Finite Group Databasebeta

← #10804153

Subgroups — #10804153

Order 768.

1294 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1294 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C2×#5526C211yesmaximal, characteristic
2#74053C212yesmaximal, complemented
3C3×#1820C211yesmaximal, characteristic, complemented
4#71594C211yesmaximal, characteristic, complemented
5#69537C212yesmaximal
6#68954C218yesmaximal, complemented
7#21751—31nomaximal, Sylow
8C22×#235C2211yescharacteristic
9C23×Dic6C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11#5526—28no—
12C2×#897C414yescomplemented
13C2×#886C2214yes—
14C2×#877C2214yes—
15C2×#875C412yescomplemented
16C2×#869C412yescomplemented
17C3×#381C2212yes—
18#5211C2212yes—
19C2×#806C2218yes—
20#4943C2214yes—
21#4646—28no—
22C3×#342C2214yes—
23#4526C2214yes—
24C2×#536—31no—
25#2310—32no—
26#1899—38no—
27#1820—31no—
28#1820D311yescharacteristic, complemented
29#1782—32no—
30C23×C3×C4C2311yescharacteristic
31C22×#67C2312yes—
32C23×Dic3C2312yes—
33C2×#235—24no—
34C2×#235—44no—
35C22×D12C2×C411yescharacteristic
36C22×C4×D3C2×C412yes—
37C22×Dic6—44no—
38C22×Dic6C2×C411yescharacteristic
39C22×Dic6C2×C412yes—
40C22×Dic6C2×C412yes—
41C22×Dic6D418yes—
42C3×#117—24no—
43C2×C3×C4⋊C4C2×C412yes—
44C2×C3×C22⋊C4C2314yes—
45C2×C3×C22⋊C4C2312yes—
46C2×C3×C42C2×C412yes—
47C2×#217C2314yes—
48C2×(C2×D3)⋊C4C2×C418yes—
49C2×C4⋊Dic3C2×C418yes—
50C2×Dic3⋊C4—24no—
51C2×C4×Dic3—24no—
52#897—216no—
53#886—28no—
54#886—28no—
55#877—48no—
56#875—28no—
57#869—28no—
58C3×#95—24no—
59C3⋊#95—24no—
60C3⋊#95—28no—
61#806—48no—
62C22×#41—31no—
63C23×Q8—31no—
64#536—68no—
65C2×C4⋊Q8—32no—
66C2×#124—32no—
67C2×#123—34no—
68C2×C22⋊Q8—34no—
69C2×#117—34no—
70C2×#117C2×D311yescharacteristic
71C2×#99—38no—
72#391—68no—
73#381—32no—
74#381—34no—
75#381C2×D312yes—
76#342—34no—
77#342C2×D314yes—
78C24×C3C2411yescharacteristic
79C23×D3C22×C412yes—
80C22×C3×C4—24no—
81C22×C3×C4—24no—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C22×C411yescharacteristic
84C22×C3×C4C22×C414yes—
85C22×C3×C4C2×D414yes—
86C2×#67—24no—
87C2×#67—28no—
88C22×Dic3—28no—
89C22×Dic3—24no—
90C22×Dic3C22×C412yes—
91C22×Dic3C22×C414yes—
92C22×Dic3C2×D414yes—
93C22×Dic3C2×D414yes—
94#235—416no—
95C2×D12—22no—
96C2×D12—42no—
97C2×C4×D3—24no—
98C2×C4×D3—44no—
99C2×Dic6—22no—
100C2×Dic6—44no—
101C2×Dic6—44no—
102C2×Dic6—48no—
103C2×Dic6—48no—
104C2×Dic6—42no—
105C2×Dic6C22⋊C418yes—
106C2×Dic6C22⋊C418yes—
107C3×C4⋊C4—24no—
108C3×C22⋊C4—24no—
109C3×C22⋊C4—28no—
110C3×C42—24no—
111#217—28no—
112(C2×D3)⋊C4—216no—
113C4⋊Dic3—216no—
114Dic3⋊C4—48no—
115C4×Dic3—48no—
116C2×#41—64no—
117C2×#41—124no—
118C22×Q8—32no—
119C22×Q8—32no—
120C22×Q8—31no—
121C22×Q8—38no—
122C22×Q8—124no—
123C22×D4—32no—
124C22×D4—31no—
125C23×C4—32no—
126C23×C4—32no—
127C23×C4C22×D311yescharacteristic
128C4⋊Q8—68no—
129#124—68no—
130#123—68no—
131#123—68no—
132C22⋊Q8—616no—
133#117—24no—
134#117—128no—
135C2×C4⋊C4—38no—
136C2×C4⋊C4—64no—
137C2×C4⋊C4C4×D312yes—
138C2×C22⋊C4—38no—
139C2×C22⋊C4—34no—
140C2×C22⋊C4C22×D314yes—
141C2×C22⋊C4C22×D312yes—
142C2×C42—64no—
143C2×C42C4×D312yes—
144#99—128no—
145#95—24no—
