Finite Group Databasebeta

← #10816037

Subgroups — #10816037

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83797C211yesmaximal, characteristic
2C2×#4825C211yesmaximal, characteristic
3#78831C211yesmaximal, characteristic
4#78821C211yesmaximal, characteristic, complemented
5#78571C211yesmaximal, characteristic, complemented
6C3×#1764C211yesmaximal, characteristic, complemented
7#72019C211yesmaximal, characteristic, complemented
8#70364C211yesmaximal, characteristic, complemented
9#69892C211yesmaximal, characteristic, complemented
10#69789C211yesmaximal, characteristic, complemented
11#69773C211yesmaximal, characteristic, complemented
12#68949C211yesmaximal, characteristic, complemented
13#68931C211yesmaximal, characteristic, complemented
14#68772C211yesmaximal, characteristic, complemented
15#68766C211yesmaximal, characteristic, complemented
16#22896—31nomaximal, Sylow
17C2×#1009C2211yescharacteristic
18C2×C3×#131C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#951C2211yescharacteristic
21C2×#909C2211yescharacteristic
22C2×#901C2211yescharacteristic
23#5523C2211yescharacteristic
24#5516C2211yescharacteristic, complemented
25C2×#875C2211yescharacteristic
26C2×C3×#104C2211yescharacteristic
27C3×#416C2211yescharacteristic
28C3×#405C2211yescharacteristic, complemented
29C2×C3×C4×C8C2211yescharacteristic, complemented
30#5232C2211yescharacteristic
31C2×#824C2211yescharacteristic, complemented
32#5133—21no—
33#5133C2211yescharacteristic
34#5131—21no—
35#5131C2211yescharacteristic
36C2×#823C2211yescharacteristic
37#5121C2211yescharacteristic, complemented
38#5113—21no—
39#5113C2211yescharacteristic
40C2×#821—21no—
41C2×#821C2211yescharacteristic
42C2×#821C2211yescharacteristic
43C2×#816C2211yescharacteristic, complemented
44C2×#816C2211yescharacteristic, complemented
45#5101C2211yescharacteristic, complemented
46#5001C2211yescharacteristic
47#4995C2211yescharacteristic, complemented
48#4936C2211yescharacteristic, complemented
49C2×#805C2211yescharacteristic
50#4825C418yescomplemented
51#4804—21no—
52#4803—21no—
53#4790—21no—
54#4789—21no—
55#4770—21no—
56D3×#102—21no—
57#4758—21no—
58#4757—21no—
59#4716—21no—
60#4710—21no—
61C4×Dic12—21no—
62C4×#204—21no—
63C3×#341C2211yescharacteristic, complemented
64C3×#339C2211yescharacteristic, complemented
65#4554C2211yescharacteristic, complemented
66#4554C2211yescharacteristic, complemented
67#4552C2211yescharacteristic, complemented
68#4523C2211yescharacteristic, complemented
69C2×#455—31no—
70#2947—31no—
71#2947—31no—
72#2897—31no—
73#2897—31no—
74#1958—31no—
75#1931—31no—
76#1931—31no—
77#1908—31no—
78#1908—31no—
79#1888—31no—
80#1860—31no—
81#1860—31no—
82#1764—31no—
83#1764D311yescharacteristic, complemented
84C2×C3×#41C2311yescharacteristic
85#1009—22no—
86#1009—22no—
87#1009—41no—
88#1009C2×C414yes—
89C2×#239—21no—
90C2×D3×D4—21no—
91C2×D3×D4—41no—
92C2×#235—21no—
93C2×#235C2311yescharacteristic
94C22×D12—21no—
95C22×D12C2311yescharacteristic
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#951—22no—
108#951C2×C414yescomplemented
109Q8×Dic3—21no—
110C2×#216—21no—
111D4×Dic3—21no—
112D4×Dic3—41no—
113C2×#214—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#909—22no—
123#909C2×C414yescomplemented
124C2×Dic12—21no—
125C2×#204—21no—
126#901—22no—
127#901—41no—
128C2×ΓL(1,𝔽25)—21no—
129C2×D3×C8—21no—
130#892—21no—
131D3×C4⋊C4—21no—
132#887—21no—
133#887—21no—
134#67⋊C4—41no—
