Finite Group Databasebeta

← #10763242

Subgroups — #10763242

Order 768.

950 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5688C211yesmaximal, characteristic
2#79331C212yesmaximal
3C3×#1534C211yesmaximal, characteristic, complemented
4#71736C211yesmaximal, characteristic, complemented
5#70814C212yesmaximal, complemented
6#68369C214yesmaximal, complemented
7#68362C214yesmaximal, complemented
8#17664—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5688—22no—
11#5688—22no—
12#5688C412yes—
13#5688C412yes—
14#5688C2212yes—
15#5688C2212yes—
16C2×#896—22no—
17C2×#892C412yes—
18C2×#888C412yes—
19C2×#887—22no—
20C2×C42×D3C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C3×#432C2212yes—
23#4926C2212yes—
24#4837—22no—
25#4833—22no—
26#4726—22no—
27#4718—22no—
28#4707C2214yes—
29#4705C2214yes—
30C3×#330C2214yescomplemented
31#4543C2214yescomplemented
32#4465C2218yescomplemented
33C2×#533—31no—
34#2920—32no—
35#1537—34no—
36#1534—32no—
37#1534—31no—
38#1534D311yescharacteristic, complemented
39#1526—34no—
40C22×C4×D3C2×C411yescharacteristic
41C22×C4×D3D412yes—
42C22×Dic6—22no—
43C3×#125—22no—
44C3×#125C2×C412yes—
45C3×#125C2312yes—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4C2×C412yes—
48C2×C3×C42C2311yescharacteristic
49C2×(C2×D3)⋊C4—22no—
50C2×C4⋊Dic3C2×C412yes—
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4—22no—
53C2×C4×Dic3—22no—
54C2×C4×Dic3C2×C411yescharacteristic
55C2×C4×Dic3D412yes—
56C2×ΓL(1,𝔽25)—24no—
57#896—24no—
58#896—24no—
59#896—24no—
60#896—24no—
61#892—22no—
62#892—22no—
63#892C2×C414yes—
64#892C2×C414yes—
65#888—22no—
66#888—22no—
67#888C2×C414yes—
68#888C2×C414yes—
69#887—24no—
70#887—24no—
71#887—24no—
72#887—24no—
73C42×D3C2×C411yescharacteristic
74C42×D3C2×C411yescharacteristic
75C42×D3D412yes—
76C42×D3D412yes—
77C42×D3C2311yescharacteristic
78C42×D3C2311yescharacteristic
79#870—22no—
80#870C2×C412yes—
81#870C2312yes—
82C3×C4⋊C8—22no—
83C3×C22⋊C8—22no—
84C3×#97C2314yes—
85#830—22no—
86(C2×D3)⋊C8—24no—
87Dic3⋊C8—24no—
88C8×Dic3—24no—
89C4⋊(C3⋊C8)—22no—
90#803C2314yes—
91#533—32no—
92#533—32no—
93#533—32no—
94#533—32no—
95#533—62no—
96#533—62no—
97C2×C4⋊Q8—32no—
98C2×#125—31no—
99C2×#125C2×D311yescharacteristic
100C2×C22⋊Q8—62no—
101C2×C4×Q8—62no—
102C2×#117—32no—
103C22×C42—31no—
104#432—32no—
105#432—34no—
106#432C2×D312yes—
107#431—62no—
108#431—62no—
109#423—62no—
110#407—62no—
111C4×M16—34no—
112#330—34no—
113#330—38no—
114#330C2×D314yescomplemented
115C23×D3C22⋊C411yescharacteristic
116C22×C3×C4—22no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D412yes—
119C22×Dic3—22no—
120C22×Dic3C22⋊C412yes—
121C22×Dic3C22⋊C411yescharacteristic
122C2×C4×D3—24no—
123C2×C4×D3—44no—
124C2×C4×D3C22⋊C412yes—
125C2×C4×D3C22⋊C411yescharacteristic
126C2×C4×D3C22⋊C411yescharacteristic
127C2×C4×D3C22⋊C412yes—
128C2×C4×D3C22×C411yescharacteristic
129C2×C4×D3C22×C411yescharacteristic
130C2×C4×D3C2×D412yes—
131C2×C4×D3C2×D412yes—
132C2×Dic6—24no—
133C2×Dic6—24no—
134C2×Dic6—48no—
135C2×C3×C8—24no—
136C3×C4⋊C4—24no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—24no—
139C3×C4⋊C4C22×C414yes—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D412yes—
142C3×C42C2411yescharacteristic
143(C2×D3)⋊C4—24no—
144(C2×D3)⋊C4—24no—
145C4⋊Dic3—22no—
146C4⋊Dic3C22×C414yes—
147Dic3⋊C4—24no—
148Dic3⋊C4—24no—
149Dic3⋊C4—24no—
150Dic3⋊C4—24no—
151C4×Dic3—24no—
152C4×Dic3—24no—
153C4×Dic3C22⋊C412yes—
154C4×Dic3C22⋊C412yes—
155C4×Dic3C22⋊C412yes—
156C4×Dic3C22×C411yescharacteristic
157C4×Dic3C22×C411yescharacteristic
158C4×Dic3C2×D412yes—
159C4×Dic3C2×D412yes—
160C2×C3⋊C8—24no—
161ΓL(1,𝔽25)—48no—
162C22×Q8—62no—
163C23×C4—31no—
164C23×C4—32no—
165C2×M16—64no—
166C4⋊Q8—34no—
167C4⋊Q8—34no—
168C4⋊Q8—62no—
169C4⋊Q8—62no—
170#125—22no—
171#125—32no—
172#125—32no—
173#125—62no—
174#125C4×D312yes—
175#125C22×D312yes—
176C22⋊Q8—64no—
177C22⋊Q8—64no—
178C22⋊Q8—64no—
179C22⋊Q8—64no—
180C4×Q8—64no—
181C4×Q8—64no—
182C4×Q8—64no—
183C4×Q8—64no—
184#117—34no—
185#117—34no—
