Finite Group Databasebeta

← #10816028

Subgroups — #10816028

Order 768.

1008 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83806C211yesmaximal, characteristic, complemented
2C2×#4827C211yesmaximal, characteristic
3#78832C211yesmaximal, characteristic, complemented
4#78820C211yesmaximal, characteristic, complemented
5#78560C211yesmaximal, characteristic, complemented
6C3×#1764C211yesmaximal, characteristic, complemented
7#72018C211yesmaximal, characteristic, complemented
8#70364C211yesmaximal, characteristic, complemented
9#69891C211yesmaximal, characteristic, complemented
10#69789C211yesmaximal, characteristic, complemented
11#69773C211yesmaximal, characteristic, complemented
12#68948C211yesmaximal, characteristic, complemented
13#68930C211yesmaximal, characteristic, complemented
14#68772C211yesmaximal, characteristic, complemented
15#68759C211yesmaximal, characteristic, complemented
16#22895—31nomaximal, Sylow
17C2×#1010C2211yescharacteristic
18C2×C3×#131C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#949C2211yescharacteristic
21C2×#908C2211yescharacteristic
22C2×#901C2211yescharacteristic
23#5523C2211yescharacteristic
24#5521C2211yescharacteristic
25C2×#875C2211yescharacteristic
26C2×C3×#104C2211yescharacteristic
27C3×#416C2211yescharacteristic
28C3×#405C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30#5232C2211yescharacteristic
31C2×#824C2211yescharacteristic
32#5133—21no—
33#5133C2211yescharacteristic
34C2×#823C2211yescharacteristic
35#5121C2211yescharacteristic
36#5120—21no—
37#5120C2211yescharacteristic
38#5113—21no—
39#5113C2211yescharacteristic
40C2×#821C2211yescharacteristic
41C2×#821C2211yescharacteristic
42C2×#816—21no—
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45#5101C2211yescharacteristic
46#5004C2211yescharacteristic
47#4995C2211yescharacteristic
48#4936C2211yescharacteristic
49C2×#805C2211yescharacteristic
50#4827C418yescomplemented
51#4817—21no—
52#4802—21no—
53D3×#103—21no—
54#4788—21no—
55#4785—21no—
56#4772—21no—
57#4771—21no—
58#4756—21no—
59#4711—21no—
60#4710—21no—
61C4×D24—21no—
62C4×#204—21no—
63C3×#341C2211yescharacteristic
64C3×#339C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4552C2211yescharacteristic
68#4523C2211yescharacteristic
69C2×#454—31no—
70#2946—31no—
71#2946—31no—
72#2898—31no—
73#2898—31no—
74#1958—31no—
75#1930—31no—
76#1930—31no—
77#1907—31no—
78#1907—31no—
79#1885—31no—
80#1860—31no—
81#1860—31no—
82#1764—31no—
83#1764D311yescharacteristic, complemented
84C2×C3×#41C2311yescharacteristic
85#1010—22no—
86#1010—22no—
87#1010—41no—
88#1010C2×C414yescomplemented
89C2×D3×Q8—21no—
