Finite Group Databasebeta

← #10804147

Subgroups — #10804147

Order 768.

1606 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1606 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×#5830C211yesmaximal, characteristic
2#74119C212yesmaximal, complemented
3C3×#1782C211yesmaximal, characteristic, complemented
4#72139C211yesmaximal, characteristic, complemented
5#70036C218yesmaximal, complemented
6#69534C212yesmaximal, complemented
7#21750—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C23×D12C2211yescharacteristic
10#5830—28no—
11C2×#947C412yescomplemented
12C2×#940C412yescomplemented
13C2×#937C2214yes—
14C2×#930C414yescomplemented
15C2×#926C2211yescharacteristic
16C2×D3×C22⋊C4C2214yes—
17C2×C3×#99C2214yes—
18C3×#381C2212yes—
19C2×#834C2214yes—
20#5211C2212yes—
21#4955C2214yes—
22#4665—28no—
23#4527C2218yes—
24C2×#535—31no—
25#2308—32no—
26#1898—38no—
27#1820—32no—
28#1782—31no—
29#1782D311yescharacteristic, complemented
30C24×D3C2312yes—
31C2×C3×#41—24no—
32C2×C3×#41—44no—
33C22×C3×Q8C2×C411yescharacteristic, complemented
34C22×C3×D4C2×C411yescharacteristic, complemented
35C22×C3×D4C2312yes—
36C23×C3×C4C2×C412yescomplemented
37C23×C3×C4C2311yescharacteristic
38C22×#67C2×C418yescomplemented
39C22×D12—44no—
40C22×D12C2×C412yes—
41C22×D12C2×C412yes—
42C22×D12D418yes—
43C22×C4×D3—24no—
44C2×C3×C22⋊C4C2314yes—
45C2×#217C2314yes—
46C2×#217C2312yes—
47#947—28no—
48#940—28no—
49#937—28no—
50#937—28no—
51#930—216no—
52C2×(C2×D3)⋊C4—24no—
53C2×(C2×D3)⋊C4—28no—
54C2×(C2×D3)⋊C4C2×C418yescomplemented
55#926—24no—
56C2×C4⋊Dic3C2×C412yescomplemented
57C2×C4×Dic3C2×C412yescomplemented
58D3×C22⋊C4—48no—
59C3×#99—44no—
60C3×#95—24no—
61#834—44no—
62C3⋊#95—24no—
63C22×#41C2×D311yescharacteristic
64C23×D4—31no—
65#535—68no—
66C2×#127—32no—
67C2×#124—32no—
68C2×#121—34no—
69C2×#120—34no—
70C2×#117—31no—
71C22×C22⋊C4—34no—
72C2×#99—34no—
73C2×#99C2×D314yes—
74#386—68no—
75#381—32no—
76#381C2×D312yes—
77#379—34no—
78#342—38no—
79C24×C3C22×C412yes—
80C24×C3C2411yescharacteristic
81C23×D3—28no—
82C23×D3—84no—
83C23×D3C22×C414yes—
84C23×D3C2×D414yes—
85C23×D3C2×D414yes—
86C3×#41—416no—
87C2×C3×Q8—22no—
88C2×C3×Q8—42no—
89C2×C3×D4—24no—
90C2×C3×D4—24no—
91C2×C3×D4—24no—
92C2×C3×D4—22no—
93C2×C3×D4—42no—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—24no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4C22×C412yes—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D414yes—
103C2×#67—216no—
104C2×#67—88no—
105C22×Dic3—24no—
106C22×Dic3C22×C414yes—
107C2×D12—44no—
108C2×D12—44no—
109C2×D12—48no—
110C2×D12—48no—
111C2×D12C22⋊C418yes—
112C2×D12C22⋊C418yes—
113C2×C4×D3—48no—
114C2×C4×D3—88no—
115C3×C22⋊C4—28no—
116#217—24no—
117#217—28no—
118(C2×D3)⋊C4—216no—
119(C2×D3)⋊C4—48no—
120(C2×D3)⋊C4—88no—
121C4⋊Dic3—24no—
122C4×Dic3—24no—
123C25—32no—
124C2×#41—24no—
125C2×#41—44no—
126C22×Q8C4×D311yescharacteristic, complemented
127C22×D4—38no—
128C22×D4—32no—
129C22×D4—32no—
130C22×D4—38no—
131C22×D4—124no—
132C22×D4C4×D311yescharacteristic, complemented
133C22×D4C22×D312yes—
134C23×C4—64no—
135C23×C4C4×D312yescomplemented
136C23×C4C22×D311yescharacteristic
137#127—68no—
138#124—68no—
139#121—616no—
140#120—68no—
141#120—68no—
142#117—64no—
143C2×C4⋊C4—32no—
144C2×C22⋊C4—32no—
145C2×C22⋊C4—38no—
146C2×C22⋊C4—34no—
147C2×C22⋊C4—64no—
148C2×C22⋊C4—68no—
149C2×C22⋊C4—128no—
150C2×C22⋊C4C22×D314yes—
151C2×C42—32no—
152#99—44no—
153#99—124no—
154#95—24no—
155#95—64no—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—42no—
160C23×C3—44no—
161C23×C3C23×C411yescharacteristic
162C23×C3C22×D412yes—
163C23×C3H(5,𝔽2)14yes—
164C22×D3—24no—
165C22×D3—24no—
