Finite Group Databasebeta

← #10742702

Subgroups — #10742702

Order 768.

3730 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic, complemented
2C22×#937C211yesmaximal, characteristic
3C2×#5213C212yesmaximal, complemented
4C2×#4955C212yesmaximal, complemented
5#72131C211yesmaximal, characteristic, complemented
6#71570C214yesmaximal, complemented
7#71550C212yesmaximal, complemented
8#71548C212yesmaximal, complemented
9#15948—31nomaximal, Sylow
10C25×D3C2211yescharacteristic
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic, complemented
13C23×#67C2212yes—
14C2×C3×#123—24no—
15C2×C3×#123—28no—
16C22×C3×C4⋊C4C2212yescomplemented
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic, complemented
19C22×#217C2211yescharacteristic
20C22×#217C2212yes—
21C22×#217C2212yescomplemented
22C2×#937—24no—
23C2×#937—28no—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2214yes—
26C22×(C2×D3)⋊C4C2214yes—
27#5213—28no—
28C2×C3⋊#95C2214yescomplemented
29C2×C3⋊#95C2214yescomplemented
30C2×C3⋊#95C2212yescomplemented
31#4955—28no—
32C22×#123D311yescharacteristic, complemented
33C22×#120—31no—
34C2×#386—32no—
35C2×#379—32no—
36#1440—32no—
37#1439—34no—
38#1438—31no—
39#1438—32no—
40C25×C3C2311yescharacteristic
41C25×C3C2311yescharacteristic
42C24×D3—24no—
43C24×D3—22no—
44C24×D3—24no—
45C24×D3—28no—
46C24×D3—42no—
47C24×D3—44no—
48C24×D3C2312yes—
49C22×C3×D4—24no—
50C22×C3×D4—24no—
51C22×C3×D4—42no—
52C22×C3×D4D414yes—
53C23×C3×C4—22no—
54C23×C3×C4—24no—
55C23×C3×C4—24no—
56C23×C3×C4—42no—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C22×#67—24no—
61C22×#67—24no—
62C22×#67—28no—
63C22×#67—44no—
64C22×#67D418yes—
65C23×Dic3C2314yes—
66C23×Dic3C2312yes—
67C3×#123—232no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4D418yescomplemented
76C2×#217—22no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—44no—
81C2×#217—44no—
82#937—232no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—28no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—28no—
90C2×(C2×D3)⋊C4—44no—
91C2×(C2×D3)⋊C4—44no—
92C3⋊#95—28no—
93C3⋊#95—44no—
94C3⋊#95—42no—
95C26—31no—
96C23×D4—32no—
97C23×D4C2×D311yescharacteristic
98C24×C4C2×D311yescharacteristic, complemented
99C2×#123—24no—
100C2×#123—28no—
101C2×#120—68no—
102C2×#120—64no—
103C22×C4⋊C4C2×D312yescomplemented
104C22×C22⋊C4—31no—
105C22×C22⋊C4—32no—
106C22×C22⋊C4—34no—
107C22×C22⋊C4—34no—
108C22×C22⋊C4—32no—
109C22×C22⋊C4—32no—
110C22×C22⋊C4C2×D312yes—
111C22×C22⋊C4C2×D311yescharacteristic, complemented
112#386—68no—
113#379—68no—
114C2×#95—34no—
115C2×#95—32no—
116C2×#95—34no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—24no—
120C24×C3—24no—
121C24×C3—24no—
122C24×C3—24no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3C2×D414yes—
126C24×C3C2×D412yes—
127C24×C3C2×D414yes—
128C24×C3C2×D412yes—
129C24×C3C2411yescharacteristic
130C23×D3—24no—
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—24no—
134C23×D3—24no—
135C23×D3—24no—
136C23×D3—24no—
137C23×D3—28no—
138C23×D3—216no—
139C23×D3—416no—
140C23×D3—48no—
141C23×D3—416no—
142C23×D3—416no—
143C23×D3—416no—
144C23×D3—48no—
145C23×D3—416no—
146C23×D3—44no—
147C23×D3—48no—
148C23×D3—48no—
149C23×D3—44no—
150C23×D3—48no—
151C23×D3C2×D414yes—
152C23×D3#4118yes—
153C2×C3×D4—28no—
154C2×C3×D4—28no—
155C2×C3×D4—44no—
156C2×C3×D4—416no—
157C22×C3×C4—28no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—416no—
170C22×C3×C4—44no—
171C22×C3×C4—44no—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D412yes—
175C2×#67—216no—
176C2×#67—216no—
177C2×#67—48no—
178C2×#67—416no—
179C2×#67—416no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3C2×D414yes—
