Finite Group Databasebeta

← #10822593

Subgroups — #10822593

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic, complemented
2#82267C211yesmaximal, characteristic
3#82267C212yesmaximal
4C3×#1740C211yesmaximal, characteristic, complemented
5#71593C212yesmaximal, complemented
6#68971C214yesmaximal
7#68956C214yesmaximal, complemented
8#23467—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×#117C2212yes—
11C2×C3×#117C2211yescharacteristic
12C2×#926C2211yescharacteristic
13C2×#926C2212yes—
14#5519—24no—
15#5517—24no—
16#5500—22no—
17#5483C2214yes—
18#5470—22no—
19#4987—22no—
20#4987—24no—
21#4955C2212yes—
22#4955C2214yes—
23#4943C2212yes—
24#4943C2214yes—
25C4×(C2×D3)⋊C4—22no—
26C4×(C2×D3)⋊C4—24no—
27C3×#342C2214yes—
28#4527C2218yes—
29#3458—31no—
30#2280—32no—
31#2280—31no—
32#2030—34no—
33#2029—34no—
34#1740—32no—
35#1740D311yescharacteristic, complemented
36C24×D3C2311yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3C2311yescharacteristic
39C22×C4×D3—21no—
40C22×C4×D3—22no—
41C22×C4×D3—21no—
42C22×C4×D3—22no—
43C22×C4×D3—24no—
44C22×C4×D3—22no—
45C22×C4×D3Q814yescomplemented
46C3×#117—22no—
47C3×#117—22no—
48C3×#117—24no—
49C3×#117—24no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4D412yes—
52C2×C3×C22⋊C4C2314yes—
53C2×C3×C22⋊C4C2312yes—
54C2×C3×C42—22no—
55C2×C3×C42D412yes—
56C2×#217C2312yes—
57C2×#217C2314yes—
58C2×(C2×D3)⋊C4—24no—
59C2×(C2×D3)⋊C4—28no—
60#926—22no—
61#926—22no—
62#926—22no—
63#926—22no—
64#926—24no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D414yes—
67#924—28no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3D414yes—
70#890—28no—
71#888—28no—
72#878—216no—
73#870—24no—
74#869—24no—
75C4×Dic6—28no—
76C3⋊#95—24no—
77C3⋊#95—28no—
78C24×C4—31no—
79#544—64no—
80#544—62no—
81#543—34no—
82#533—64no—
83#533—62no—
84C2×#117—32no—
85C2×#117—31no—
86C2×#117C2×D312yes—
87C2×#117C2×D311yescharacteristic
88#381—34no—
89#381—32no—
90#380—64no—
91#380—62no—
92#379—34no—
93#379—32no—
94C4×C22⋊C4—64no—
95C4×C22⋊C4—62no—
96#342—38no—
97#342C2×D314yes—
98C24×C3C2411yescharacteristic
99C23×D3—21no—
100C23×D3—22no—
101C23×D3—22no—
102C23×D3—21no—
103C23×D3—44no—
104C23×D3C2×Q812yes—
105C22×C3×C4—24no—
106C22×C3×C4C2×D414yes—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×Q812yes—
112C22×C3×C4#4114yes—
113C22×C3×C4#4112yes—
114C22×Dic3—21no—
115C22×Dic3—22no—
116C22×Dic3—22no—
117C22×Dic3—21no—
118C22×Dic3—22no—
119C22×Dic3C2×D418yes—
120C22×Dic3C2×Q812yes—
121C2×C4×D3—28no—
122C2×C4×D3—28no—
123C2×C4×D3—24no—
124C2×C4×D3—24no—
125C2×C4×D3—42no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—42no—
129C2×C4×D3—42no—
130C2×C4×D3—44no—
131C2×C4×D3—44no—
132C2×Dic6—28no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4—44no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—24no—
