Finite Group Databasebeta

← #10944099

Subgroups — #10944099

Order 768.

1546 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic
2C2×#5218C212yesmaximal
3C2×#5019C212yesmaximal, complemented
4#82262C211yesmaximal, characteristic, complemented
5#71612C214yesmaximal, complemented
6#71605C212yesmaximal, complemented
7#71585C212yesmaximal, complemented
8#71583C211yesmaximal, characteristic, complemented
9#32914—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C3×#518—28no—
12C2×C3×C4×D4—22no—
13C2×C3×C4×D4C2212yes—
14C2×C3×C4×D4C2212yes—
15C2×C3×#117C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C22⋊C4—22no—
18C22×#214C2214yes—
19C2×#924C2212yes—
20C2×#869C2211yescharacteristic, complemented
21#5218—28no—
22C2×#832C2214yes—
23C2×#830C2212yes—
24#5019—28no—
25#4981—22no—
26C2×#810C2214yescomplemented
27#4953—22no—
28#4936C2211yescharacteristic, complemented
29#4929C2212yescomplemented
30C2×C4⋊(C3⋊C8)C2212yescomplemented
31#4529C2214yescomplemented
32#4524C2212yescomplemented
33C2×#518D311yescharacteristic
34C2×#461—32no—
35C2×#450—32no—
36#2621—31no—
37#1877—34no—
38#1873—32no—
39#1804—32no—
40#1798—31no—
41C25×C3—21no—
42C22×C3×D4—24no—
43C22×C3×D4—24no—
44C22×C3×D4—21no—
45C22×C3×D4—22no—
46C22×C3×D4D414yes—
47C22×C3×D4C2312yes—
48C23×C3×C4—22no—
49C23×C3×C4—22no—
50C23×C3×C4C2311yescharacteristic
51C22×Dic6C2312yes—
52C3×C4×D4—216no—
53C3×C4×D4—24no—
54C3×C4×D4—24no—
55C3×C4×D4—24no—
56C3×C4×D4D418yes—
57C3×#117—24no—
58C3×#117—24no—
59C2×C3×C4⋊C4D412yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C22⋊C4—216no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4D412yes—
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×#217—22no—
70C2×#214—28no—
71C2×#214—28no—
72C2×#214—28no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3C2312yes—
75#924—24no—
76#924—24no—
77C2×Dic3⋊C4—22no—
78C22×C3⋊C8C2314yes—
79#869—24no—
80#832—28no—
81C3⋊#95—24no—
82C3⋊#95—22no—
83#830—24no—
84#810—28no—
85C4⋊(C3⋊C8)—24no—
86C23×D4C2×D311yescharacteristic
87C22×SD16—34no—
88C2×C4⋊Q8—31no—
89C2×C22⋊Q8—32no—
90#518—28no—
91C2×C4×D4—22no—
92C2×C4×D4C2×D312yes—
93C2×C4×D4C2×D312yes—
94C2×#117C2×D311yescharacteristic
95C2×#117C2×D311yescharacteristic
96C22×C22⋊C4—22no—
97#461—68no—
98#450—68no—
99C2×C4⋊C8—32no—
100C2×#103—34no—
101C2×#102—34no—
102C2×C22⋊C8—32no—
103#398—62no—
104#397—62no—
105#391—32no—
106#382—31no—
107#340—34no—
108#336—32no—
109C24×C3—24no—
110C24×C3—24no—
111C24×C3—22no—
112C24×C3—24no—
113C24×C3—84no—
114C24×C3C2×D411yescharacteristic
115C2×C3×D4—24no—
116C2×C3×D4—24no—
117C2×C3×D4—216no—
118C2×C3×D4—216no—
119C2×C3×D4—24no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4C2×D414yes—
124C2×C3×D4C2×D414yes—
125C2×C3×D4#4114yes—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—28no—
130C22×C3×C4—28no—
131C22×C3×C4—84no—
132C22×C3×C4—84no—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4112yes—
138C22×C3×C4C2411yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—22no—
141C2×Dic6—24no—
142C2×Dic6—22no—
143C2×Dic6—22no—
144C2×Dic6—24no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4C2×D414yes—
148C3×C22⋊C4—24no—
149C3×C22⋊C4—28no—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—28no—
152C3×C42—24no—
153C3×C42—22no—
154C3×C42C2×D414yes—
155#217—44no—
156#214—416no—
157C4⋊Dic3—22no—
158C4⋊Dic3—22no—
159C4⋊Dic3—44no—
160Dic3⋊C4—44no—
161C2×C3⋊C8—24no—
162C2×C3⋊C8—24no—
163C2×C3⋊C8—24no—
164C25—21no—
165C22×Q8—32no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4D1214yes—
171C22×D4C22×D312yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—68no—
176C2×SD16—68no—
177C2×SD16—68no—
178C22×C8—34no—
179C4⋊Q8—64no—
180C22⋊Q8—64no—
181C22⋊Q8—64no—
182C4×D4—216no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4#6718yes—
187#117—24no—
188#117—24no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4#6712yes—
193C2×C4⋊C4C22×D312yes—
194C2×C22⋊C4—216no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4#6712yes—
200C2×C42—21no—
201C2×C42C22×D311yescharacteristic
