Finite Group Databasebeta

← #10742706

Subgroups — #10742706

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×#120C211yesmaximal, characteristic
2C22×#935C211yesmaximal, characteristic, complemented
3C2×#5247C212yesmaximal, complemented
4C2×#5220C212yesmaximal, complemented
5#72131C212yesmaximal, complemented
6#72131C211yesmaximal, characteristic, complemented
7#71549C212yesmaximal, complemented
8#71547C214yesmaximal, complemented
9#15948—31nomaximal, Sylow
10C26×C3C2211yescharacteristic
11C23×C3×D4C2212yes—
12C23×C3×D4C2211yescharacteristic
13C24×Dic3C2211yescharacteristic, complemented
14C2×C3×#120—28no—
15C2×C3×#120—24no—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2214yes—
19C22×#217C2214yes—
20C22×#217C2212yes—
21C22×#217C2212yes—
22C22×#217C2211yescharacteristic, complemented
23C22×#217C2212yescomplemented
24C2×#935—24no—
25C2×#935—28no—
26C22×Dic3⋊C4C2212yescomplemented
27#5247—28no—
28#5220—28no—
29C2×C3⋊#95C2214yescomplemented
30C2×C3⋊#95C2212yescomplemented
31C2×C3⋊#95C2214yescomplemented
32C22×#123—31no—
33C22×#120D311yescharacteristic
34C2×#386—32no—
35C2×#379—32no—
36#1440—32no—
37#1439—34no—
38#1438—32no—
39#1438—31no—
40C25×C3—24no—
41C25×C3—22no—
42C25×C3—24no—
43C25×C3—28no—
44C25×C3—44no—
45C25×C3—42no—
46C25×C3C2311yescharacteristic
47C25×C3C2311yescharacteristic
48C25×C3C2312yes—
49C22×C3×D4—24no—
50C22×C3×D4—24no—
51C22×C3×D4—24no—
52C22×C3×D4—24no—
53C22×C3×D4—28no—
54C22×C3×D4—44no—
55C22×C3×D4—42no—
56C22×C3×D4D418yes—
57C22×C3×D4D414yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×Dic3—22no—
61C23×Dic3—24no—
62C23×Dic3—24no—
63C23×Dic3—42no—
64C23×Dic3C2314yes—
65C23×Dic3C2312yes—
66C23×Dic3C2312yes—
67C3×#120—232no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×#217—24no—
74C2×#217—24no—
75C2×#217—22no—
76C2×#217—24no—
77C2×#217—28no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—28no—
81C2×#217—28no—
82C2×#217—28no—
83C2×#217—44no—
84C2×#217—44no—
85C2×#217—44no—
86C2×#217—44no—
87C2×#217D418yescomplemented
88#935—232no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C3⋊#95—28no—
93C3⋊#95—44no—
94C3⋊#95—42no—
95C26C2×D311yescharacteristic
96C23×D4C2×D311yescharacteristic
97C23×D4C2×D312yes—
98C24×C4—31no—
99C2×#123—68no—
100C2×#123—64no—
101C2×#120—24no—
102C2×#120—28no—
103C22×C4⋊C4—32no—
104C22×C22⋊C4—32no—
105C22×C22⋊C4—31no—
106C22×C22⋊C4—34no—
107C22×C22⋊C4—34no—
108C22×C22⋊C4—32no—
109C22×C22⋊C4—32no—
110C22×C22⋊C4C2×D312yes—
111C22×C22⋊C4C2×D311yescharacteristic
112#386—68no—
113#379—68no—
114C2×#95—34no—
115C2×#95—32no—
116C2×#95—34no—
117C24×C3—24no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—24no—
123C24×C3—24no—
124C24×C3—24no—
125C24×C3—24no—
126C24×C3—24no—
127C24×C3—24no—
128C24×C3—22no—
129C24×C3—24no—
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—216no—
133C24×C3—28no—
134C24×C3—416no—
135C24×C3—416no—
136C24×C3—416no—
137C24×C3—416no—
138C24×C3—48no—
139C24×C3—44no—
140C24×C3—48no—
141C24×C3—48no—
142C24×C3—416no—
143C24×C3—48no—
144C24×C3—44no—
145C24×C3—48no—
146C24×C3C2×D414yes—
147C24×C3C2×D412yes—
148C24×C3C2×D412yes—
149C24×C3C2×D414yes—
150C24×C3C2×D414yes—
151C24×C3#4118yes—
152C24×C3C2411yescharacteristic
153C2×C3×D4—28no—
154C2×C3×D4—216no—
155C2×C3×D4—28no—
156C2×C3×D4—216no—
157C2×C3×D4—48no—
158C2×C3×D4—44no—
159C2×C3×D4—416no—
160C2×C3×D4—416no—