135#877—41no—
136#875—41no—
137#875C2×C414yescomplemented
138C4×D12—21no—
139C4×Dic6—21no—
140C4×Dic6—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#824—41no—
154#823C2×C414yescomplemented
155#821—21no—
156#821—21no—
157#821—21no—
158#821—21no—
159#821—21no—
160#821—21no—
161#821—21no—
162#821D411yescharacteristic
163#821D411yescharacteristic
164#818—21no—
165#817—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#809—21no—
176#809—21no—
177#808—21no—
178#807—21no—
179#805C2×C414yescomplemented
180C2×H(5,𝔽2)—31no—
181C2×#137—31no—
182C2×#137—31no—
183C2×#131—31no—
184C2×#131C2×D311yescharacteristic
185C2×#124—31no—
186#520—31no—
187#520—31no—
188#519—31no—
189#455—38no—
190#441—61no—
191#441—61no—
192#440—61no—
193#440—61no—
194C4×Q16—61no—
195C4×Q16—61no—
196C4×SD16—61no—
197C4×SD16—61no—
198#428—31no—
199#428—31no—
200C2×#104—31no—
201C2×#104C2×D311yescharacteristic
202#418—31no—
203#418—31no—
204#418—61no—
205#418—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×#102—31no—
217C2×#102—31no—
218C2×#102—61no—
219C2×#102—61no—
220C2×#101—31no—
221C2×#100—31no—
222#405—31no—
223#405C2×D311yescharacteristic, complemented
224C2×C4×C8C2×D311yescharacteristic, complemented
225#382—31no—
226#343—31no—
227#343—31no—
228#341—31no—
229#341C2×D311yescharacteristic, complemented
230#339—31no—
231#339C2×D311yescharacteristic, complemented
232C23×D3—21no—
233C23×D3—41no—
234C3×#41—21no—
235C3×#41—21no—
236C3×#41C22×C412yes—
237C3×#41#4112yes—
238C2×C3×Q8C2×D411yescharacteristic
239C2×C3×D4—21no—
240C2×C3×D4C2×D411yescharacteristic
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4C2411yescharacteristic
244C2×#67—21no—
245C2×#67—41no—
246C22×Dic3—21no—
247C22×Dic3—21no—
248C22×Dic3—21no—
249C22×Dic3—41no—
250#239—24no—
251#239—42no—
252D3×D4—24no—
253D3×D4—42no—
254D3×D4—44no—
255D3×D4—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—24no—
267C2×D12—21no—
268C2×D12—42no—
269C2×D12—41no—
270C2×D12C22×C412yes—
271C2×D12C2×D411yescharacteristic
272C2×C4×D3—21no—
273C2×C4×D3—41no—
274C2×C4×D3—41no—
275C2×Dic6—21no—
276C2×Dic6—21no—
277C2×Dic6—21no—
278C2×Dic6—21no—
279C2×Dic6—21no—
280C2×Dic6C22×C412yes—
281C2×Dic6C2×D411yescharacteristic
282C3×M16—21no—
283C3×M16—21no—
284C3×M16—21no—
285C3×M16—42no—
286C3×M16C22×C412yes—
287C2×C3×C8—21no—
288C2×C3×C8—21no—
289C2×C3×C8—21no—
290C2×C3×C8—21no—
291C2×C3×C8—21no—
292C2×C3×C8—21no—
293C2×C3×C8—42no—
294C2×C3×C8C2×D411yescharacteristic
295C2×C3×C8C2×D411yescharacteristic
296C2×C3×C8C2×D411yescharacteristic
297C2×C3×C8C2×D411yescharacteristic
298C3×C4⋊C4C2×D411yescharacteristic
299C3×C4⋊C4C2×D411yescharacteristic
300C3×C22⋊C4—21no—
301C3×C42—21no—
302C3×C42—21no—
303C3×C42C22×C412yes—
304#217—21no—
305#217—41no—
306#216—24no—
307#214—24no—
308(C2×D3)⋊C4—44no—
309(C2×D3)⋊C4—41no—
310C4⋊Dic3—21no—
311C4⋊Dic3—21no—
312C4⋊Dic3—21no—
313C4⋊Dic3—21no—
314C4⋊Dic3—41no—
315C4⋊Dic3C2×D411yescharacteristic
316C4⋊Dic3C2×D411yescharacteristic
317Dic3⋊C4—41no—
318Dic3⋊C4—41no—
319Dic3⋊C4—41no—
320C4×Dic3—21no—
321C4×Dic3—41no—
322C4×Dic3—41no—
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—
331Dic12—24no—
332#204—24no—
333ΓL(1,𝔽25)—42no—
334D3×C8—42no—
335H(5,𝔽2)—34no—
336H(5,𝔽2)—62no—
337H(5,𝔽2)—62no—
338H(5,𝔽2)—121no—
339C2×#41—31no—
340C2×#41—61no—
341C2×#41—61no—
342C2×#41C22×D311yescharacteristic
343C22×Q8—31no—
344C22×D4—31no—
345C22×D4—61no—
346C22×D4—61no—
347C22×D4—121no—
