Finite Group Databasebeta

← #10726580

Subgroups — #10726580

Order 768.

3654 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C22⋊C4C211yesmaximal, characteristic
2C22×#924C211yesmaximal, characteristic, complemented
3C2×#5247C212yesmaximal
4C2×#4947C212yesmaximal, complemented
5#72127C212yesmaximal, complemented
6#72127C211yesmaximal, characteristic, complemented
7#71547C214yesmaximal, complemented
8#71546C212yesmaximal, complemented
9#14717—31nomaximal, Sylow
10C26×C3C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C23×Dic6C2211yescharacteristic, complemented
14C22×C3×C22⋊C4—24no—
15C22×C3×C22⋊C4—28no—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2212yes—
18C22×#217C2212yes—
19C22×#217C2214yes—
20C22×#217C2214yes—
21C22×#217C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic, complemented
23C2×#924—24no—
24C2×#924—28no—
25C22×Dic3⋊C4C2212yescomplemented
26#5247—28no—
27C2×C3⋊#95C2212yescomplemented
28C2×C3⋊#95C2214yescomplemented
29C2×C3⋊#95C2214yescomplemented
30C2×C3⋊#95C2212yescomplemented
31#4947—28no—
32C22×C22⋊Q8—31no—
33C23×C22⋊C4D311yescharacteristic
34C2×#393—32no—
35C2×#379—32no—
36#1439—32no—
37#1439—34no—
38C22⋊#95—32no—
39C22⋊#95—31no—
40C25×C3—24no—
41C25×C3—22no—
42C25×C3—24no—
43C25×C3—22no—
44C25×C3—24no—
45C25×C3—24no—
46C25×C3—42no—
47C25×C3—42no—
48C25×C3Q814yescomplemented
49C25×C3C2312yes—
50C25×C3C2311yescharacteristic
51C23×C3×C4—24no—
52C23×C3×C4—22no—
53C23×C3×C4—24no—
54C23×C3×C4—24no—
55C23×C3×C4—24no—
56C23×C3×C4—42no—
57C23×C3×C4—42no—
58C23×C3×C4D414yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×Dic3C2314yes—
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C22×Dic6—22no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6—42no—
69C2×C3×C22⋊C4—232no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4D418yes—
76C2×#217—24no—
77C2×#217—24no—
78C2×#217—28no—
79C2×#217—24no—
80C2×#217—28no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—28no—
84C2×#217—28no—
85C2×#217—44no—
86C2×#217—44no—
87C2×#217D418yes—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—24no—
90#924—232no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C3⋊#95—28no—
95C3⋊#95—24no—
96C3⋊#95—44no—
97C3⋊#95—42no—
98C26C2×D311yescharacteristic
99C23×Q8—31no—
100C24×C4C2×D311yescharacteristic
101C24×C4C2×D311yescharacteristic
102C2×C22⋊Q8—68no—
103C2×C22⋊Q8—64no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4—31no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—32no—
109C22×C22⋊C4—32no—
110C22×C22⋊C4—34no—
111C22×C22⋊C4—34no—
112C22×C22⋊C4C2×D312yes—
113C22×C22⋊C4C2×D312yes—
114#393—68no—
115#379—68no—
116C2×#95—34no—
117C2×#95—32no—
118C2×#95—34no—
119C2×#95—32no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—28no—
125C24×C3—216no—
126C24×C3—28no—
127C24×C3—28no—
128C24×C3—216no—
129C24×C3—24no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—24no—
135C24×C3—24no—
136C24×C3—24no—
137C24×C3—216no—
138C24×C3—48no—
139C24×C3—416no—
140C24×C3—416no—
141C24×C3—416no—
142C24×C3—416no—
143C24×C3—44no—
144C24×C3—44no—
145C24×C3—48no—
146C24×C3—44no—
147C24×C3—48no—
148C24×C3—44no—
149C24×C3C2×D414yes—
150C24×C3C2×D414yes—
151C24×C3C2×D412yes—
152C24×C3C2×Q814yes—
153C24×C3C2×Q812yes—
154C24×C3#4114yes—
155C24×C3#4112yes—
156C24×C3#4114yes—
157C24×C3C2411yescharacteristic
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—28no—
161C22×C3×C4—28no—
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×C4—28no—
170C22×C3×C4—44no—
171C22×C3×C4—416no—
172C22×C3×C4—416no—
173C22×C3×C4—44no—
174C22×C3×C4—44no—
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D412yes—
179C22×Dic3—24no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3C2×D414yes—
195C22×Dic3C2×D414yes—
196C2×Dic6—28no—
197C2×Dic6—416no—
198C2×Dic6—44no—
199C2×Dic6—44no—
200C3×C22⋊C4—216no—
201C3×C22⋊C4—216no—
202#217—216no—
203#217—416no—
204#217—416no—