192C3×C4⋊C4—216no—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—216no—
195#217—28no—
196#217—48no—
197#217—48no—
198(C2×D3)⋊C4—216no—
199(C2×D3)⋊C4—416no—
200(C2×D3)⋊C4—416no—
201C25—32no—
202C25—64no—
203C25—68no—
204C25—62no—
205C25—64no—
206C25—122no—
207C25—124no—
208C25C22×D311yescharacteristic
209C25C22×D311yescharacteristic
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—38no—
213C22×D4—42no—
214C22×D4—68no—
215C22×D4—64no—
216C22×D4—64no—
217C22×D4—124no—
218C22×D4#6714yes—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—32no—
223C23×C4—34no—
224C23×C4—42no—
225C23×C4C22×D312yes—
226C23×C4C22×D312yes—
227C23×C4C22×D311yescharacteristic
228#123—232no—
229#120—632no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—64no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4—68no—
243C2×C22⋊C4—68no—
244C2×C22⋊C4—64no—
245C2×C22⋊C4—64no—
246C2×C22⋊C4—68no—
247C2×C22⋊C4—68no—
248C2×C22⋊C4—124no—
249C2×C22⋊C4—124no—
250C2×C22⋊C4—124no—
251C2×C22⋊C4—124no—
252C2×C22⋊C4#6718yescomplemented
253#95—68no—
254#95—122no—
255#95—124no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—24no—
264C23×C3—28no—
265C23×C3—28no—
266C23×C3—28no—
267C23×C3—28no—
268C23×C3—28no—
269C23×C3—28no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—44no—
275C23×C3—44no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3#12014yes—
283C23×C3#12014yes—
284C23×C3C22×D412yes—
285C23×C3C22×D411yescharacteristic
286C23×C3C22×D412yes—
287C23×C3C2×#4114yes—
288C23×C3H(5,𝔽2)12yes—
289C23×C3H(5,𝔽2)14yes—
290C22×D3—24no—
291C22×D3—28no—
292C22×D3—24no—
293C22×D3—28no—
294C22×D3—28no—
295C22×D3—28no—
296C22×D3—28no—
297C22×D3—28no—
298C22×D3—28no—
299C22×D3—28no—
300C22×D3—432no—
301C22×D3—464no—
302C22×D3—416no—
303C22×D3—432no—
304C22×D3—464no—
305C22×D3—432no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—44no—
311C22×D3—44no—
312C22×D3—44no—
313C22×D3—44no—
314C22×D3—44no—
315C22×D3—44no—
316C22×D3#12318yescomplemented
317C3×D4—416no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—28no—
324C2×C3×C4—28no—
325C2×C3×C4—416no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—44no—
331C2×C3×C4—44no—
332C2×C3×C4—44no—
333#67—432no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—44no—
338C2×Dic3—44no—
339C2×Dic3—44no—
340C2×Dic3—44no—
341C2×Dic3—44no—
342C2×Dic3—44no—
343C2×Dic3—44no—
344C2×Dic3—44no—
345C2×Dic3—44no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—34no—
355C24—38no—
356C24—68no—
357C24—616no—
358C24—62no—
359C24—64no—
360C24—64no—
361C24—64no—
362C24—64no—
363C24—64no—
364C24—62no—
365C24—1216no—
366C24—1216no—
367C24—1216no—
368C24—124no—
369C24—128no—
370C24—128no—
371C24—1216no—
372C24—128no—
373C24—128no—
374C24—1216no—
375C24—128no—
376C24—124no—
377C24D3×D414yes—
378C24C2×#6712yes—
379C24C2×#6712yes—
380C24C2×#6714yes—
381C24C23×D311yescharacteristic
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—44no—
385C2×D4—416no—
386C2×D4—616no—
387C2×D4—616no—
388C2×D4—128no—
389C2×D4—1216no—
390C2×D4—1216no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—34no—
404C22×C4—416no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—62no—
408C22×C4—64no—
409C22×C4—64no—
410C22×C4—62no—
411C22×C4—64no—
412C22×C4—64no—
413C22×C4—64no—
414C22×C4—64no—
415C22×C4—64no—
416C22×C4—64no—
417C22×C4—64no—
418C22×C4C2×#6714yes—
419C22×C4C2×#6714yes—
420C22×C4C2×#6712yes—
421C4⋊C4—216no—
422C22⋊C4—28no—
423C22⋊C4—216no—
424C22⋊C4—616no—
425C22⋊C4—68no—
426C22⋊C4—1216no—
427C22⋊C4—1216no—
428C22⋊C4—128no—
429C22⋊C4—128no—
430C22×C3—24no—
431C22×C3—24no—
