Finite Group Databasebeta

← #10944008

Subgroups — #10944008

Order 768.

1868 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×#5218C212yesmaximal
3C2×#5019C212yesmaximal, complemented
4#82076C211yesmaximal, characteristic, complemented
5#71612C214yesmaximal, complemented
6#71604C212yesmaximal, complemented
7#71559C212yesmaximal, complemented
8#71530C211yesmaximal, characteristic, complemented
9#32905—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4—21no—
12C2×C3×C4×D4—28no—
13C2×C3×C4×D4—21no—
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2212yes—
16C2×C3×C4×D4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4—21no—
19C22×C3×C42C2211yescharacteristic
20C22×#214C2214yes—
21C2×#924C2212yes—
22C2×#869C2211yescharacteristic, complemented
23#5218—28no—
24C2×#832C2214yes—
25#5212—21no—
26#5212—21no—
27C2×#830C2212yes—
28#5019—28no—
29C2×#810C2214yescomplemented
30#4935C2211yescharacteristic, complemented
31#4929C2212yescomplemented
32#4928—21no—
33#4928—21no—
34C2×C4⋊(C3⋊C8)C2212yescomplemented
35#4529C2214yescomplemented
36#4524C2212yescomplemented
37C22×C4×D4D311yescharacteristic
38C2×#461—32no—
39C2×#450—32no—
40#2857—31no—
41#1877—34no—
42#1872—32no—
43#1802—32no—
44#1797—31no—
45C25×C3—21no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4—21no—
49C22×C3×D4—22no—
50C22×C3×D4D414yes—
51C22×C3×D4C2312yes—
52C23×C3×C4—28no—
53C23×C3×C4—21no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—44no—
57C23×C3×C4—44no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×Dic6C2312yes—
62C3×C4×D4—28no—
63C3×C4×D4—28no—
64C3×C4×D4—24no—
65C3×C4×D4—24no—
66C3×C4×D4—24no—
67C3×C4×D4D418yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—28no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—24no—
75C2×C3×C42D411yescharacteristic
76C2×C3×C42D411yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×#217—22no—
80C2×#214—28no—
81C2×#214—28no—
82C2×#214—28no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2312yes—
88#924—24no—
89#924—24no—
90C2×Dic3⋊C4—22no—
91C22×C3⋊C8C2314yes—
92#869—24no—
93#832—28no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96#830—24no—
97#810—28no—
98C4⋊(C3⋊C8)—24no—
99C23×D4C2×D311yescharacteristic
100C24×C4—21no—
101C22×SD16—34no—
102C2×C4⋊Q8—31no—
103C2×C22⋊Q8—32no—
104C2×C4×D4—28no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2×D312yes—
108C2×C4×D4C2×D312yes—
109C22×C4⋊C4C2×D311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C42C2×D311yescharacteristic
112#461—68no—
113#450—68no—
114C2×C4⋊C8—32no—
115C2×#103—34no—
116C2×#102—34no—
117C2×C22⋊C8—32no—
118#391—32no—
119#389—61no—
120#389—61no—
121#384—31no—
122#380—61no—
123#380—61no—
124#340—34no—
125#336—32no—
126C24×C3—24no—
127C24×C3—24no—
128C24×C3—24no—
129C24×C3—22no—
130C24×C3—42no—
131C24×C3—42no—
132C24×C3—42no—
133C24×C3—42no—
134C24×C3C2×D411yescharacteristic
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—216no—
138C2×C3×D4—216no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4C2×D414yes—
144C2×C3×D4C2×D414yes—
145C2×C3×D4#4114yes—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—28no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—28no—
156C22×C3×C4—41no—
157C22×C3×C4—44no—
158C22×C3×C4—432no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—44no—
162C22×C3×C4—44no—
163C22×C3×C4—44no—
164C22×C3×C4SD1614yescomplemented
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—22no—
175C2×Dic6—24no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4C2×D414yes—
182C3×C22⋊C4—28no—
183C3×C22⋊C4—28no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42C2×D414yes—
188#217—44no—
189#214—416no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195C4⋊Dic3—42no—
196Dic3⋊C4—44no—
197C2×C3⋊C8—24no—
198C2×C3⋊C8—24no—
199C2×C3⋊C8—24no—
200C25—21no—
201C22×Q8—32no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—21no—
206C22×D4D1214yes—
207C22×D4C22×D312yes—
208C23×C4—28no—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4#6711yescharacteristic
215C23×C4#6711yescharacteristic
216C23×C4C22×D311yescharacteristic
217C2×SD16—68no—
218C2×SD16—68no—
219C2×SD16—68no—
220C22×C8—34no—
221C4⋊Q8—64no—
222C22⋊Q8—64no—
223C22⋊Q8—64no—
224C4×D4—28no—
225C4×D4—28no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4#6718yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—62no—
243C2×C42—24no—
244C2×C42#6711yescharacteristic
245C2×C42#6711yescharacteristic
246C2×C42C22×D311yescharacteristic
247C2×C42C22×D311yescharacteristic
248C4⋊C8—64no—
249#103—68no—
250#102—68no—
251C22⋊C8—64no—
252#95—62no—
253#95—62no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—216no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—48no—
263C23×C3—48no—
264C23×C3—48no—
265C23×C3—48no—
266C23×C3—48no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—48no—
272C23×C3#12112yes—
273C23×C3C2×SD1612yes—
274C23×C3#13712yes—
275C23×C3C22×D411yescharacteristic
276C3×D4—216no—
277C3×D4—24no—
278C3×D4—24no—