205#217—48no—
206C4⋊Dic3—28no—
207Dic3⋊C4—216no—
208C25—24no—
209C25—22no—
210C25—24no—
211C25—22no—
212C25—24no—
213C25—24no—
214C25—42no—
215C25—42no—
216C25Dic614yescomplemented
217C25C22×D312yes—
218C25C22×D311yescharacteristic
219C22×Q8—64no—
220C22×Q8—62no—
221C22×Q8—64no—
222C22×Q8—122no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—32no—
229C23×C4—32no—
230C23×C4—34no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4#6714yes—
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D312yes—
236C23×C4C22×D311yescharacteristic
237C22⋊Q8—632no—
238C2×C4⋊C4—64no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—62no—
243C2×C22⋊C4—232no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—38no—
250C2×C22⋊C4—64no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—68no—
253C2×C22⋊C4—68no—
254C2×C22⋊C4—68no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—68no—
259C2×C22⋊C4—124no—
260C2×C22⋊C4—124no—
261C2×C22⋊C4#6718yes—
262#95—68no—
263#95—64no—
264#95—124no—
265#95—122no—
266C23×C3—22no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—24no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—28no—
275C23×C3—28no—
276C23×C3—28no—
277C23×C3—28no—
278C23×C3—232no—
279C23×C3—24no—
280C23×C3—28no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—28no—
287C23×C3—28no—
288C23×C3—28no—
289C23×C3—28no—
290C23×C3—28no—
291C23×C3—464no—
292C23×C3—416no—
293C23×C3—416no—
294C23×C3—416no—
295C23×C3—416no—
296C23×C3—416no—
297C23×C3—464no—
298C23×C3—42no—
299C23×C3—44no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—44no—
304C23×C3—44no—
305C23×C3—44no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—416no—
309C23×C3#12014yes—
310C23×C3C22⋊Q818yes—
311C23×C3C22⋊Q814yes—
312C23×C3C22⋊Q818yescomplemented
313C23×C3#12418yes—
314C23×C3#12414yes—
315C23×C3C22×D412yes—
316C23×C3C22×D411yescharacteristic
317C23×C3C22×D412yes—
318C23×C3C22×Q811yescharacteristic
319C23×C3C2×#4111yescharacteristic
320C23×C3C2×#4112yes—
321C23×C3C2×#4112yes—
322C23×C3H(5,𝔽2)12yes—
323C23×C3H(5,𝔽2)12yes—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—28no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—28no—
332C2×C3×C4—416no—
333C2×C3×C4—416no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4#12014yes—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—28no—
342C2×Dic3—28no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—44no—
348C2×Dic3—44no—
349C2×Dic3—44no—
350C2×Dic3—44no—
351C2×Dic3—44no—
352C2×Dic3—44no—
353C2×Dic3—44no—
354C2×Dic3—44no—
355C2×Dic3—44no—
356Dic6—416no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—28no—
362C24—216no—
363C24—28no—
364C24—28no—
365C24—216no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—216no—
375C24—48no—
376C24—416no—
377C24—416no—
378C24—416no—
379C24—416no—
380C24—48no—
381C24—44no—
382C24—44no—
383C24—48no—
384C24—44no—
385C24—44no—
386C24C2×Dic614yes—
387C24C2×Dic612yes—
388C24#23514yes—
389C24#23512yes—
390C24D3×D414yes—
391C24#23714yes—
392C24C2×#6714yes—
393C24C2×#6712yes—
394C24C23×D311yescharacteristic
395C2×Q8—68no—
396C2×Q8—1216no—
397C2×Q8—124no—
398C2×Q8—124no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—34no—
412C22×C4—34no—
413C22×C4—44no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—64no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—64no—
422C22×C4—64no—
423C22×C4—64no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4C2×#6714yes—
434C22×C4C2×#6712yes—
435C22×C4C2×#6714yes—
436C22×C4C2×#6712yes—
437C4⋊C4—616no—
438C4⋊C4—68no—
439C22⋊C4—216no—
440C22⋊C4—216no—
441C22⋊C4—616no—
442C22⋊C4—1216no—
443C22⋊C4—1216no—
444C22⋊C4—128no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—24no—
448C22×C3—24no—
449C22×C3—24no—
450C22×C3—24no—
451C22×C3—216no—
452C22×C3—216no—
453C22×C3—28no—
454C22×C3—216no—
455C22×C3—464no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—48no—
