Finite Group Databasebeta

← #10804152

Subgroups — #10804152

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×#5825C211yesmaximal, characteristic
2#74053C212yesmaximal, complemented
3C3×#1782C211yesmaximal, characteristic
4#72139C211yesmaximal, characteristic
5#70029C218yesmaximal, complemented
6#69534C212yesmaximal, complemented
7#21751—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C23×Dic6C2211yescharacteristic
10#5825—28no—
11C2×#944C412yescomplemented
12C2×#936C412yescomplemented
13C2×#935C2214yes—
14C2×#926C2211yescharacteristic
15C2×#924C414yescomplemented
16C2×#877C2214yes—
17C2×C3×#99C2214yes—
18C3×#381C2212yes—
19C2×#834C2214yes—
20#5211C2212yes—
21#4943C2214yes—
22#4646—28no—
23#4527C2218yes—
24C2×#536—31no—
25#2310—32no—
26#1899—38no—
27#1820—32no—
28#1782—31no—
29#1782D311yescharacteristic
30C2×C3×#41—24no—
31C2×C3×#41—44no—
32C22×C3×Q8C2×C411yescharacteristic
33C22×C3×D4C2×C411yescharacteristic
34C22×C3×D4C2312yes—
35C23×C3×C4C2×C412yes—
36C23×C3×C4C2311yescharacteristic
37C23×Dic3C2312yes—
38C22×Dic6—44no—
39C22×Dic6C2×C412yes—
40C22×Dic6C2×C412yes—
41C22×Dic6D418yes—
42C2×C3×C22⋊C4C2314yes—
43C2×#217C2×C418yes—
44C2×#217C2314yes—
45C2×#217C2312yes—
46#944—28no—
47#936—28no—
48#935—28no—
49#935—28no—
50#926—24no—
51C2×C4⋊Dic3C2×C412yes—
52#924—216no—
53C2×Dic3⋊C4—24no—
54C2×Dic3⋊C4C2×C418yes—
55C2×C4×Dic3—24no—
56C2×C4×Dic3C2×C412yes—
57#877—48no—
58C3×#99—44no—
59C3×#95—24no—
60#834—44no—
61C3⋊#95—24no—
62C3⋊#95—28no—
63C22×#41C2×D311yescharacteristic
64C23×Q8—31no—
65#536—68no—
66C2×C4⋊Q8—32no—
67C2×#124—32no—
68C2×#123—34no—
69C2×C22⋊Q8—34no—
70C2×#117—31no—
71C2×#117—34no—
72C2×#99—34no—
73C2×#99C2×D314yes—
74#391—68no—
75#381—32no—
76#381—34no—
77#381C2×D312yes—
78#342—38no—
79C24×C3C22×C412yes—
80C24×C3C2411yescharacteristic
81C3×#41—416no—
82C2×C3×Q8—22no—
83C2×C3×Q8—42no—
84C2×C3×D4—24no—
85C2×C3×D4—24no—
86C2×C3×D4—24no—
87C2×C3×D4—22no—
88C2×C3×D4—42no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4C22×C412yes—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D414yes—
98C22×Dic3—28no—
99C22×Dic3—24no—
100C22×Dic3C22×C414yes—
101C22×Dic3C22×C414yes—
102C22×Dic3C2×D414yes—
103C22×Dic3C2×D414yes—
104C2×Dic6—44no—
105C2×Dic6—44no—
106C2×Dic6—48no—
107C2×Dic6—48no—
108C2×Dic6C22⋊C418yes—
109C2×Dic6C22⋊C418yes—
110C3×C22⋊C4—28no—
111#217—24no—
112#217—216no—
113#217—28no—
114C4⋊Dic3—24no—
115Dic3⋊C4—216no—
116Dic3⋊C4—48no—
117C4×Dic3—24no—
118C4×Dic3—48no—
119C2×#41—24no—
120C2×#41—44no—
121C22×Q8—32no—
122C22×Q8—32no—
123C22×Q8—38no—
124C22×Q8—124no—
125C22×Q8C4×D311yescharacteristic
126C22×D4C4×D311yescharacteristic
127C22×D4C22×D312yes—
128C23×C4—32no—
129C23×C4C4×D312yes—
130C23×C4C22×D311yescharacteristic
131C4⋊Q8—68no—
132#124—68no—
133#123—68no—
134#123—68no—
135C22⋊Q8—616no—
136#117—64no—
137#117—128no—
138C2×C4⋊C4—32no—
139C2×C4⋊C4—38no—
140C2×C4⋊C4—64no—
141C2×C22⋊C4—38no—
142C2×C22⋊C4—32no—
143C2×C22⋊C4—34no—
144C2×C22⋊C4C22×D314yes—
145C2×C42—32no—
146C2×C42—64no—
147#99—44no—
148#99—124no—
149#95—24no—
150#95—68no—
151#95—64no—
152C23×C3—22no—
153C23×C3—22no—
154C23×C3—22no—
155C23×C3—42no—