90C2×#237—21no—
91C2×#237—41no—
92C2×#235—21no—
93C2×#235C2311yescharacteristic
94C22×D12C2311yescharacteristic
95C22×Dic6—21no—
96C22×Dic6C2311yescharacteristic
97C3×#131—22no—
98C3×#131—41no—
99C2×C3×M16—21no—
100C2×C3×M16C2311yescharacteristic
101C2×C3×M16C2311yescharacteristic
102C22×C3×C8—21no—
103C22×C3×C8C2311yescharacteristic
104C22×C3×C8C2311yescharacteristic
105C3×#117C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107#949—22no—
108#949C2×C414yescomplemented
109Q8×Dic3—21no—
110C2×#215—21no—
111D4×Dic3—21no—
112D4×Dic3—41no—
113C2×#213—21no—
114C2×(C2×D3)⋊C4—41no—
115#926—21no—
116#926C2311yescharacteristic
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×#208C2311yescharacteristic
121C2×#208C2311yescharacteristic
122#908—22no—
123#908C2×C414yescomplemented
124C2×D24—21no—
125C2×#204—21no—
126#901—22no—
127#901—41no—
128C2×ΓL(1,𝔽25)—21no—
129C2×D3×C8—21no—
130#893—21no—
131#893—21no—
132#892—21no—
133D3×C4⋊C4—21no—
134#67⋊C4—41no—
135D3×C22⋊C4—41no—
136#875—41no—
137#875C2×C414yescomplemented
138C4×D12—21no—
139C4×D12—21no—
140C4×D12—21no—
141C4×Dic6—21no—
142C3×#104C2×C414yescomplemented
143C3×#103D411yescharacteristic
144C3×#103D411yescharacteristic
145C3×#102D411yescharacteristic
146C3×#102D411yescharacteristic
147C3×#97—21no—
148C3×C4×C8—21no—
149C3×C4×C8—41no—
150#835—21no—
151#832—21no—
152C3⋊#95—41no—
153#824C2×C414yescomplemented
154#823—41no—
155#821—21no—
156#821—21no—
157#821—21no—
158#821—21no—
159#821—21no—
160#821D411yescharacteristic
161#821D411yescharacteristic
162#818—21no—
163#817—21no—
164#816—21no—
165#816—21no—
166#816—21no—
167#816—21no—
168#816—21no—
169#816—21no—
170#816—21no—
171#816D411yescharacteristic
172#816D411yescharacteristic
173#815—21no—
174C8×Dic3—21no—
175#810—21no—
176#810—21no—
177#808—21no—
178#807—21no—
179#805C2×C414yescomplemented
180C2×2−1+4—31no—
181C2×AGL(1,ℤ/8)—31no—
182C2×AGL(1,ℤ/8)—31no—
183C2×#131—31no—
184C2×#131C2×D311yescharacteristic
185C2×#124—31no—
186#520—31no—
187#520—31no—
188#518—31no—
189#454—38no—
190#442—61no—
191#442—61no—
192#440—61no—
193#440—61no—
194C4×SD16—61no—
195C4×SD16—61no—
196C4×D8—61no—
197C4×D8—61no—
198#428—31no—
199#428—31no—
200C2×#104—31no—
201C2×#104C2×D311yescharacteristic
202#419—31no—
203#419—31no—
204#419—61no—
205#419—61no—
206#417—31no—
207#417—31no—
208#417—61no—
209#417—61no—
210#416—31no—
211#416—61no—
212#416—61no—
213#416C2×D311yescharacteristic
214C2×#103—31no—
215C2×#103—31no—
216C2×#103—61no—
217C2×#103—61no—
218C2×#102—31no—
219C2×#102—31no—
220C2×#101—31no—
221C2×#100—31no—
222#405—31no—
223#405C2×D311yescharacteristic
224C2×C4×C8C2×D311yescharacteristic
