect
20. #1306D3semidirect
273. C3#8752semidirect

Maximal subgroups

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

subgroupindex
#741192
#699882
C3×#13062
#676862
#676292
#87523