146#95—68no—
147#95—64no—
148C23×C3—42no—
149C23×C3C23×C411yescharacteristic
150C23×C3C22×D412yes—
151C23×C3H(5,𝔽2)14yes—
152C22×D3—22no—
153C22×D3—24no—
154C22×D3—44no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—28no—
158C2×C3×C4—24no—
159C2×C3×C4—44no—
160C2×C3×C4—48no—
161C2×C3×C4—44no—
162C2×C3×C4C2×C22⋊C412yes—
163C2×C3×C4C2×C22⋊C412yes—
164#67—48no—
165C2×Dic3—28no—
166C2×Dic3—24no—
167C2×Dic3—24no—
168C2×Dic3—28no—
169C2×Dic3—22no—
170C2×Dic3—24no—
171C2×Dic3—48no—
172C2×Dic3—44no—
173C2×Dic3C2×C22⋊C418yes—
174D12—44no—
175C4×D3—48no—
176Dic6—48no—
177Dic6—48no—
178Dic6—44no—
179C24—32no—
180C24C23×D311yescharacteristic
181#41—1216no—
182C2×Q8—38no—
183C2×Q8—38no—
184C2×Q8—62no—
185C2×Q8—128no—
186C2×Q8—128no—
187C2×Q8—124no—
188C2×Q8—124no—
189C2×Q8—122no—
190C2×D4—64no—
191C2×D4—68no—
192C2×D4—62no—
193C2×D4—122no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—34no—
197C22×C4—34no—
198C22×C4—34no—
199C22×C4—32no—
200C22×C4—68no—
201C22×C4—64no—
202C22×C4—64no—
203C22×C4—124no—
204C22×C4C2×C4×D311yescharacteristic
205C22×C4C2×C4×D311yescharacteristic
206C22×C4C2×C4×D314yes—
207C22×C4D3×D414yes—
208C4⋊C4—24no—
209C4⋊C4—616no—
210C4⋊C4—128no—
211C22⋊C4—24no—
212C22⋊C4—28no—
213C22⋊C4—616no—
214C22⋊C4—68no—
215C42—24no—
216C42—128no—
217C22×C3—24no—
218C22×C3—22no—
219C22×C3—44no—
220C22×C3C22×C22⋊C411yescharacteristic
221C22×C3#51812yes—
222C22×C3#53414yes—
223C2×D3—42no—
224C2×D3—44no—
225C2×D3—84no—
226C3×C4—22no—
227C3×C4—22no—
228C3×C4—44no—
229C3×C4—84no—
230Dic3—28no—
231Dic3—42no—
232Dic3—84no—
233C23—42no—
234C23—62no—
235C23—64no—
236C23—124no—
237C23C22×C4×D311yescharacteristic
238C23C2×D3×D412yes—
239C23#100914yes—
240Q8—128no—
241Q8—128no—
242Q8—124no—
243D4—124no—
244D4—128no—
245C2×C4—28no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—38no—
250C2×C4—44no—
251C2×C4—48no—
252C2×C4—44no—
253C2×C4—64no—
254C2×C4—64no—
255C2×C4—68no—
256C2×C4—68no—
257C2×C4—62no—
258C2×C4—64no—
259C2×C4—128no—
260C2×C4—128no—
261C2×C4—124no—
262C2×C4D3×C22⋊C412yes—
263C2×C4D3×C22⋊C412yes—
264C2×C3—24no—
265C2×C3—22no—
266C2×C3#230811yescharacteristic
267C2×C3#288412yes—
268D3—82no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—124no—
273C22—122no—
274C22—244no—
275C22C2×D3×C22⋊C411yescharacteristic
276C22#552112yes—
277C22#565314yes—
278C4—22no—
279C4—22no—
280C4—44no—
281C4—68no—
282C4—84no—
283C4—122no—
284C4—244no—
285C3#2175111yescharacteristic, complemented, Sylow
286C2—24no—
287C2—22no—
288C2—242no—
289C2#7411911yescharacteristic
290C2#7868912yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1820C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#235C22
cosocleC24×C3C24
hypercenter#1820D3
hypocenterC3#21751
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #74053C2semidirect
3. C3×#1820C2semidirect
4. #71594C2semidirect
6. #68954C2semidirect
12. C2×#897C4semidirect
15. C2×#875C4semidirect
16. C2×#869C4semidirect
28. #1820D3semidirect
285. C3#21751semidirect

Maximal subgroups

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

subgroupindex
C2×#55262
#740532
C3×#18202
#715942
#695372
#689542
#217513