156C23×C3—44no—
157C23×C3C23×C411yescharacteristic
158C23×C3C22×D412yes—
159C23×C3H(5,𝔽2)14yes—
160C3×Q8—44no—
161C3×D4—44no—
162C3×D4—48no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—28no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—48no—
170C2×C3×C4—48no—
171C2×C3×C4—44no—
172C2×C3×C4C2×C22⋊C412yes—
173C2×C3×C4C2×C22⋊C412yes—
174C2×Dic3—28no—
175C2×Dic3—24no—
176C2×Dic3—24no—
177C2×Dic3—28no—
178C2×Dic3—24no—
179C2×Dic3—44no—
180C2×Dic3—48no—
181C2×Dic3—44no—
182C2×Dic3C2×C22⋊C418yes—
183Dic6—48no—
184Dic6—48no—
185C24C2×C4×D312yes—
186C24C23×D311yescharacteristic
187#41—416no—
188C2×Q8—22no—
189C2×Q8—38no—
190C2×Q8—38no—
191C2×Q8—42no—
192C2×Q8—124no—
193C2×Q8—124no—
194C2×Q8—128no—
195C2×Q8—128no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—42no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—34no—
205C22×C4—34no—
206C22×C4—34no—
207C22×C4—34no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—68no—
211C22×C4—64no—
212C22×C4C2×C4×D312yes—
213C22×C4C2×C4×D311yescharacteristic
214C22×C4C2×C4×D311yescharacteristic
215C22×C4D3×D414yes—
216C4⋊C4—616no—
217C4⋊C4—64no—
218C4⋊C4—128no—
219C22⋊C4—28no—
220C22⋊C4—616no—
221C22⋊C4—64no—
222C22⋊C4—68no—
223C42—64no—
224C42—128no—
225C22×C3—24no—
226C22×C3—22no—
227C22×C3—44no—
228C22×C3—42no—
229C22×C3—44no—
230C22×C3—82no—
231C22×C3—82no—
232C22×C3C22×C22⋊C411yescharacteristic
233C22×C3#51812yes—
234C22×C3#53414yes—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—42no—
238C3×C4—84no—
239Dic3—28no—
240Dic3—44no—
241Dic3—84no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—42no—
246C23—44no—
247C23C22×C4×D311yescharacteristic
248C23C2×D3×D412yes—
249C23#100414yes—
250Q8—44no—
251Q8—128no—
252Q8—128no—
253D4—44no—
254D4—48no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—28no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—38no—
262C2×C4—48no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—68no—
266C2×C4—64no—
267C2×C4—64no—
268C2×C4—68no—
269C2×C4—64no—
270C2×C4—124no—
271C2×C4—128no—
272C2×C4—124no—
273C2×C4D3×C22⋊C412yes—
274C2×C4D3×C22⋊C412yes—
275C2×C3—24no—
276C2×C3—22no—
277C2×C3—82no—
278C2×C3#230811yescharacteristic
279C2×C3#288412yes—
280C22—24no—
281C22—22no—
282C22—44no—
283C22—44no—
284C22—42no—
285C22—82no—
286C22—82no—
287C22C2×D3×C22⋊C411yescharacteristic
288C22#548612yes—
289C22#558914yes—
290C4—22no—
291C4—22no—
292C4—42no—
293C4—68no—
294C4—84no—
295C4—124no—
296C4—244no—
297C3#2175111yescharacteristic, complemented, Sylow
298C2—24no—
299C2—22no—
300C2—82no—
301C2#7384911yescharacteristic
302C2#7868912yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1782C2
socleC22×C3C22×C22⋊C4
prime radicalC22×C3×#41C22
cosocleC24×C3C24
hypercenter#1782D3
hypocenterC3#21751
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. #74053C2semidirect
5. #70029C2semidirect
6. #69534C2semidirect
11. C2×#944C4semidirect
12. C2×#936C4semidirect
15. C2×#924C4semidirect
297. C3#21751semidirect

Maximal subgroups

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

subgroupindex
C2×#58252
#740532
C3×#17822
#721392
#700292
#695342
#217513