161C2×C3×D4—416no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2×D412yes—
171C22×Dic3—28no—
172C22×Dic3—24no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—24no—
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—416no—
188C22×Dic3—44no—
189C22×Dic3—44no—
190C22×Dic3C2×D414yes—
191C22×Dic3C2×D414yes—
192C3×C22⋊C4—216no—
193C3×C22⋊C4—28no—
194#217—28no—
195#217—216no—
196#217—416no—
197#217—416no—
198#217—48no—
199#217—48no—
200Dic3⋊C4—216no—
201C25—24no—
202C25—28no—
203C25—24no—
204C25—22no—
205C25—42no—
206C25—44no—
207C25C22×D311yescharacteristic
208C25C22×D312yes—
209C25C22×D311yescharacteristic
210C22×D4—24no—
211C22×D4—28no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—44no—
216C22×D4—42no—
217C22×D4#6714yes—
218C22×D4#6718yes—
219C23×C4—32no—
220C23×C4—32no—
221C23×C4—34no—
222C23×C4—64no—
223C23×C4—62no—
224C23×C4—64no—
225C23×C4—122no—
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D312yes—
228#123—632no—
229#120—232no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—64no—
232C2×C4⋊C4—64no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—38no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—64no—
243C2×C22⋊C4—68no—
244C2×C22⋊C4—68no—
245C2×C22⋊C4—64no—
246C2×C22⋊C4—68no—
247C2×C22⋊C4—68no—
248C2×C22⋊C4—64no—
249C2×C22⋊C4—124no—
250C2×C22⋊C4—124no—
251C2×C22⋊C4—124no—
252C2×C22⋊C4—124no—
253#95—68no—
254#95—124no—
255#95—122no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—28no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—28no—
268C23×C3—28no—
269C23×C3—28no—
270C23×C3—28no—
271C23×C3—28no—
272C23×C3—28no—
273C23×C3—28no—
274C23×C3—28no—
275C23×C3—28no—
276C23×C3—28no—
277C23×C3—28no—
278C23×C3—28no—
279C23×C3—28no—
280C23×C3—416no—
281C23×C3—432no—
282C23×C3—432no—
283C23×C3—464no—
284C23×C3—464no—
285C23×C3—432no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3—44no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—44no—
307C23×C3—42no—
308C23×C3#12014yes—
309C23×C3#12014yes—
310C23×C3#12318yescomplemented
311C23×C3C22×D412yes—
312C23×C3C22×D411yescharacteristic
313C23×C3C22×D412yes—
314C23×C3C2×#4114yes—
315C23×C3H(5,𝔽2)14yes—
316C23×C3H(5,𝔽2)12yes—
317C3×D4—416no—
318C3×D4—432no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—44no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—28no—
331C2×Dic3—28no—
332C2×Dic3—416no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
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—24no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—28no—
362C24—216no—
363C24—416no—
364C24—416no—
365C24—416no—
366C24—48no—
367C24—416no—
368C24—48no—
369C24—44no—
370C24—48no—
371C24—416no—
372C24—48no—
373C24—48no—
374C24—44no—
375C24D3×D414yes—
376C24#23718yes—
377C24C2×#6714yes—
378C24C2×#6712yes—
379C24C2×#6714yes—
380C24C2×#6712yes—
381C24C23×D311yescharacteristic
382C2×D4—216no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—216no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—416no—
389C2×D4—416no—
390C2×D4—416no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—34no—
399C22×C4—34no—
400C22×C4—68no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4—64no—
412C22×C4—64no—
413C22×C4—64no—
414C22×C4—64no—
415C22×C4—64no—
416C22×C4—1216no—
417C22×C4—124no—
418C22×C4—124no—
419C22×C4C2×#6714yes—
420C22×C4C2×#6712yes—