432C22×C3—28no—
433C22×C3—28no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—48no—
437C22×C3—48no—
438C22×C3—48no—
439C22×C3—48no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3C2×#12011yescharacteristic
443C22×C3C2×#12011yescharacteristic
444C22×C3C2×#12312yes—
445C22×C3#52512yes—
446C22×C3#53412yes—
447C22×C3#53411yescharacteristic
448C22×C3#53512yes—
449C22×C3#53814yes—
450C22×C3D4214yes—
451C22×C3#54614yes—
452C22×C3#54614yes—
453C22×C3#54614yes—
454C22×C3#55112yes—
455C22×C3#56012yes—
456C2×D3—28no—
457C2×D3—216no—
458C2×D3—464no—
459C2×D3—44no—
460C2×D3—48no—
461C2×D3—44no—
462C2×D3—48no—
463C2×D3—48no—
464C2×D3—48no—
465C2×D3—48no—
466C3×C4—44no—
467C3×C4—42no—
468C3×C4—44no—
469Dic3—44no—
470Dic3—84no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—22no—
475C23—28no—
476C23—28no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—22no—
485C23—38no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—42no—
498C23—64no—
499C23—68no—
500C23—68no—
501C23—68no—
502C23—68no—
503C23—64no—
504C23—68no—
505C23—68no—
506C23—68no—
507C23—68no—
508C23—1216no—
509C23—1232no—
510C23—1232no—
511C23—1264no—
512C23—1264no—
513C23—1232no—
514C23—122no—
515C23—122no—
516C23—124no—
517C23—124no—
518C23—124no—
519C23—122no—
520C23—124no—
521C23—124no—
522C23—124no—
523C23—122no—
524C23#93714yes—
525C23#95314yes—
526C23C2×D3×D412yes—
527C23#100412yes—
528C23D3×#4114yes—
529C23#100914yes—
530C23C22×#6711yescharacteristic
531C23C22×#6712yes—
532D4—416no—
533D4—1232no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—124no—
553C2×C4—124no—
554C2×C4—124no—
555C2×C4—124no—
556C2×C4—124no—
557C2×C4—124no—
558C2×C4—124no—
559C2×C4—124no—
560C2×C4—124no—
561C2×C3—44no—
562C2×C3—44no—
563C2×C3#240311yescharacteristic
564C2×C3#240412yes—
565C2×C3#240512yes—
566C2×C3#241814yes—
567C2×C3#267814yes—
568C2×C3#269712yes—
569D3—48no—
570C22—24no—
571C22—24no—
572C22—28no—
573C22—28no—
574C22—44no—
575C22—44no—
576C22—48no—
577C22—48no—
578C22—48no—
579C22—48no—
580C22—44no—
581C22—44no—
582C22—68no—
583C22—616no—
584C22—1264no—
585C22—124no—
586C22—128no—
587C22—124no—
588C22—128no—
589C22—128no—
590C22—128no—
591C22—128no—
592C22D3×#12312yes—
593C22#565312yes—
594C22#565414yes—
595C22#565714yes—
596C22#565912yes—
597C22#566612yes—
598C22C2×#93711yescharacteristic
599C22D4×#6714yes—
600C22#580411yescharacteristic
601C22#582812yes—
602C22#582914yes—
603C22#582914yes—
604C22#583012yes—
605C22C2×#95311yescharacteristic
606C4—44no—
607C4—42no—
608C4—44no—
609C4—124no—
610C4—244no—
611C3#1594811yescharacteristic, complemented, Sylow
612C2—44no—
613C2—44no—
614C2—128no—
615C2#8289312yes—
616C2#8289614yes—
617C2#8290714yes—
618C2#8290812yes—
619C2#8405211yescharacteristic
620C2#8409612yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#123C2
prime radicalC22×#937C2
hypercenterC22×#123D3
hypocenterC3#15948
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C22×C3×#123C2semidirect
3. C2×#5213C2semidirect
4. C2×#4955C2semidirect
5. #72131C2semidirect
6. #71570C2semidirect
7. #71550C2semidirect
8. #71548C2semidirect
12. C24×C3×C4C22semidirect
16. C22×C3×C4⋊C4C22semidirect
18. C22×C3×C22⋊C4C22semidirect
21. C22×#217C22semidirect
28. C2×C3⋊#95C22semidirect
29. C2×C3⋊#95C22semidirect
30. C2×C3⋊#95C22semidirect
32. C22×#123D3semidirect
75. C2×C3×C22⋊C4D4semidirect
98. C24×C4C2×D3semidirect
103. C22×C4⋊C4C2×D3semidirect
111. C22×C22⋊C4C2×D3semidirect
252. C2×C22⋊C4#67semidirect
316. C22×D3#123semidirect
611. C3#15948semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C22×#9372
C2×#52132
C2×#49552
#721312
#715702
#715502
#715482
#159483