Finite Group Databasebeta

← #11659419

Subgroups — #11659419

Order 768.

2742 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5751C211yesmaximal, characteristic
2#85887C211yesmaximal, characteristic, complemented
3#85886C211yesmaximal, characteristic, complemented
4C3×#2912C211yesmaximal, characteristic, complemented
5Dic3×#131C211yesmaximal, characteristic, complemented
6#82570C211yesmaximal, characteristic, complemented
7#82563C211yesmaximal, characteristic, complemented
8#79202C213yesmaximal, complemented
9#79201C213yesmaximal, complemented
10#79200C213yesmaximal, complemented
11#79199C213yesmaximal, complemented
12#79168C213yesmaximal, complemented
13#79165C213yesmaximal, complemented
14#79106C213yesmaximal, complemented
15C2×#4852C213yesmaximal, complemented
16C4×#577—31nomaximal, Sylow
17C2×#1010C2211yescharacteristic
18C2×#1009C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20C3×#520C2211yescharacteristic
21C3×#520C2211yescharacteristic
22Dic3×#41C2211yescharacteristic
23C2×#948C2211yescharacteristic
24#5751C4116yescomplemented
25C2×#909C2213yes—
26C2×#908C2213yes—
27C2×#904C2213yes—
28C22×#204C2213yes—
29#5574—21no—
30C4×#239—21no—
31#5572—21no—
32C4×D3×Q8—21no—
33#5546—23no—
34#5545—23no—
35C4×D3×D4—23no—
36C4×#237—23no—
37#5522C2213yes—
38#5522C2213yescomplemented
39#5521C2213yes—
40#5516C2213yescomplemented
41#5504C2213yes—
42#5504C2213yescomplemented
43C2×#893C2213yes—
44C2×#887C2213yescomplemented
45C3×#428C2213yescomplemented
46C3×#428C2213yescomplemented
47C3×#427C2213yescomplemented
48C2×C3×#106C2213yescomplemented
49#5122C2213yescomplemented
50Dic3×M16C2213yescomplemented
51#5101C2213yescomplemented
52C2×C8×Dic3C2213yescomplemented
53#5004C2213yescomplemented
54#5002C2213yes—
55#5002C2213yescomplemented
56#5001C2213yes—
57C2×#810C2213yescomplemented
58#4966C2213yes—
59#4966C2213yescomplemented
60C2×#809C2213yes—
61#4889C2216yescomplemented
62#4889C2216yescomplemented
63#4873C2213yescomplemented
64#4873C2213yescomplemented
65#4872C2213yescomplemented
66#4872C2213yescomplemented
67#4869C2216yescomplemented
68#4853C2216yescomplemented
69#4852C2213yescomplemented
70#4852C2216yescomplemented
71#4852C2211yescharacteristic, complemented
72#4852C2213yescomplemented
73#4852C2213yescomplemented
74#4852C2216yescomplemented
75#4852C2216yescomplemented
76C2×#577—31no—
77C4×2−1+4—31no—
78C4×H(5,𝔽2)—31no—
79#2952—33no—
80#2949—33no—
81#2947—33no—
82#2946—33no—
83C4×#137—33no—
84C4×AGL(1,ℤ/8)—33no—
85C4×#135—33no—
86C2×C4×SD16—33no—
87#2912D311yescharacteristic, complemented
88#2898—31no—
89#2897—31no—
90C4×#131—31no—
91C2×C3×#41C2311yescharacteristic