421C4⋊C4—616no—
422C22⋊C4—28no—
423C22⋊C4—216no—
424C22⋊C4—68no—
425C22⋊C4—616no—
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—28no—
435C22×C3—216no—
436C22×C3—464no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—48no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—48no—
443C22×C3—48no—
444C22×C3—48no—
445C22×C3—48no—
446C22×C3—48no—
447C22×C3—48no—
448C22×C3—48no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—48no—
452C22×C3C2×#12011yescharacteristic
453C22×C3C2×#12011yescharacteristic
454C22×C3C2×#12312yes—
455C22×C3#52512yes—
456C22×C3#53411yescharacteristic
457C22×C3#53412yes—
458C22×C3#53512yes—
459C22×C3#53814yes—
460C22×C3D4214yes—
461C22×C3#54614yes—
462C22×C3#54614yes—
463C22×C3#54614yes—
464C22×C3#55112yes—
465C22×C3#56012yes—
466C3×C4—42no—
467C3×C4—44no—
468Dic3—44no—
469Dic3—44no—
470Dic3—84no—
471C23—24no—
472C23—22no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—28no—
495C23—432no—
496C23—416no—
497C23—432no—
498C23—464no—
499C23—464no—
500C23—432no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23#93518yescomplemented
524C23#93714yes—
525C23#95314yes—
526C23C2×D3×D412yes—
527C23C2×#23714yes—
528C23#100412yes—
529C23#100414yes—
530C23C22×#6712yes—
531C23C22×#6711yescharacteristic
532D4—432no—
533D4—416no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—68no—
546C2×C4—68no—
547C2×C4—1216no—
548C2×C4—124no—
549C2×C4—122no—
550C2×C4—124no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
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—48no—
564C2×C3#240311yescharacteristic
565C2×C3#240412yes—
566C2×C3#240512yes—
567C2×C3#241814yes—
568C2×C3#267814yes—
569C2×C3#269712yes—
570C22—28no—
571C22—28no—
572C22—24no—
573C22—24no—
574C22—28no—
575C22—216no—
576C22—464no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—48no—
581C22—44no—
582C22—48no—
583C22—48no—
584C22—48no—
585C22—48no—
586C22—48no—
587C22—48no—
588C22—48no—
589C22—44no—
590C22—48no—
591C22—44no—
592C22#558612yes—
593C22#558712yes—
594C22#559114yes—
595C22#559212yes—
596C22#559314yes—
597C22#559512yes—
598C22C2×#93512yes—
599C22C2×#93711yescharacteristic
600C22D4×#6714yes—
601C22#580412yes—
602C22#580411yescharacteristic
603C22#580614yes—
604C22#580614yes—
605C22C2×#95311yescharacteristic
606C4—44no—
607C4—42no—
608C4—124no—
609C4—124no—
610C4—244no—
611C3#1594811yescharacteristic, complemented, Sylow
612C2—48no—
613C2—44no—
614C2—44no—
615C2#8258112yes—
616C2#8258514yes—
617C2#8259614yes—
618C2#8260012yes—
619C2#8404712yes—
620C2#8405211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
Fitting, prime radicalC22×C3×#120C2
hypercenterC22×#120D3
hypocenterC3#15948
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#120

Complements

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

NH ≅ G/Ntype
2. C22×#935C2semidirect
3. C2×#5247C2semidirect
4. C2×#5220C2semidirect
5. #72131C2semidirect
6. #72131C2semidirect
7. #71549C2semidirect
8. #71547C2semidirect
13. C24×Dic3C22semidirect
22. C22×#217C22semidirect
23. C22×#217C22semidirect
26. C22×Dic3⋊C4C22semidirect
29. C2×C3⋊#95C22semidirect
30. C2×C3⋊#95C22semidirect
31. C2×C3⋊#95C22semidirect
87. C2×#217D4semidirect
310. C23×C3#123semidirect
523. C23#935semidirect
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×#1202
C22×#9352
C2×#52472
C2×#52202
#721312
#715492
#715472
#159483