Finite Group Databasebeta

← #10804107

Subgroups — #10804107

Order 768.

1470 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1470 conjugacy classes of subgroups — too many to draw. Only its 251 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×#5524C211yesmaximal, characteristic
2C2×#4726C212yesmaximal, complemented
3C3×#1818C211yesmaximal, characteristic, complemented
4#71584C211yesmaximal, characteristic, complemented
5#69956C212yesmaximal, complemented
6#68669C218yesmaximal, complemented
7#21743—31nomaximal, Sylow
8C22×#235C2211yescharacteristic
9C23×C4×D3C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C22×ΓL(1,𝔽25)C2214yes—
12#5524—28no—
13C2×#897C412yes—
14C2×#895C412yes—
15C2×#886C412yes—
16C2×#882C412yes—
17C2×C4×D12C2214yes—
18C2×C3×C22⋊C8C2212yes—
19C2×#830C2212yes—
20C2×(C2×D3)⋊C8C2214yes—
21C2×(C2×D3)⋊C8C2218yescomplemented
22#4726—28no—
23C3×#336C2214yescomplemented
24#4524C2214yescomplemented
25C2×#525—31no—
26C2×#407—32no—
27#1870—38no—
28#1818—31no—
29#1818D311yescharacteristic, complemented
30#1780—32no—
31C24×D3C2×C411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C22×#67C2×C412yes—
34C23×Dic3C2×C411yescharacteristic
35C2×#235—24no—
36C2×#235—44no—
37C22×D12C2×C411yescharacteristic
38C22×D12C2×C414yes—
39C22×C4×D3—44no—
40C22×C4×D3C2×C412yes—
41C22×C4×D3D418yes—
42C22×C4×D3C2312yes—
43C22×C4×D3C2312yes—
44C22×Dic6C2×C411yescharacteristic
45C22×C3×C8C2314yes—
46C3×#117—24no—
47C2×C3×C4⋊C4C2×C412yes—
48C2×C3×C22⋊C4C2×C412yes—
49C2×C3×C42C2312yes—
50C2×(C2×D3)⋊C4C2×C414yes—
51C2×(C2×D3)⋊C4C2×C414yes—
52C2×C4⋊Dic3C2×C414yes—
53C22×C3⋊C8C2314yes—
54C2×ΓL(1,𝔽25)—28no—
55C2×ΓL(1,𝔽25)—48no—
56#897—28no—
57#895—28no—
58#886—28no—
59#882—28no—
60C4×D12—28no—
61C4×D12—28no—
62C3×C22⋊C8—24no—
63#830—24no—
64(C2×D3)⋊C8—28no—
65(C2×D3)⋊C8—48no—
66C22×#41—31no—
67C24×C4—31no—
68C22×M16—34no—
69#525—68no—
70C2×#123—32no—
71C2×C22⋊Q8—32no—
72C2×#121—32no—
73C2×#120—32no—
74C2×C4×D4—34no—
75C2×#117C2×D311yescharacteristic
76#407—68no—
77C2×C22⋊C8—32no—
78C2×C22⋊C8—34no—
79C2×C22⋊C8—38no—
80C2×C22⋊C8C2×D312yes—
81#336—34no—
82#336C2×D314yescomplemented
83C24×C3C22×C411yescharacteristic
84C23×D3—42no—
85C23×D3—44no—
86C23×D3C22⋊C414yes—
87C23×D3C22×C412yes—
88C23×D3C22×C412yes—
89C22×C3×C4—24no—
90C22×C3×C4C22×C414yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D414yes—
93C22×C3×C4C2411yescharacteristic
94C2×#67—24no—
95C2×#67—44no—
96C22×Dic3—42no—
97C22×Dic3C22⋊C414yes—
98C22×Dic3C22×C412yes—
99C22×Dic3C22×C412yes—
100#235—416no—
101C2×D12—28no—
102C2×D12—22no—
103C2×D12—48no—
104C2×D12—42no—
105C2×C4×D3—28no—
106C2×C4×D3—24no—
107C2×C4×D3—28no—
108C2×C4×D3—416no—
109C2×C4×D3—44no—
110C2×C4×D3C22⋊C418yes—
111C2×C4×D3C2×D414yes—
112C2×C4×D3C2×D414yes—
113C2×Dic6—22no—
114C2×Dic6—42no—
115C2×C3×C8—24no—
116C2×C3×C8—28no—
117C3×C4⋊C4—24no—
118C3×C22⋊C4—24no—
119C3×C42—24no—
120(C2×D3)⋊C4—28no—
121(C2×D3)⋊C4—28no—
122C4⋊Dic3—28no—
123C2×C3⋊C8—24no—
124C2×C3⋊C8—28no—
125ΓL(1,𝔽25)—416no—
126C25—31no—
127C2×#41—64no—
128C2×#41—124no—
129C22×Q8—31no—
130C22×D4—32no—
131C22×D4—34no—
132C22×D4—31no—
133C23×C4—31no—
134C23×C4—32no—
135C23×C4—32no—
136C23×C4—38no—
137C23×C4—32no—
138C23×C4—124no—
139C23×C4C22×D311yescharacteristic
140C2×M16—68no—
141C2×M16—128no—
142C22×C8—34no—
143C22×C8C22×D314yes—
144#123—68no—
145C22⋊Q8—68no—
146#121—68no—
147#120—68no—
148C4×D4—68no—
149C4×D4—68no—
150#117—24no—
151C2×C4⋊C4—34no—
152C2×C4⋊C4C4×D312yes—
153C2×C22⋊C4—34no—
154C2×C22⋊C4—34no—
155C2×C22⋊C4C4×D312yes—
156C2×C42C22×D312yes—