92#1010—24no—
93#1010C2×C418yescomplemented
94#1009—24no—
95#1009C2×C418yescomplemented
96C2×#239—21no—
97C2×D3×Q8—21no—
98C2×#237—23no—
99C2×D3×D4—23no—
100C2×#235C2313yes—
101C2×#235C2313yes—
102C22×D12C2313yes—
103C22×C4×D3—23no—
104C22×Dic6C2313yes—
105C3×#131C2×C418yescomplemented
106C2×C3×M16C2313yes—
107C22×C3×C8C2313yes—
108C3×C4×Q8—21no—
109C3×C4×Q8—21no—
110C3×C4×D4—23no—
111C3×C4×D4—23no—
112C3×#117C2313yes—
113C3×#117C2313yes—
114C2×C3×C4⋊C4C2313yes—
115C2×C3×C4⋊C4C2313yes—
116#948—24no—
117Q8×Dic3D411yescharacteristic
118Q8×Dic3D411yescharacteristic
119D4×Dic3D413yes—
120D4×Dic3D413yes—
121C4×#67—26no—
122C4×#67—26no—
123C2×(C2×D3)⋊C4—23no—
124#926C2313yes—
125C2×Dic3⋊C4—23no—
126C2×C4×Dic3—23no—
127C2×C4×Dic3C2313yes—
128C2×#208C2313yes—
129C22×C3⋊C8C2313yes—
130#909C2×C4124yescomplemented
131#908C2×C4124yescomplemented
132C2×Dic12C2313yes—
133#904C2×C4124yescomplemented
134C2×D24C2313yes—
135C2×#204C2×C4124yescomplemented
136C2×#204C2316yes—
137C2×#204C2313yes—
138C2×#204C2311yescharacteristic
139#893—23no—
140#893—23no—
141#893C2313yes—
142#893C2316yes—
143#893C2311yescharacteristic
144#893C2313yes—
145#893C2313yes—
146#892—23no—
147#892—23no—
148D3×C4⋊C4—23no—
149D3×C4⋊C4—23no—
150#887—23no—
151#887—23no—
152#887C2311yescharacteristic
153#887C2313yes—
154#887C2313yes—
155#887C2313yes—
156#887C2316yes—
157#67⋊C4—23no—
158#67⋊C4—23no—
159#67⋊C4—23no—
160#67⋊C4—23no—
161D3×C22⋊C4—26no—
162#877—26no—
163C4×D12—23no—
164C4×D12—23no—
165#872—23no—
166#872—23no—
167C42×D3—23no—
168C42×D3—23no—
169C4×Dic6—23no—
170C4×Dic6—23no—
171C3×#107C2313yes—
172C3×#107C2313yes—
173C3×#106C2316yes—
174C3×#106C2313yes—
175C3×#106C2311yescharacteristic
176#815C2316yes—
177C8×Dic3C2313yes—
178C8×Dic3C2313yes—
179C8×Dic3C2311yescharacteristic
180C8×Dic3C2313yes—
181#810C2311yescharacteristic
182#810C2313yes—
183#810C2313yes—
184#810C2313yes—
185#810C2316yes—
186#809C2313yes—
187#809C2316yes—
188#809C2311yescharacteristic
189#809C2313yes—
190#809C2313yes—
191C2×2−1+4—31no—
192C2×H(5,𝔽2)—31no—
193#577—316no—
194C2×#137—33no—
195C2×AGL(1,ℤ/8)—33no—
196C2×#135—33no—
197C22×SD16—33no—
198C2×#131—31no—
199C2×#131C2×D311yescharacteristic
200#520—33no—
201#520—33no—
202#520—63no—
203#520—61no—
204#520C2×D311yescharacteristic
205#520C2×D311yescharacteristic
206#519—33no—
207#519—61no—
208#518—33no—
209#518—63no—
210C4×#41—31no—
211C4×#41—33no—
212C4×#41—33no—
213C4×#41—63no—
214C4×#41—61no—
215C2×C4×Q8—33no—
216C2×C4×Q8—61no—
217C2×C4×D4—33no—
218C2×C4×D4—63no—
219#442—36no—
220#441—36no—
221#440—36no—
222#440—36no—
223C4×Q16—33no—
224C4×Q16—33no—
225C4×SD16—33no—
226C4×SD16—36no—