463C22×C3—48no—
464C22×C3—48no—
465C22×C3#39318yescomplemented
466C22×C3C2×#12011yescharacteristic
467C22×C3C2×#12011yescharacteristic
468C22×C3C2×C22⋊Q812yes—
469C22×C3C2×C22⋊Q811yescharacteristic
470C22×C3C2×C22⋊Q812yes—
471C22×C3#52511yescharacteristic
472C22×C3#52512yes—
473C22×C3C2×#12412yes—
474C22×C3C2×#12411yescharacteristic
475C22×C3#53411yescharacteristic
476C22×C3#53511yescharacteristic
477C22×C3#53812yes—
478C22×C3#54312yes—
479C22×C3D4214yes—
480C22×C3#54614yes—
481C22×C3#54614yes—
482C22×C3#55114yes—
483C3×C4—24no—
484C3×C4—42no—
485C3×C4—44no—
486Dic3—44no—
487Dic3—44no—
488Dic3—84no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—22no—
496C23—28no—
497C23—22no—
498C23—22no—
499C23—28no—
500C23—22no—
501C23—232no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—24no—
514C23—464no—
515C23—416no—
516C23—416no—
517C23—464no—
518C23—416no—
519C23—416no—
520C23—416no—
521C23—416no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23#87818yes—
533C23#88518yes—
534C23#92414yes—
535C23#92418yescomplemented
536C23#93614yes—
537C23#93714yes—
538C23C22×Dic611yescharacteristic
539C23C2×#23512yes—
540C23C2×#23511yescharacteristic
541C23C2×D3×D412yes—
542C23C2×#23712yes—
543C23#100412yes—
544C23#100412yes—
545C23C22×#6711yescharacteristic
546C23C22×#6712yes—
547Q8—1216no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—68no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—68no—
566C2×C4—124no—
567C2×C4—122no—
568C2×C4—124no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—124no—
573C2×C4—124no—
574C2×C4—124no—
575C2×C4—124no—
576C2×C4—124no—
577C2×C4—124no—
578C2×C4—124no—
579C2×C4#95314yes—
580C2×C3—28no—
581C2×C3—48no—
582C2×C3C2×#39311yescharacteristic
583C2×C3#240311yescharacteristic
584C2×C3#240411yescharacteristic
585C2×C3#243511yescharacteristic
586C2×C3#243714yes—
587C2×C3#247214yes—
588C2×C3#256111yescharacteristic
589C2×C3#262611yescharacteristic
590C2×C3#284811yescharacteristic
591C22—24no—
592C22—24no—
593C22—24no—
594C22—216no—
595C22—216no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—28no—
600C22—216no—
601C22—464no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—48no—
606C22—48no—
607C22—48no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22#494718yescomplemented
612C22C2×#87812yes—
613C22#548312yes—
614C22#549014yes—
615C22C2×#88512yes—
616C22#549412yes—
617C22#549514yes—
618C22C2×#92411yescharacteristic
619C22C2×#92412yes—
620C22#579112yes—
621C22#579111yescharacteristic
622C22C2×#93611yescharacteristic
623C22C2×#93711yescharacteristic
624C22D4×#6714yes—
625C22#580411yescharacteristic
626C22#580511yescharacteristic
627C22#580614yes—
628C22C2×#95311yescharacteristic
629C4—24no—
630C4—44no—
631C4—42no—
632C4—124no—
633C4—124no—
634C4—244no—
635C3#1471711yescharacteristic, complemented, Sylow
636C2—28no—
637C2—48no—
638C2C2×#494711yescharacteristic
639C2#8211014yes—
640C2#8211114yes—
641C2#8211611yescharacteristic
642C2#8211711yescharacteristic
643C2#8212511yescharacteristic
644C2#8403111yescharacteristic
645C2#8405011yescharacteristic
646C2#8405211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC23×C3×C22⋊C4C2
prime radicalC26×C3C22
hypercenterC23×C22⋊C4D3
hypocenterC3#14717
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#924C2semidirect
4. C2×#4947C2semidirect
5. #72127C2semidirect
6. #72127C2semidirect
7. #71547C2semidirect
8. #71546C2semidirect
13. C23×Dic6C22semidirect
22. C22×C4⋊Dic3C22semidirect
25. C22×Dic3⋊C4C22semidirect
27. C2×C3⋊#95C22semidirect
28. C2×C3⋊#95C22semidirect
29. C2×C3⋊#95C22semidirect
30. C2×C3⋊#95C22semidirect
48. C25×C3Q8semidirect
216. C25Dic6semidirect
312. C23×C3C22⋊Q8semidirect
465. C22×C3#393semidirect
535. C23#924semidirect
611. C22#4947semidirect
635. C3#14717semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C42
C22×#9242
C2×#52472
C2×#49472
#721272
#715472
#715462
#147173