138C3×C42—22no—
139C3×C42—22no—
140C3×C42—44no—
141#217—24no—
142#217—24no—
143#217—24no—
144(C2×D3)⋊C4—416no—
145(C2×D3)⋊C4—48no—
146C4⋊Dic3—28no—
147C4⋊Dic3—24no—
148C4⋊Dic3—24no—
149C4⋊Dic3—42no—
150Dic3⋊C4—216no—
151C4×Dic3—24no—
152C4×Dic3—24no—
153C4×Dic3—42no—
154C25—31no—
155C23×C4—31no—
156C23×C4—34no—
157C23×C4—61no—
158C23×C4—61no—
159C23×C4—64no—
160C23×C4—62no—
161C23×C4—62no—
162C23×C4—62no—
163C23×C4C22×D311yescharacteristic
164C4⋊Q8—64no—
165C4⋊Q8—68no—
166#125—64no—
167#125—68no—
168C22⋊Q8—616no—
169C22⋊Q8—68no—
170C4×Q8—68no—
171#117—22no—
172#117—22no—
173#117—24no—
174#117—24no—
175#117—62no—
176#117—62no—
177#117—62no—
178#117—62no—
179#117—64no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—34no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4D1212yes—
184C2×C22⋊C4—34no—
185C2×C22⋊C4—32no—
186C2×C22⋊C4—68no—
187C2×C22⋊C4—64no—
188C2×C22⋊C4C22×D314yes—
189C2×C22⋊C4C22×D312yes—
190C2×C42—22no—
191C2×C42—34no—
192C2×C42—61no—
193C2×C42D1212yes—
194#95—68no—
195#95—64no—
196C23×C3—42no—
197C23×C3C22×D411yescharacteristic
198C23×C3C22×D412yes—
199C23×C3C22×Q811yescharacteristic
200C23×C3C2×#4112yes—
201C23×C3C2×#4111yescharacteristic
202C22×D3—24no—
203C22×D3—28no—
204C22×D3—24no—
205C22×D3—22no—
206C22×D3—44no—
207C22×D3—48no—
208C22×D3—44no—
209C22×D3—42no—
210C22×D3—44no—
211C22×D3—41no—
212C22×D3—41no—
213C22×D3—42no—
214C2×C3×C4—24no—
215C2×C3×C4—44no—
216C2×C3×C4—48no—
217C2×C3×C4—44no—
218C2×C3×C4#12012yes—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4C22⋊Q814yes—
222C2×C3×C4C22⋊Q814yes—
223C2×C3×C4C22⋊Q812yes—
224C2×C3×C4C22⋊Q812yes—
225C2×C3×C4#12412yes—
226C2×C3×C4#12412yes—
227C2×C3×C4#12414yes—
228C2×C3×C4#12414yes—
229C2×Dic3—28no—
230C2×Dic3—24no—
231C2×Dic3—22no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—48no—
236C2×Dic3—44no—
237C2×Dic3—42no—
238C4×D3—48no—
239C4×D3—44no—
240C4×D3—44no—
241Dic6—88no—
242C24—32no—
243C24—61no—
244C24—62no—
245C24—62no—
246C24—61no—
247C24—124no—
248C24C23×D311yescharacteristic
249C2×Q8—68no—
250C22×C4—24no—
251C22×C4—32no—
252C22×C4—38no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—62no—
256C22×C4—61no—
257C22×C4—68no—
258C22×C4—68no—
259C22×C4—64no—
260C22×C4—64no—
261C22×C4—62no—
262C22×C4—122no—
263C22×C4—122no—
264C22×C4—121no—
265C22×C4—122no—
266C22×C4—121no—
267C22×C4—124no—
268C22×C4—124no—
269C22×C4C2×D1214yes—
270C22×C4C2×D1211yescharacteristic
271C22×C4C2×D1211yescharacteristic
272C22×C4#23514yes—
273C22×C4D3×D412yes—
274C22×C4D3×D412yes—
275C22×C4#23712yes—
276C22×C4D3×Q812yes—
277C4⋊C4—24no—
278C4⋊C4—44no—
279C4⋊C4—616no—
280C4⋊C4—68no—
281C4⋊C4—64no—
282C4⋊C4—64no—
283C4⋊C4—122no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—64no—