227C4×SD16—36no—
228C4×SD16—36no—
229C4×SD16—31no—
230C4×SD16—33no—
231C4×SD16—33no—
232C4×D8—33no—
233C4×D8—33no—
234#428C2×D313yescomplemented
235#428C2×D313yescomplemented
236#427C2×D313yescomplemented
237C2×#106C2×D313yescomplemented
238#419—33no—
239#418—33no—
240#417—33no—
241#417—33no—
242#416—33no—
243#416—33no—
244C2×#103—33no—
245C2×#102—33no—
246#405—33no—
247#405—33no—
248C4×M16—33no—
249C2×C4×C8—33no—
250C23×D3—23no—
251C3×#41C22×C414yes—
252C3×#41#4114yes—
253C2×C3×Q8C2×D411yescharacteristic
254C2×C3×D4C2×D413yes—
255C22×C3×C4—23no—
256C22×C3×C4—23no—
257C22×C3×C4C2413yes—
258C2×#67—23no—
259C2×#67—23no—
260C22×Dic3—23no—
261C22×Dic3C2×D413yes—
262C22×Dic3C2×D413yes—
263#239—28no—
264D3×Q8—28no—
265#237—224no—
266D3×D4—224no—
267#235—26no—
268#235—26no—
269#235C22×C4112yes—
270#235C22×C4112yescomplemented
271C2×D12—23no—
272C2×D12—26no—
273C2×D12C22×C4112yes—
274C2×D12C2413yes—
275C2×D12C2411yescharacteristic
276C2×C4×D3—23no—
277C2×C4×D3—23no—
278C2×C4×D3—21no—
279C2×C4×D3—23no—
280C2×C4×D3—23no—
281C2×C4×D3—23no—
282C2×C4×D3—26no—
283C2×C4×D3—86no—
284C2×Dic6—23no—
285C2×Dic6—26no—
286C2×Dic6C22×C4112yescomplemented
287C2×Dic6C2413yes—
288C2×Dic6C2411yescharacteristic
289C3×M16C22×C4112yescomplemented
290C2×C3×C8C22×C4112yescomplemented
291C2×C3×C8C2411yescharacteristic
292C2×C3×C8C2413yes—
293C3×C4⋊C4—23no—
294C3×C4⋊C4—23no—
295C3×C4⋊C4C2413yes—
296C3×C4⋊C4C2411yescharacteristic
297C3×C4⋊C4C2413yes—
298C3×C4⋊C4C2411yescharacteristic
299C3×C22⋊C4—23no—
300C3×C22⋊C4—23no—
301C3×C42—23no—
302C3×C42—23no—
303#217C2×D416yes—
304(C2×D3)⋊C4—21no—
305(C2×D3)⋊C4—23no—
306(C2×D3)⋊C4—23no—
307(C2×D3)⋊C4—23no—
308(C2×D3)⋊C4—26no—
309C4⋊Dic3C2×D413yes—
310C4⋊Dic3C2×D413yes—
311Dic3⋊C4—23no—
312Dic3⋊C4—26no—
313Dic3⋊C4—21no—
314Dic3⋊C4—23no—
315Dic3⋊C4—23no—
316C4×Dic3—23no—
317C4×Dic3—26no—
318C4×Dic3—21no—
319C4×Dic3—23no—
320C4×Dic3—23no—
321C4×Dic3C2×D413yes—
322C4×Dic3C2×D413yes—
323C4×Dic3C2411yescharacteristic
324C4×Dic3C2413yes—
325#208—26no—
326C2×C3⋊C8—26no—
327C2×C3⋊C8C2411yescharacteristic
328C2×C3⋊C8C2413yes—
329Dic12C22×C4112yescomplemented
330D24C22×C4112yescomplemented
331#204C22×C4112yescomplemented
332#204C22×C4124yescomplemented
333#204C22×C414yescomplemented
3342−1+4—38no—
3352−1+4—64no—
336H(5,𝔽2)—38no—
337H(5,𝔽2)—64no—
338C2×#41—33no—
339C2×#41—33no—
340C2×#41—61no—
341C2×#41—63no—
342C2×#41C22×D311yescharacteristic
343C22×Q8—33no—
344C22×Q8—61no—
345C22×D4—33no—
346C22×D4—63no—
347C23×C4—63no—
348#137—324no—
349AGL(1,ℤ/8)—324no—
350#135—324no—
351C2×Q16—33no—
352C2×SD16—324no—
353C2×SD16—33no—
354C2×SD16—36no—
355C2×SD16—31no—