186#117—62no—
187#117—62no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4C4×D312yes—
193C2×C22⋊C4—62no—
194C2×C42—31no—
195C2×C42—32no—
196C2×C42—31no—
197C2×C42—31no—
198C2×C42—32no—
199C2×C42—32no—
200C2×C42—31no—
201C2×C42—31no—
202C2×C42—62no—
203C2×C42C22×D311yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—64no—
206C4⋊C8—62no—
207C22⋊C8—22no—
208C22⋊C8—64no—
209C22⋊C8—62no—
210#97—34no—
211#97C22×D314yes—
212C4×C8—64no—
213C23×C3C2×C22⋊C411yescharacteristic
214C22×D3—22no—
215C22×D3—42no—
216C22×D3C2×C22⋊C411yescharacteristic
217C22×D3C2×C22⋊C411yescharacteristic
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—42no—
221C2×C3×C4—44no—
222C2×C3×C4C2×C22⋊C412yes—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C23×C411yescharacteristic
225C2×C3×C4C22×D412yes—
226C2×Dic3—24no—
227C2×Dic3—24no—
228C2×Dic3—44no—
229C2×Dic3#10114yes—
230C2×Dic3C2×C22⋊C412yes—
231C2×Dic3C2×C22⋊C411yescharacteristic
232C2×Dic3C2×C22⋊C411yescharacteristic
233C2×Dic3C2×C22⋊C412yes—
234C4×D3—22no—
235C4×D3—22no—
236C4×D3—22no—
237C4×D3—22no—
238C4×D3—24no—
239C4×D3—24no—
240C4×D3—42no—
241C4×D3—42no—
242Dic6—48no—
243Dic6—48no—
244C3×C8—44no—
245C3⋊C8—44no—
246C24—31no—
247C2×Q8—64no—
248C2×Q8—64no—
249C2×Q8—128no—
250C22×C4—22no—
251C22×C4—31no—
252C22×C4—32no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4—32no—
260C22×C4—32no—
261C22×C4—64no—
262C22×C4—62no—
263C22×C4—124no—
264C22×C4C2×C4×D311yescharacteristic
265C22×C4D3×D412yes—
266M16—128no—
267C2×C8—24no—
268C2×C8—64no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—34no—
273C4⋊C4—64no—
274C4⋊C4—64no—
275C4⋊C4—62no—
276C4⋊C4—64no—
277C4⋊C4—64no—
278C4⋊C4C2×C4×D314yes—
279C22⋊C4—64no—
280C22⋊C4—64no—
281C42—32no—
282C42—32no—
283C42—32no—
284C42—31no—
285C42—31no—
286C42—32no—
287C42—32no—
288C42—64no—
289C42—64no—
290C42C2×C4×D311yescharacteristic
291C42D3×D412yes—
292C42C23×D311yescharacteristic
293C22×C3—22no—
294C22×C3C2×#10112yes—
295C22×C3C22×C22⋊C411yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—22no—
299C2×D3—22no—
300C2×D3—44no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—44no—
306Dic3—22no—
307Dic3—44no—
308Dic3C2×#10112yes—
309Dic3C2×#10112yes—
310Dic3#42114yes—
311C23—31no—
312C23—31no—
313C23—62no—
314C23—122no—
315C23D3×C22⋊C411yescharacteristic
316Q8—128no—
317Q8—128no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—32no—
323C2×C4—34no—
324C2×C4—32no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—64no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—64no—
333C2×C4—64no—
334C2×C4—64no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—124no—
338C2×C4D3×C22⋊C412yes—
339C2×C4D3×C22⋊C411yescharacteristic
340C2×C4C22×C4×D311yescharacteristic
341C2×C4C2×D3×D412yes—
342C8—44no—
343C8—124no—
344C2×C3—22no—
345C2×C3C22×#10111yescharacteristic
346C2×C3C2×#42112yes—
347D3—41no—
348D3—41no—
349C22—22no—
350C22—61no—
351C22—61no—
352C22—62no—
353C22—62no—
354C22—124no—
355C22D3×#10112yes—
356C22C2×D3×C22⋊C411yescharacteristic
357C4—21no—
358C4—21no—
359C4—22no—
360C4—22no—
361C4—32no—
362C4—32no—
363C4—34no—
364C4—44no—
365C4—62no—
366C4—124no—
367C3#1766411yescharacteristic, complemented, Sylow
368C2—22no—
369C2—121no—
370C2—121no—
371C2C2×D3×#10111yescharacteristic
372C2D3×#42112yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1534C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC3×C42C24
hypercenter#1534D3
hypocenterC3#17664
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1534

Complements

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

NH ≅ G/Ntype
3. C3×#1534C2semidirect
4. #71736C2semidirect
5. #70814C2semidirect
6. #68369C2semidirect
7. #68362C2semidirect
30. C3×#330C22semidirect
31. #4543C22semidirect
32. #4465C22semidirect
38. #1534D3semidirect
114. #330C2×D3semidirect
367. C3#17664semidirect

Maximal subgroups

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

subgroupindex
C2×#56882
#793312
C3×#15342
#717362
#708142
#683692
#683622
#176643