225#382—31no—
226#343—31no—
227#343—31no—
228#341—31no—
229#341C2×D311yescharacteristic
230#339—31no—
231#339C2×D311yescharacteristic
232C23×D3—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41C22×C412yes—
236C3×#41#4112yes—
237C2×C3×Q8C2×D411yescharacteristic
238C2×C3×D4—21no—
239C2×C3×D4C2×D411yescharacteristic
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4C2411yescharacteristic
243C2×#67—21no—
244C2×#67—41no—
245C22×Dic3—21no—
246C22×Dic3—21no—
247C22×Dic3—21no—
248C22×Dic3—41no—
249C22×Dic3—41no—
250D3×Q8—24no—
251D3×Q8—42no—
252#237—24no—
253#237—42no—
254#237—44no—
255#237—44no—
256#235—21no—
257#235—21no—
258#235—24no—
259#235—42no—
260#235C22×C412yes—
261#235#4112yes—
262C2×D12—21no—
263C2×D12—21no—
264C2×D12—21no—
265C2×D12—21no—
266C2×D12—21no—
267C2×D12C22×C412yes—
268C2×D12C2×D411yescharacteristic
269C2×C4×D3—21no—
270C2×C4×D3—41no—
271C2×C4×D3—41no—
272C2×C4×D3—41no—
273C2×C4×D3—41no—
274C2×Dic6—21no—
275C2×Dic6—21no—
276C2×Dic6—21no—
277C2×Dic6—21no—
278C2×Dic6—24no—
279C2×Dic6—21no—
280C2×Dic6—42no—
281C2×Dic6—41no—
282C2×Dic6C22×C412yes—
283C2×Dic6C2×D411yescharacteristic
284C3×M16—21no—
285C3×M16—21no—
286C3×M16—21no—
287C3×M16—42no—
288C3×M16C22×C412yes—
289C2×C3×C8—21no—
290C2×C3×C8—21no—
291C2×C3×C8—21no—
292C2×C3×C8—21no—
293C2×C3×C8—21no—
294C2×C3×C8—21no—
295C2×C3×C8—42no—
296C2×C3×C8C2×D411yescharacteristic
297C2×C3×C8C2×D411yescharacteristic
298C2×C3×C8C2×D411yescharacteristic
299C2×C3×C8C2×D411yescharacteristic
300C3×C4⋊C4C2×D411yescharacteristic
301C3×C4⋊C4C2×D411yescharacteristic
302C3×C22⋊C4—21no—
303C3×C42—21no—
304C3×C42—21no—
305C3×C42C22×C412yes—
306#217—21no—
307#217—41no—
308#215—24no—
309#213—24no—
310(C2×D3)⋊C4—44no—
311(C2×D3)⋊C4—41no—
312(C2×D3)⋊C4—41no—
313(C2×D3)⋊C4—41no—
314C4⋊Dic3—21no—
315C4⋊Dic3—21no—
316C4⋊Dic3—21no—
317C4⋊Dic3—21no—
318C4⋊Dic3—41no—
319C4⋊Dic3C2×D411yescharacteristic
320C4⋊Dic3C2×D411yescharacteristic
321Dic3⋊C4—41no—
322C4×Dic3—21no—
323C4×Dic3—41no—
324C4×Dic3—41no—
325#208—21no—
326#208—21no—
327#208—21no—
328#208C22×C412yes—
329C2×C3⋊C8—21no—
330C2×C3⋊C8—21no—
331D24—24no—
332#204—24no—
333ΓL(1,𝔽25)—42no—
334D3×C8—42no—
3352−1+4—34no—
3362−1+4—62no—
3372−1+4—62no—
3382−1+4—121no—
339C2×#41—31no—
340C2×#41—61no—
341C2×#41—61no—
342C2×#41—121no—
343C2×#41C22×D311yescharacteristic
344C22×Q8—31no—
345C22×Q8—61no—
346C22×Q8—61no—
347C22×D4—31no—
348AGL(1,ℤ/8)—34no—
349AGL(1,ℤ/8)—34no—
350AGL(1,ℤ/8)—62no—
351AGL(1,ℤ/8)—62no—
352C2×SD16—61no—
353C2×SD16—61no—
354C2×D8—61no—
355C2×D8—61no—
356#131—22no—
357#131—41no—
358#131—62no—
359#131—121no—
360C2×M16—21no—
361C2×M16—31no—
362C2×M16—31no—