356C2×D8—33no—
357#131—64no—
358#131C4×D318yescomplemented
359C2×M16—33no—
360C2×M16C22×D313yes—
361C22×C8—33no—
362C22×C8C22×D313yes—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—31no—
366C4×Q8—31no—
367C4×Q8—33no—
368C4×Q8—36no—
369C4×Q8—31no—
370C4×Q8—33no—
371C4×Q8—33no—
372C4×Q8—63no—
373C4×Q8—63no—
374C4×Q8—63no—
375C4×Q8—63no—
376C4×D4—23no—
377C4×D4—23no—
378C4×D4—33no—
379C4×D4—33no—
380C4×D4—33no—
381C4×D4—36no—
382C4×D4—31no—
383C4×D4—33no—
384C4×D4—33no—
385C4×D4—63no—
386C4×D4—63no—
387C4×D4—63no—
388C4×D4—63no—
389C4×D4—63no—
390C4×D4—63no—
391C4×D4—63no—
392C4×D4—63no—
393C4×D4—66no—
394C4×D4—66no—
395#117—33no—
396#117—63no—
397#117—63no—
398#117—66no—
399#117—63no—
400#117—63no—
401#117C22×D313yes—
402#117C22×D313yes—
403C2×C4⋊C4—63no—
404C2×C4⋊C4—63no—
405C2×C4⋊C4—63no—
406C2×C4⋊C4C22×D313yes—
407C2×C4⋊C4C22×D313yes—
408C2×C22⋊C4—66no—
409C2×C22⋊C4—63no—
410C2×C42—33no—
411C2×C42—63no—
412C2×C42—63no—
413C2×C42—63no—
414#107C22×D313yes—
415#107C22×D313yes—
416#106C22×D313yes—
417#106C22×D316yes—
418#106C22×D311yescharacteristic
419#103—33no—
420#103—36no—
421#103—31no—
422#103—33no—
423#103—33no—
424#102—33no—
425#102—36no—
426#102—31no—
427#102—33no—
428#102—33no—
429#97—36no—
430C4×C8—31no—
431C4×C8—33no—
432C4×C8—33no—
433C4×C8—33no—
434C23×C3C22×D413yes—
435C22×D3—21no—
436C22×D3—23no—
437C22×D3—212no—
438C22×D3—83no—
439C22×D3—83no—
440C3×Q8C4×D414yes—
441C3×D4C4×D4112yes—
442C2×C3×C4—21no—
443C2×C3×C4—23no—
444C2×C3×C4—21no—
445C2×C3×C4—23no—
446C2×C3×C4—83no—
447C2×C3×C4—83no—
448C2×C3×C4C23×C416yes—
449C2×C3×C4C22×D413yes—
450C2×C3×C4C2×#4116yes—
451C2×C3×C4C2511yescharacteristic
452#67—212no—
453#67—212no—
454C2×Dic3—212no—
455C2×Dic3—23no—
456C2×Dic3—21no—
457C2×Dic3—26no—
458C2×Dic3—43no—
459C2×Dic3C22×D413yes—
460C2×Dic3C22×D413yes—
461C2×Dic3C22×D411yescharacteristic
462C2×Dic3C22×D411yescharacteristic
463D12—212no—
464D12—23no—
465D12—21no—
466D12C23×C416yes—
467D12C23×C412yes—
468C4×D3—212no—
469C4×D3—212no—
470C4×D3—42no—
471C4×D3—46no—
472C4×D3—83no—
473C4×D3—83no—
474C4×D3—86no—
475Dic6—212no—
476Dic6—23no—
477Dic6—21no—
478Dic6C23×C416yes—
479Dic6C23×C412yes—
480C3×C8C23×C412yes—
481C3×C8C23×C416yes—
482C3⋊C8—21no—
483C3⋊C8—23no—
484C24—63no—
485#41—312no—
486#41—312no—
487#41—68no—
488#41—624no—
489#41—66no—
490#41—66no—
491#41C2×C4×D314yes—
492#41#23914yes—
493C2×Q8—312no—
494C2×Q8—33no—
495C2×Q8—31no—
496C2×Q8—68no—
497C2×Q8—63no—
498C2×Q8—66no—
499C2×Q8D3×D411yescharacteristic
500C2×D4—312no—
501C2×D4—31no—
502C2×D4—33no—
503C2×D4—63no—
504C2×D4—63no—
505C2×D4—624no—
506C2×D4—63no—
507C2×D4—66no—
508C2×D4D3×D413yes—