279C3×D4—24no—
280C3×D4#12118yes—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—28no—
290C2×C3×C4—21no—
291C2×C3×C4—24no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—44no—
295C2×C3×C4—416no—
296C2×C3×C4—48no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—416no—
300C2×C3×C4#12012yes—
301C2×C3×C4#12012yes—
302C2×C3×C4#12112yes—
303C2×C3×C4C2×SD1612yes—
304C2×C3×C4C2×SD1612yes—
305C2×C3×C4AGL(1,ℤ/8)12yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D412yes—
308C2×C3×C4C2×#4112yes—
309C2×C3×C4C2×#4111yescharacteristic
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316Dic6—44no—
317Dic6—44no—
318C3⋊C8—44no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24C2×#6711yescharacteristic
328C2×Q8—62no—
329C2×Q8—62no—
330C2×Q8—64no—
331C2×Q8—64no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—216no—
335C2×D4—216no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4C2×D1214yes—
341C2×D4#23514yes—
342C2×D4C2×#6714yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—41no—
354C22×C4—44no—
355C22×C4—432no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4#21414yescomplemented
364C22×C4C2×D1212yes—
365C22×C4#23512yes—
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C23×D311yescharacteristic
372SD16—1216no—
373C2×C8—64no—
374C2×C8—64no—
375C2×C8—64no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—122no—
381C4⋊C4—124no—
382C4⋊C4—122no—
383C4⋊C4—122no—
384C4⋊C4—122no—
385C4⋊C4C2×#6714yes—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—124no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42C2×#6714yes—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—28no—
400C22×C3—432no—
401C22×C3—44no—
402C22×C3—44no—
403C22×C3—44no—
404C22×C3—44no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3#46114yescomplemented
408C22×C3C2×#12011yescharacteristic
409C22×C3C2×#12111yescharacteristic
410C22×C3#52511yescharacteristic
411C22×C3C22×SD1611yescharacteristic
412C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C3C2×#13711yescharacteristic
414C22×C3#57511yescharacteristic
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—44no—
419C3×C4—41no—
420C3×C4#45014yescomplemented
421C3×C4C2×#12112yes—
422C3×C4#52512yes—
423Dic3—82no—
424Dic3—82no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—216no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—48no—
434C23—48no—
435C23—48no—
436C23—48no—
437C23—48no—
438C23—48no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23#93012yes—
444C23C2×#21412yes—
445C23#95112yes—
446C23C22×#6711yescharacteristic
447Q8—124no—
448Q8—124no—
449D4—216no—
450D4—24no—
451D4—24no—
452D4—24no—
453D4#93018yes—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—44no—
468C2×C4—416no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—416no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4#93012yes—
480C2×C4#93212yes—
481C2×C4C2×#21412yes—
482C2×C4C2×#21412yes—
483C2×C4#95312yes—
484C2×C4#95312yes—
485C2×C4C22×D1211yescharacteristic
486C2×C4C2×#23512yes—
487C2×C4C2×#23511yescharacteristic
488C2×C4C22×#6712yes—
489C8—124no—
490C2×C3—24no—
491C2×C3—48no—
492C2×C3#284711yescharacteristic
493C2×C3C2×#45011yescharacteristic
494C2×C3#300711yescharacteristic
495C2×C3C2×#46111yescharacteristic
496C2×C3#303611yescharacteristic
497C2×C3#310111yescharacteristic
498C2×C3#311011yescharacteristic
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—28no—
506C22—432no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22#501914yescomplemented
514C22C2×#93011yescharacteristic
515C22#579111yescharacteristic
516C22C2×#93211yescharacteristic
517C22C22×#21411yescharacteristic
518C22#582211yescharacteristic
519C22C2×#95111yescharacteristic
520C22C2×#95311yescharacteristic
521C4—22no—
522C4—22no—
523C4—22no—
524C4—44no—
525C4—41no—
526C4—242no—
527C4—242no—
528C4#521814yescomplemented
529C4C2×#93012yes—
530C4#579112yes—
531C3#3290511yescharacteristic, complemented, Sylow
532C2—24no—
533C2—48no—
534C2#8230711yescharacteristic
535C2C2×#501911yescharacteristic
536C2#8231411yescharacteristic
537C2#8231511yescharacteristic
538C2#8403211yescharacteristic
539C2C2×#521811yescharacteristic
540C2#8408411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C4×D4C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenterC22×C4×D4D3
hypocenterC3#32905
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
3. C2×#5019C2semidirect
4. #82076C2semidirect
5. #71612C2semidirect
6. #71604C2semidirect
7. #71559C2semidirect
8. #71530C2semidirect
22. C2×#869C22semidirect
29. C2×#810C22semidirect
30. #4935C22semidirect
31. #4929C22semidirect
34. C2×C4⋊(C3⋊C8)C22semidirect
35. #4529C22semidirect
36. #4524C22semidirect
164. C22×C3×C4SD16semidirect
363. C22×C4#214semidirect
407. C22×C3#461semidirect
420. C3×C4#450semidirect
513. C22#5019semidirect
528. C4#5218semidirect
531. C3#32905semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×#52182
C2×#50192
#820762
#716122
#716042
#715592
#715302
#329053