363C2×M16—61no—
364C2×M16C22×D311yescharacteristic
365C2×M16C22×D311yescharacteristic
366C22×C8—21no—
367C22×C8—61no—
368C22×C8C22×D311yescharacteristic
369C22×C8C22×D311yescharacteristic
370#124—34no—
371#124—121no—
372C4×Q8—61no—
373C4×Q8—61no—
374C4×D4—61no—
375C4×D4—61no—
376C4×D4—61no—
377C4×D4—61no—
378C4×D4—61no—
379C4×D4—61no—
380C4×D4—121no—
381C4×D4—121no—
382#117—31no—
383#117—61no—
384#117—61no—
385#117C22×D311yescharacteristic
386C2×C4⋊C4—31no—
387C2×C4⋊C4—31no—
388C2×C4⋊C4—61no—
389C2×C4⋊C4—61no—
390C2×C22⋊C4—121no—
391C2×C22⋊C4—121no—
392C2×C42C22×D311yescharacteristic
393#107—61no—
394#107—61no—
395#106—61no—
396#106—61no—
397#104—34no—
398#104C4×D314yescomplemented
399#103—31no—
400#103—31no—
401#103—61no—
402#103—61no—
403#103—61no—
404#103—61no—
405#103—61no—
406#103—61no—
407#103—61no—
408#103—61no—
409#103—61no—
410#103—61no—
411#103D1211yescharacteristic
412#103D1211yescharacteristic
413#102—31no—
414#102—31no—
415#102—61no—
416#102—61no—
417#102—61no—
418#102—61no—
419#102—61no—
420#102—61no—
421#102D1211yescharacteristic
422#102D1211yescharacteristic
423#101—34no—
424#100—121no—
425#97—21no—
426#97—61no—
427C4×C8—21no—
428C4×C8—41no—
429C4×C8—61no—
430#95—121no—
431C23×C3—21no—
432C23×C3H(5,𝔽2)11yescharacteristic
433C22×D3—21no—
434C22×D3—22no—
435C22×D3—41no—
436C22×D3—41no—
437C22×D3—81no—
438C22×D3—81no—
439C3×Q8—21no—
440C3×Q8C4×D412yes—
441C3×D4—22no—
442C3×D4—22no—
443C3×D4—21no—
444C3×D4C4×D412yes—
445C2×C3×C4—21no—
446C2×C3×C4—22no—
447C2×C3×C4—22no—
448C2×C3×C4—22no—
449C2×C3×C4—21no—
450C2×C3×C4—21no—
451C2×C3×C4—41no—
452C2×C3×C4—41no—
453C2×C3×C4C23×C411yescharacteristic
454C2×C3×C4C22×D411yescharacteristic
455C2×C3×C4C22×D411yescharacteristic
456C2×C3×C4C2×#4111yescharacteristic
457C2×C3×C4C2×#4111yescharacteristic
458C2×C3×C4H(5,𝔽2)11yescharacteristic
459#67—22no—
460#67—22no—
461#67—44no—
462C2×Dic3—21no—
463C2×Dic3—22no—
464C2×Dic3—21no—
465C2×Dic3—21no—
466C2×Dic3—44no—
467C2×Dic3—41no—
468C2×Dic3—41no—
469C2×Dic3—41no—
470C2×Dic3—41no—
471C2×Dic3—81no—
472C2×Dic3—81no—
473C2×Dic3—161no—
474D12—22no—
475D12—22no—
476D12—21no—
477D12—41no—
478D12—41no—
479D12C4×D412yes—
480C4×D3—22no—
481C4×D3—22no—
482C4×D3—44no—
483C4×D3—161no—
484C4×D3—161no—
485C4×D3—161no—
486Dic6—21no—
487Dic6—22no—
488Dic6—22no—
489Dic6—44no—
490Dic6—41no—
491Dic6—41no—
492Dic6C4×D412yes—
493C3×C8—22no—
494C3×C8—21no—
495C3×C8—21no—
496C3×C8—21no—
497C3×C8—21no—
498C3×C8—41no—
499C3⋊C8—22no—
500C3⋊C8—41no—
501C24—61no—
502#41—21no—
503#41—21no—
504#41—32no—
505#41—32no—
506#41—64no—
507#41—61no—
508#41—61no—
509#41—64no—
510#41—122no—
511#41—124no—