509C22×C4—23no—
510C22×C4—23no—
511C22×C4—33no—
512C22×C4—33no—
513C22×C4—61no—
514C22×C4—63no—
515C22×C4—63no—
516C22×C4—63no—
517C22×C4—66no—
518C22×C4—63no—
519C22×C4—63no—
520C22×C4—63no—
521C22×C4—246no—
522C22×C4C23×D313yes—
523Q16—312no—
524SD16—312no—
525SD16—324no—
526SD16—34no—
527D8—312no—
528M16—66no—
529M16C2×C4×D3112yescomplemented
530C2×C8—31no—
531C2×C8—33no—
532C2×C8—66no—
533C2×C8C2×C4×D3112yescomplemented
534C2×C8C23×D311yescharacteristic
535C2×C8C23×D313yes—
536C4⋊C4—23no—
537C4⋊C4—23no—
538C4⋊C4—33no—
539C4⋊C4—33no—
540C4⋊C4—63no—
541C4⋊C4—63no—
542C4⋊C4—66no—
543C4⋊C4—61no—
544C4⋊C4—63no—
545C4⋊C4C23×D313yes—
546C4⋊C4C23×D311yescharacteristic
547C4⋊C4C23×D313yes—
548C4⋊C4C23×D311yescharacteristic
549C22⋊C4—23no—
550C22⋊C4—23no—
551C22⋊C4—36no—
552C22⋊C4—63no—
553C22⋊C4—63no—
554C22⋊C4—66no—
555C22⋊C4—61no—
556C22⋊C4—63no—
557C42—23no—
558C42—23no—
559C42—33no—
560C42—33no—
561C42—31no—
562C42—33no—
563C42—61no—
564C42—63no—
565C42—63no—
566C42—63no—
567C42—66no—
568C22×C3—23no—
569C22×C3—23no—
570C22×C3C2×C4×D416yes—
571C22×C3C23×D411yescharacteristic
572C2×D3—22no—
573C2×D3—26no—
574C2×D3—41no—
575C2×D3—43no—
576C2×D3—43no—
577C2×D3—41no—
578C2×D3—812no—
579C3×C4—41no—
580C3×C4—43no—
581C3×C4—43no—
582C3×C4—41no—
583C3×C4C2×C4×D416yes—
584C3×C4C24×C411yescharacteristic
585C3×C4C22×#4111yescharacteristic
586Dic3—23no—
587Dic3—21no—
588Dic3—26no—
589Dic3—22no—
590Dic3—23no—
591Dic3#57712yescomplemented
592C23—61no—
593C23—63no—
594C23—612no—
595C23—243no—
596C23—243no—
597C23C2×D3×D413yes—
598Q8—36no—
599Q8—32no—
600Q8—63no—
601Q8—61no—
602Q8—612no—
603Q8#89314yes—
604D4—32no—
605D4—36no—
606D4—612no—
607D4—612no—
608D4—63no—
609D4—61no—
610D4—612no—
611D4#893112yes—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—23no—
615C2×C4—23no—
616C2×C4—31no—
617C2×C4—31no—
618C2×C4—33no—
619C2×C4—33no—
620C2×C4—66no—
621C2×C4—61no—
622C2×C4—63no—
623C2×C4—612no—
624C2×C4—612no—
625C2×C4—612no—
626C2×C4—83no—
627C2×C4—83no—
628C2×C4—126no—
629C2×C4—122no—
630C2×C4—123no—
631C2×C4—246no—
632C2×C4—243no—
633C2×C4—243no—
634C2×C4C22×C4×D316yes—
635C2×C4C2×D3×D413yes—
636C2×C4C2×#23916yes—
637C2×C4C24×D311yescharacteristic
638C8—61no—
639C8—63no—
640C8C22×C4×D312yes—
641C8C22×C4×D316yes—
642C2×C3—26no—
643C2×C3C22×C4×D411yescharacteristic
644C2×C3C2×#57711yescharacteristic
645C2×C3#358411yescharacteristic
646D3—42no—
647D3—46no—
648C22—23no—
649C22—23no—
650C22—62no—
651C22—66no—
652C22—123no—
653C22—121no—
654C22—121no—
655C22—123no—
656C22—2412no—
657C22C2×#89316yes—
658C22C22×D3×D411yescharacteristic
659C4—32no—
660C4—43no—
661C4—41no—
662C4—43no—