202C2×C42C22×D311yescharacteristic
203C4⋊C8—64no—
204#103—68no—
205#102—68no—
206C22⋊C8—64no—
207#95—62no—
208#95—64no—
209C23×C3—216no—
210C23×C3—21no—
211C23×C3—24no—
212C23×C3—24no—
213C23×C3—21no—
214C23×C3—24no—
215C23×C3—24no—
216C23×C3—44no—
217C23×C3—44no—
218C23×C3—88no—
219C23×C3—88no—
220C23×C3—88no—
221C23×C3#12112yes—
222C23×C3C22×D411yescharacteristic
223C3×D4—216no—
224C3×D4—24no—
225C3×D4—24no—
226C3×D4—24no—
227C3×D4#12118yes—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—44no—
237C2×C3×C4—88no—
238C2×C3×C4—88no—
239C2×C3×C4#12014yes—
240C2×C3×C4#12112yes—
241C2×C3×C4C22×D411yescharacteristic
242C2×C3×C4C22×D412yes—
243C2×C3×C4C2×#4112yes—
244C2×C3×C4C2×#4111yescharacteristic
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—44no—
250Dic6—44no—
251Dic6—44no—
252C3⋊C8—44no—
253C24—24no—
254C24—24no—
255C24—22no—
256C24—24no—
257C24—84no—
258C24C2×#6711yescharacteristic
259C2×Q8—62no—
260C2×Q8—62no—
261C2×Q8—64no—
262C2×Q8—64no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—216no—
266C2×D4—216no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4C2×D1214yes—
272C2×D4#23514yes—
273C2×D4C2×#6714yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—84no—
282C22×C4—84no—
283C22×C4C2×D1212yes—
284C22×C4#23512yes—
285C22×C4C2×#6712yes—
286C22×C4C2×#6711yescharacteristic
287C22×C4C2×#6712yes—
288C22×C4C23×D311yescharacteristic
289SD16—1216no—
290C2×C8—64no—
291C2×C8—64no—
292C2×C8—64no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—62no—
296C4⋊C4—62no—
297C4⋊C4—124no—
298C4⋊C4—124no—
299C4⋊C4C2×#6714yes—
300C22⋊C4—24no—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—124no—
305C42—24no—
306C42—22no—
307C42C2×#6714yes—
308C22×C3—22no—
309C22×C3—28no—
310C22×C3—24no—
311C22×C3—42no—
312C22×C3—48no—
313C22×C3—44no—
314C22×C3—44no—
315C22×C3—42no—
316C22×C3—48no—
317C22×C3—816no—
318C22×C3C2×#12011yescharacteristic
319C22×C3C2×#12111yescharacteristic
320C22×C3#52511yescharacteristic
321C22×C3#57714yes—
322C3×C4—22no—
323C3×C4—24no—
324C3×C4—42no—
325C3×C4—44no—
326C3×C4C2×#12112yes—
327C3×C4#52512yes—
328Dic3—82no—
329Dic3—82no—
330C23—216no—
331C23—24no—
332C23—21no—
333C23—24no—
334C23—24no—
335C23—21no—
336C23—24no—
337C23—44no—
338C23—44no—
339C23—88no—
340C23—88no—
341C23—88no—
342C23#93012yes—
343C23C22×#6711yescharacteristic
344Q8—124no—
345Q8—124no—
346D4—216no—
347D4—24no—
348D4—24no—
349D4—24no—
350D4#93018yes—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—44no—
360C2×C4—88no—
361C2×C4—88no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—124no—
366C2×C4—122no—
367C2×C4#93012yes—
368C2×C4#95314yes—
369C2×C4C22×D1211yescharacteristic
370C2×C4C2×#23512yes—
371C2×C4C2×#23511yescharacteristic
372C2×C4C22×#6712yes—
373C8—124no—
374C2×C3—42no—
375C2×C3—48no—
376C2×C3#284711yescharacteristic
377C2×C3#300912yes—
378C2×C3#304212yes—
379C2×C3#312312yes—
380C22—22no—
381C22—28no—
382C22—24no—
383C22—48no—
384C22—42no—
385C22—48no—
386C22—44no—
387C22—44no—
388C22—42no—
389C22—816no—
390C22C2×#93011yescharacteristic
391C22#579111yescharacteristic
392C22#583614yes—
393C22C2×#95311yescharacteristic
394C4—22no—
395C4—24no—
396C4—42no—
397C4—44no—
398C4—242no—
399C4—242no—
400C4C2×#93012yes—
401C4#579112yes—
402C3#3291411yescharacteristic, complemented, Sylow
403C2—42no—
404C2—48no—
405C2#8254412yes—
406C2#8254512yes—
407C2#8403211yescharacteristic
408C2#8410512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#518C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenterC2×#518D3
hypocenterC3#32914
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
3. C2×#5019C2semidirect
4. #82262C2semidirect
5. #71612C2semidirect
6. #71605C2semidirect
7. #71585C2semidirect
8. #71583C2semidirect
20. C2×#869C22semidirect
26. C2×#810C22semidirect
28. #4936C22semidirect
29. #4929C22semidirect
30. C2×C4⋊(C3⋊C8)C22semidirect
31. #4529C22semidirect
32. #4524C22semidirect
402. C3#32914semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×#52182
C2×#50192
#822622
#716122
#716052
#715852
#715832
#329143