157C22⋊C8—24no—
158C22⋊C8—68no—
159C22⋊C8—64no—
160C22⋊C8—128no—
161C23×C3—22no—
162C23×C3—24no—
163C23×C3C2×C22⋊C412yes—
164C23×C3C23×C411yescharacteristic
165C22×D3—24no—
166C22×D3—24no—
167C22×D3—24no—
168C22×D3—22no—
169C22×D3—24no—
170C22×D3—28no—
171C22×D3—44no—
172C22×D3—48no—
173C22×D3—416no—
174C22×D3C2×C22⋊C414yes—
175C2×C3×C4—22no—
176C2×C3×C4—28no—
177C2×C3×C4—24no—
178C2×C3×C4—28no—
179C2×C3×C4C2×C22⋊C412yes—
180C2×C3×C4C22×D412yes—
181C2×C3×C4H(5,𝔽2)14yes—
182#67—48no—
183C2×Dic3—24no—
184C2×Dic3—22no—
185C2×Dic3—24no—
186C2×Dic3—44no—
187C2×Dic3C2×C22⋊C414yes—
188D12—44no—
189D12—416no—
190C4×D3—28no—
191C4×D3—28no—
192C4×D3—48no—
193C4×D3—48no—
194Dic6—44no—
195C3×C8—44no—
196C3⋊C8—44no—
197C24—32no—
198C24—34no—
199C24—32no—
200C24—122no—
201C24—124no—
202C24C2×C4×D311yescharacteristic
203#41—1216no—
204C2×Q8—62no—
205C2×Q8—122no—
206C2×D4—68no—
207C2×D4—64no—
208C2×D4—62no—
209C2×D4—122no—
210C2×D4—128no—
211C2×D4—124no—
212C22×C4—24no—
213C22×C4—32no—
214C22×C4—34no—
215C22×C4—34no—
216C22×C4—34no—
217C22×C4—38no—
218C22×C4—32no—
219C22×C4—68no—
220C22×C4—64no—
221C22×C4—68no—
222C22×C4—1216no—
223C22×C4—124no—
224C22×C4—122no—
225C22×C4C2×C4×D314yes—
226C22×C4C2×C4×D311yescharacteristic
227C22×C4D3×D414yes—
228C22×C4C23×D311yescharacteristic
229M16—1216no—
230C2×C8—24no—
231C2×C8—28no—
232C2×C8—64no—
233C2×C8—68no—
234C4⋊C4—24no—
235C4⋊C4—68no—
236C22⋊C4—24no—
237C22⋊C4—68no—
238C22⋊C4—68no—
239C42—24no—
240C22×C3—22no—
241C22×C3—24no—
242C22×C3—44no—
243C22×C3C22×C22⋊C411yescharacteristic
244C22×C3#51812yes—
245C22×C3#56814yes—
246C2×D3—24no—
247C2×D3—24no—
248C2×D3—24no—
249C2×D3—416no—
250C2×D3—48no—
251C2×D3—42no—
252C2×D3—44no—
253C2×D3—48no—
254C3×C4—22no—
255C3×C4—44no—
256C3×C4#53414yes—
257Dic3—24no—
258Dic3—42no—
259C23—24no—
260C23—22no—
261C23—34no—
262C23—64no—
263C23—64no—
264C23—64no—
265C23—62no—
266C23—68no—
267C23—64no—
268C23—124no—
269C23—128no—
270C23—1216no—
271C23D3×C22⋊C412yes—
272C23C22×C4×D311yescharacteristic
273Q8—124no—
274D4—124no—
275D4—1216no—
276D4—128no—
277C2×C4—22no—
278C2×C4—28no—
279C2×C4—24no—
280C2×C4—28no—
281C2×C4—34no—
282C2×C4—64no—
283C2×C4—68no—
284C2×C4—68no—
285C2×C4—62no—
286C2×C4—64no—
287C2×C4—128no—
288C2×C4—124no—
289C2×C4—128no—
290C2×C4D3×C22⋊C412yes—
291C2×C4C2×D3×D412yes—
292C2×C4#100914yes—
293C8—44no—
294C8—124no—
295C2×C3—42no—
296C2×C3#230811yescharacteristic
297C2×C3#287912yes—
298C2×C3#297814yes—
299D3—44no—
300D3—82no—
301C22—24no—
302C22—22no—
303C22—44no—
304C22—64no—
305C22—64no—
306C22—64no—
307C22—1216no—
308C22—128no—
309C22—128no—
310C22—124no—
311C22—122no—
312C22C2×D3×C22⋊C411yescharacteristic
313C22#552112yes—
314C22#575014yes—
315C4—22no—
316C4—44no—
317C4—64no—
318C4—122no—
319C4#565314yes—
320C3#2174311yescharacteristic, complemented, Sylow
321C2—42no—
322C2—124no—
323C2—242no—
324C2#7411911yescharacteristic
325C2#7866212yes—
326C2#7937814yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1818C2
prime radicalC2×#5524C2
cosocleC22×C3×C4C24
hypercenter#1818D3
hypocenterC3#21743
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. C2×#4726C2semidirect
3. C3×#1818C2semidirect
4. #71584C2semidirect
5. #69956C2semidirect
6. #68669C2semidirect
21. C2×(C2×D3)⋊C8C22semidirect
23. C3×#336C22semidirect
24. #4524C22semidirect
29. #1818D3semidirect
82. #336C2×D3semidirect
320. C3#21743semidirect

Maximal subgroups

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

subgroupindex
C2×#55242
C2×#47262
C3×#18182
#715842
#699562
#686692
#217433