663C4—41no—
664C4—63no—
665C4—61no—
666C4—63no—
667C4—62no—
668C4—66no—
669C4C2×#89316yes—
670C4C23×C4×D311yescharacteristic
671C4C22×#23911yescharacteristic
672C3C4×#57711yescharacteristic, complemented, Sylow
673C2—26no—
674C2—126no—
675C2—122no—
676C2C22×#89311yescharacteristic
677C2D3×#57711yescharacteristic
678C2#8720611yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC3×#2912C2
socleC22×C3C23×D4
prime radicalC2×#5751C2
cosocleC2×C3×C4C25
hypercenter#2912D3
hypocenterC3C4×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2912

Complements

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

NH ≅ G/Ntype
2. #85887C2semidirect
3. #85886C2semidirect
4. C3×#2912C2semidirect
5. Dic3×#131C2semidirect
6. #82570C2semidirect
7. #82563C2semidirect
8. #79202C2semidirect
9. #79201C2semidirect
10. #79200C2semidirect
11. #79199C2semidirect
12. #79168C2semidirect
13. #79165C2semidirect
14. #79106C2semidirect
15. C2×#4852C2semidirect
24. #5751C4semidirect
38. #5522C22semidirect
40. #5516C22semidirect
42. #5504C22semidirect
44. C2×#887C22semidirect
45. C3×#428C22semidirect
46. C3×#428C22semidirect
47. C3×#427C22semidirect
48. C2×C3×#106C22semidirect
49. #5122C22semidirect
50. Dic3×M16C22semidirect
51. #5101C22semidirect
52. C2×C8×Dic3C22semidirect
53. #5004C22semidirect
55. #5002C22semidirect
57. C2×#810C22semidirect
59. #4966C22semidirect
61. #4889C22semidirect
62. #4889C22semidirect
63. #4873C22semidirect
64. #4873C22semidirect
65. #4872C22semidirect
66. #4872C22semidirect
67. #4869C22semidirect
68. #4853C22semidirect
69. #4852C22semidirect
70. #4852C22semidirect
71. #4852C22semidirect
72. #4852C22semidirect
73. #4852C22semidirect
74. #4852C22semidirect
75. #4852C22semidirect
87. #2912D3semidirect
93. #1010C2×C4semidirect
95. #1009C2×C4semidirect
105. C3×#131C2×C4semidirect
130. #909C2×C4semidirect
131. #908C2×C4semidirect
133. #904C2×C4semidirect
135. C2×#204C2×C4semidirect
234. #428C2×D3semidirect
235. #428C2×D3semidirect
236. #427C2×D3semidirect
237. C2×#106C2×D3semidirect
270. #235C22×C4semidirect
286. C2×Dic6C22×C4semidirect
289. C3×M16C22×C4semidirect
290. C2×C3×C8C22×C4semidirect
329. Dic12C22×C4semidirect
330. D24C22×C4semidirect
331. #204C22×C4semidirect
332. #204C22×C4semidirect
333. #204C22×C4semidirect
358. #131C4×D3semidirect
529. M16C2×C4×D3semidirect
533. C2×C8C2×C4×D3semidirect
591. Dic3#577semidirect
672. C3C4×#577semidirect

Maximal subgroups

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

subgroupindex
C2×#57512
#858872
#858862
C3×#29122
Dic3×#1312
#825702
#825632
#792022
#792012
#792002
#791992
#791682
#791652
#791062
C2×#48522
C4×#5773