288C22⋊C4—64no—
289C22⋊C4—64no—
290C22⋊C4—128no—
291C22⋊C4—1216no—
292C42—22no—
293C42—22no—
294C42—44no—
295C42—64no—
296C42—64no—
297C42—122no—
298C22×C3—22no—
299C22×C3—24no—
300C22×C3—41no—
301C22×C3—42no—
302C22×C3—41no—
303C22×C3C2×#12011yescharacteristic
304C22×C3C2×C22⋊Q812yes—
305C22×C3C2×C22⋊Q811yescharacteristic
306C22×C3C2×#12411yescharacteristic
307C22×C3C2×#12412yes—
308C2×D3—28no—
309C2×D3—24no—
310C2×D3—44no—
311C2×D3—48no—
312C2×D3—44no—
313C2×D3—42no—
314C2×D3—42no—
315C2×D3—44no—
316C3×C4—44no—
317C3×C4—84no—
318C3×C4#39314yes—
319C3×C4#39312yes—
320C3×C4#39312yes—
321Dic3—42no—
322Dic3—48no—
323Dic3—82no—
324C23—42no—
325C23—64no—
326C23—68no—
327C23—64no—
328C23—62no—
329C23—122no—
330C23—124no—
331C23—124no—
332C23—128no—
333C23—124no—
334C23—121no—
335C23—121no—
336C23—122no—
337C23C22×D1211yescharacteristic
338C23C2×#23512yes—
339C23C2×D3×D412yes—
340C23C2×#23711yescharacteristic
341C23C2×D3×Q811yescharacteristic
342Q8—248no—
343C2×C4—24no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—68no—
348C2×C4—64no—
349C2×C4—62no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—122no—
353C2×C4—124no—
354C2×C4—128no—
355C2×C4—124no—
356C2×C4—128no—
357C2×C4—124no—
358C2×C4—122no—
359C2×C4#87512yes—
360C2×C4#87512yes—
361C2×C4#88211yescharacteristic
362C2×C4#88212yes—
363C2×C4#88211yescharacteristic
364C2×C4#88514yes—
365C2×C4#88514yes—
366C2×C4#89614yes—
367C2×C4#89614yes—
368C2×C4#89712yes—
369C2×C4#89712yes—
370C2×C3—21no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3C2×#39311yescharacteristic
375C2×C3#302512yes—
376D3—44no—
377C22—22no—
378C22—24no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—68no—
383C22—64no—
384C22—128no—
385C22—124no—
386C22—124no—
387C22—122no—
388C22—124no—
389C22—122no—
390C22C2×#87511yescharacteristic
391C22C2×#88211yescharacteristic
392C22C2×#88512yes—
393C22C2×#89612yes—
394C22C2×#89711yescharacteristic
395C4—44no—
396C4—84no—
397C4—128no—
398C4—122no—
399C4—242no—
400C4#467312yes—
401C4#467314yes—
402C4#467312yes—
403C3#2346711yescharacteristic, complemented, Sylow
404C2—22no—
405C2—21no—
406C2—22no—
407C2—21no—
408C2—124no—
409C2C2×#467311yescharacteristic
410C2#7869012yes—
subgroupG/H
centerC2×C4#882
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#1740C2
socleC22×C3C2×#120
prime radicalC24×D3C23
hypercenter#1740D3
hypocenterC3#23467
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
1. #85725C2semidirect
4. C3×#1740C2semidirect
5. #71593C2semidirect
7. #68956C2semidirect
35. #1740D3semidirect
45. C22×C4×D3Q8semidirect
403. C3#23467semidirect

Maximal subgroups

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

subgroupindex
#857252
#822672
C3×#17402
#715932
#689712
#689562
#234673