512#41—122no—
513#41—124no—
514#41C2×C4×D312yes—
515#41#23512yes—
516C2×Q8—32no—
517C2×Q8—31no—
518C2×Q8—64no—
519C2×Q8—61no—
520C2×Q8—64no—
521C2×Q8—61no—
522C2×Q8—61no—
523C2×Q8—61no—
524C2×Q8—61no—
525C2×Q8—121no—
526C2×Q8—122no—
527C2×Q8—122no—
528C2×Q8C2×D1211yescharacteristic
529C2×D4—21no—
530C2×D4—32no—
531C2×D4—31no—
532C2×D4—61no—
533C2×D4—61no—
534C2×D4—61no—
535C2×D4—61no—
536C2×D4—61no—
537C2×D4—61no—
538C2×D4—121no—
539C2×D4C2×D1211yescharacteristic
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—61no—
543C22×C4—61no—
544C22×C4—61no—
545C22×C4—61no—
546C22×C4—121no—
547C22×C4—121no—
548C22×C4—121no—
549C22×C4—121no—
550C22×C4—121no—
551C22×C4—121no—
552C22×C4C23×D311yescharacteristic
553SD16—64no—
554SD16—64no—
555D8—64no—
556D8—64no—
557M16—21no—
558M16—21no—
559M16—21no—
560M16—32no—
561M16—42no—
562M16—61no—
563M16—61no—
564M16—61no—
565M16—122no—
566M16C2×C4×D312yes—
567C2×C8—21no—
568C2×C8—21no—
569C2×C8—21no—
570C2×C8—21no—
571C2×C8—21no—
572C2×C8—21no—
573C2×C8—42no—
574C2×C8—61no—
575C2×C8—61no—
576C2×C8—122no—
577C2×C8C2×D1211yescharacteristic
578C2×C8C2×D1211yescharacteristic
579C2×C8D3×D411yescharacteristic
580C2×C8D3×D411yescharacteristic
581C4⋊C4—31no—
582C4⋊C4—31no—
583C4⋊C4—61no—
584C4⋊C4—61no—
585C4⋊C4—61no—
586C4⋊C4—61no—
587C4⋊C4—121no—
588C4⋊C4—121no—
589C4⋊C4C2×D1211yescharacteristic
590C4⋊C4C2×D1211yescharacteristic
591C22⋊C4—21no—
592C22⋊C4—61no—
593C22⋊C4—124no—
594C22⋊C4—121no—
595C22⋊C4—121no—
596C22⋊C4—121no—
597C22⋊C4—121no—
598C42—21no—
599C42—21no—
600C42—61no—
601C42—121no—
602C42—121no—
603C42C2×C4×D312yes—
604C22×C3—21no—
605C22×C3—21no—
606C22×C3—22no—
607C22×C3—41no—
608C22×C3—41no—
609C22×C3#51811yescharacteristic
610C22×C3D4211yescharacteristic
611C22×C3#55111yescharacteristic
612C2×D3—22no—
613C2×D3—41no—
614C2×D3—41no—
615C2×D3—42no—
616C2×D3—81no—
617C2×D3—81no—
618C2×D3—82no—
619C2×D3—161no—
620C3×C4—22no—
621C3×C4—42no—
622C3×C4—41no—
623C3×C4—41no—
624C3×C4C2×C4×D411yescharacteristic
625C3×C4C2×C4×D411yescharacteristic
626C3×C4#54611yescharacteristic
627C3×C4#54611yescharacteristic
628Dic3—22no—
629Dic3—42no—
630Dic3—41no—
631Dic3—41no—
632Dic3—81no—
633Dic3—81no—
634C23—21no—
635C23—61no—
636C23—62no—
637C23—121no—
638C23—121no—
639C23—241no—
640C23—241no—
641C23#100911yescharacteristic
642Q8—21no—
643Q8—32no—
644Q8—61no—
645Q8—62no—
646Q8—62no—
647Q8—124no—
648Q8—121no—
649Q8—121no—
650Q8C4×D1212yes—
651D4—22no—
652D4—22no—
653D4—21no—
654D4—32no—
655D4—62no—
656D4—62no—
657D4—62no—
658D4—61no—
659D4—62no—
660D4—124no—
661D4—121no—
662D4—121no—
663D4C4×D1212yes—
664C2×C4—21no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—61no—