166C22×D3—28no—
167C22×D3—28no—
168C22×D3—48no—
169C22×D3—48no—
170C22×D3—48no—
171C22×D3—816no—
172C22×D3C2×C22⋊C418yes—
173C3×Q8—44no—
174C3×D4—44no—
175C3×D4—48no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—28no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—48no—
183C2×C3×C4—48no—
184C2×C3×C4—44no—
185C2×C3×C4C2×C22⋊C412yes—
186C2×C3×C4C2×C22⋊C412yes—
187#67—816no—
188C2×Dic3—24no—
189C2×Dic3—44no—
190C2×Dic3—48no—
191C2×Dic3—44no—
192D12—48no—
193D12—48no—
194C4×D3—816no—
195C24—34no—
196C24—34no—
197C24—34no—
198C24—68no—
199C24—244no—
200C24C2×C4×D312yes—
201C24C23×D311yescharacteristic
202#41—416no—
203C2×Q8—22no—
204C2×Q8—42no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—38no—
210C2×D4—38no—
211C2×D4—42no—
212C2×D4—616no—
213C2×D4—124no—
214C2×D4—124no—
215C2×D4—128no—
216C2×D4—128no—
217C2×D4—248no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—34no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—64no—
225C22×C4—128no—
226C22×C4—248no—
227C22×C4C2×C4×D312yes—
228C22×C4C2×C4×D311yescharacteristic
229C22×C4C2×C4×D311yescharacteristic
230C22×C4D3×D414yes—
231C4⋊C4—64no—
232C22⋊C4—28no—
233C22⋊C4—616no—
234C22⋊C4—64no—
235C22⋊C4—68no—
236C22⋊C4—128no—
237C22⋊C4—248no—
238C42—64no—
239C22×C3—24no—
240C22×C3—22no—
241C22×C3—44no—
242C22×C3—42no—
243C22×C3—44no—
244C22×C3—82no—
245C22×C3—82no—
246C22×C3C22×C22⋊C411yescharacteristic
247C22×C3#51812yes—
248C22×C3#53414yes—
249C2×D3—28no—
250C2×D3—216no—
251C2×D3—48no—
252C2×D3—48no—
253C2×D3—88no—
254C2×D3—88no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—84no—
259Dic3—44no—
260Dic3—84no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—38no—
265C23—42no—
266C23—44no—
267C23—64no—
268C23—64no—
269C23—68no—
270C23—68no—
271C23—128no—
272C23—128no—
273C23—128no—
274C23—2416no—
275C23C22×C4×D311yescharacteristic
276C23C2×D3×D412yes—
277C23#100414yes—
278Q8—44no—
279D4—44no—
280D4—48no—
281D4—128no—
282D4—128no—
283D4—2416no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—28no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—64no—
294C2×C4—124no—
295C2×C4—128no—
296C2×C4—124no—
297C2×C4—2416no—
298C2×C4D3×C22⋊C412yes—
299C2×C4D3×C22⋊C412yes—
300C2×C3—24no—
301C2×C3—22no—
302C2×C3—82no—
303C2×C3#230811yescharacteristic
304C2×C3#288312yes—
305D3—48no—
306C22—24no—
307C22—22no—
308C22—44no—
309C22—44no—
310C22—42no—
311C22—616no—
312C22—68no—
313C22—82no—
314C22—82no—
315C22—128no—
316C22—128no—
317C22—248no—
318C22—248no—
319C22C2×D3×C22⋊C411yescharacteristic
320C22#548612yes—
321C22#558914yes—
322C4—22no—
323C4—22no—
324C4—42no—
325C4—84no—
326C4—124no—
327C4—244no—
328C3#2175011yescharacteristic, complemented, Sylow
329C2—24no—
330C2—22no—
331C2—82no—
332C2—128no—
333C2#7384911yescharacteristic
334C2#7869412yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1782C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5830C2
cosocleC24×C3C24
hypercenter#1782D3
hypocenterC3#21750
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #74119C2semidirect
3. C3×#1782C2semidirect
4. #72139C2semidirect
5. #70036C2semidirect
6. #69534C2semidirect
11. C2×#947C4semidirect
12. C2×#940C4semidirect
14. C2×#930C4semidirect
29. #1782D3semidirect
33. C22×C3×Q8C2×C4semidirect
34. C22×C3×D4C2×C4semidirect
36. C23×C3×C4C2×C4semidirect
38. C22×#67C2×C4semidirect
54. C2×(C2×D3)⋊C4C2×C4semidirect
56. C2×C4⋊Dic3C2×C4semidirect
57. C2×C4×Dic3C2×C4semidirect
126. C22×Q8C4×D3semidirect
132. C22×D4C4×D3semidirect
135. C23×C4C4×D3semidirect
328. C3#21750semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
#741192
C3×#17822
#721392
#700362
#695342
#217503