673C2×C4—62no—
674C2×C4—62no—
675C2×C4—62no—
676C2×C4—61no—
677C2×C4—61no—
678C2×C4—124no—
679C2×C4—124no—
680C2×C4—121no—
681C2×C4—121no—
682C2×C4—121no—
683C2×C4—121no—
684C2×C4—241no—
685C2×C4—241no—
686C2×C4—481no—
687C2×C4—481no—
688C2×C4—481no—
689C2×C4—481no—
690C2×C4C22×C4×D311yescharacteristic
691C2×C4C22×D1211yescharacteristic
692C2×C4C2×#23511yescharacteristic
693C2×C4C2×D3×D411yescharacteristic
694C2×C4#100411yescharacteristic
695C2×C4D3×#4111yescharacteristic
696C8—21no—
697C8—21no—
698C8—22no—
699C8—21no—
700C8—21no—
701C8—41no—
702C8—62no—
703C8—121no—
704C2×C3—21no—
705C2×C3—21no—
706C2×C3—42no—
707C2×C3#235811yescharacteristic
708C2×C3#329011yescharacteristic
709C2×C3#329011yescharacteristic
710D3—42no—
711D3—82no—
712C22—21no—
713C22—21no—
714C22—22no—
715C22—41no—
716C22—41no—
717C22—62no—
718C22—121no—
719C22—121no—
720C22—122no—
721C22—242no—
722C22—241no—
723C22—241no—
724C22—481no—
725C22#552711yescharacteristic
726C22#554511yescharacteristic
727C22D4×D1211yescharacteristic
728C4—22no—
729C4—42no—
730C4—41no—
731C4—41no—
732C4—62no—
733C4—121no—
734C4—121no—
735C4—122no—
736C4—241no—
737C4—241no—
738C4C2×C4×D1211yescharacteristic
739C4#549011yescharacteristic
740C4C4×D3×D411yescharacteristic
741C4#555411yescharacteristic
742C3#2289511yescharacteristic, complemented, Sylow
743C2—21no—
744C2—21no—
745C2—42no—
746C2—122no—
747C2—242no—
748C2#7357711yescharacteristic
749C2#7989111yescharacteristic
750C2#7989211yescharacteristic
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1764C2
socleC22×C3D42
prime radicalC2×#4827C2
cosocleC22×C3×C4C24
hypercenter#1764D3
hypocenterC3#22895
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1764

Complements

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

NH ≅ G/Ntype
1. #83806C2semidirect
3. #78832C2semidirect
4. #78820C2semidirect
5. #78560C2semidirect
6. C3×#1764C2semidirect
7. #72018C2semidirect
8. #70364C2semidirect
9. #69891C2semidirect
10. #69789C2semidirect
11. #69773C2semidirect
12. #68948C2semidirect
13. #68930C2semidirect
14. #68772C2semidirect
15. #68759C2semidirect
50. #4827C4semidirect
83. #1764D3semidirect
88. #1010C2×C4semidirect
108. #949C2×C4semidirect
123. #908C2×C4semidirect
137. #875C2×C4semidirect
142. C3×#104C2×C4semidirect
153. #824C2×C4semidirect
179. #805C2×C4semidirect
398. #104C4×D3semidirect
742. C3#22895semidirect

Maximal subgroups

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

subgroupindex
#838062
C2×#48272
#788322
#788202
#785602
C3×#17642
#720182
#703642
#698912
#697892
#697732
#689482
#689302
#687722
#687592
#228953