348#137—34no—
349#137—34no—
350#137—62no—
351#137—62no—
352C2×Q16—61no—
353C2×Q16—61no—
354C2×SD16—61no—
355C2×SD16—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×Q8—61no—
375C4×Q8—61no—
376C4×Q8—61no—
377C4×Q8—61no—
378C4×D4—61no—
379C4×D4—61no—
380C4×D4—121no—
381C4×D4—121no—
382#117—31no—
383#117—61no—
384#117—61no—
385#117—121no—
386#117C22×D311yescharacteristic
387C2×C4⋊C4—31no—
388C2×C4⋊C4—31no—
389C2×C4⋊C4—61no—
390C2×C4⋊C4—61no—
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#103D1211yescharacteristic
408#103D1211yescharacteristic
409#102—31no—
410#102—31no—
411#102—61no—
412#102—61no—
413#102—61no—
414#102—61no—
415#102—61no—
416#102—61no—
417#102—61no—
418#102—61no—
419#102—61no—
420#102—61no—
421#102D1211yescharacteristic
422#102D1211yescharacteristic
423#101—121no—
424#100—34no—
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—44no—
436C22×D3—41no—
437C22×D3—41no—
438C22×D3—81no—
439C22×D3—81no—
440C22×D3—161no—
441C22×D3—161no—
442C3×Q8—21no—
443C3×Q8C4×D412yes—
444C3×D4—22no—
445C3×D4—22no—
446C3×D4—21no—
447C3×D4C4×D412yes—
448C2×C3×C4—21no—
449C2×C3×C4—22no—
450C2×C3×C4—22no—
451C2×C3×C4—22no—
452C2×C3×C4—21no—
453C2×C3×C4—21no—
454C2×C3×C4—41no—
455C2×C3×C4—41no—
456C2×C3×C4C23×C411yescharacteristic
457C2×C3×C4C22×D411yescharacteristic
458C2×C3×C4C22×D411yescharacteristic
459C2×C3×C4C2×#4111yescharacteristic
460C2×C3×C4C2×#4111yescharacteristic
461C2×C3×C4H(5,𝔽2)11yescharacteristic
462#67—22no—
463#67—22no—
464#67—44no—
465C2×Dic3—21no—
466C2×Dic3—22no—
467C2×Dic3—21no—
468C2×Dic3—21no—
469C2×Dic3—41no—
470C2×Dic3—41no—
471C2×Dic3—41no—
472C2×Dic3—41no—
473C2×Dic3—81no—
474C2×Dic3—81no—
475C2×Dic3—161no—
476D12—22no—
477D12—22no—
478D12—21no—
479D12—44no—
480D12—41no—
481D12—41no—
482D12C4×D412yes—
483C4×D3—22no—
484C4×D3—22no—
485C4×D3—44no—
486C4×D3—161no—
487Dic6—21no—
488Dic6—22no—
489Dic6—22no—
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—
502C24—121no—
503#41—21no—
504#41—21no—
505#41—32no—
506#41—32no—
507#41—64no—
508#41—61no—
509#41—61no—
510#41—64no—
511#41—122no—
512#41—122no—
513#41C2×C4×D312yes—
514#41#23512yes—
515C2×Q8—32no—
516C2×Q8—31no—
517C2×Q8—61no—
518C2×Q8—61no—
519C2×Q8—61no—
520C2×Q8—61no—
521C2×Q8—61no—
522C2×Q8C2×D1211yescharacteristic
523C2×D4—21no—
524C2×D4—32no—
525C2×D4—31no—
526C2×D4—64no—
527C2×D4—61no—
528C2×D4—64no—
529C2×D4—61no—
530C2×D4—61no—
531C2×D4—61no—
532C2×D4—61no—
533C2×D4—61no—
534C2×D4—122no—
535C2×D4—121no—
536C2×D4—122no—
537C2×D4—121no—
538C2×D4—124no—
539C2×D4—124no—
540C2×D4C2×D1211yescharacteristic
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—61no—
544C22×C4—61no—
545C22×C4—61no—
546C22×C4—61no—
547C22×C4—121no—
548C22×C4—121no—
549C22×C4—121no—
550C22×C4C23×D311yescharacteristic
551Q16—64no—
552Q16—64no—
553SD16—64no—
554SD16—64no—
555M16—21no—
556M16—21no—
557M16—21no—
558M16—32no—
559M16—42no—
560M16—61no—
561M16—61no—
562M16—61no—
563M16—122no—
564M16C2×C4×D312yes—
565C2×C8—21no—
566C2×C8—21no—
567C2×C8—21no—
568C2×C8—21no—
569C2×C8—21no—
570C2×C8—21no—
571C2×C8—42no—
572C2×C8—61no—
573C2×C8—61no—
574C2×C8—122no—
575C2×C8C2×D1211yescharacteristic
576C2×C8C2×D1211yescharacteristic
577C2×C8D3×D411yescharacteristic
578C2×C8D3×D411yescharacteristic
579C4⋊C4—31no—
580C4⋊C4—31no—
581C4⋊C4—61no—
582C4⋊C4—61no—
583C4⋊C4—61no—
584C4⋊C4—61no—
585C4⋊C4—121no—
586C4⋊C4—121no—
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—
596C42—21no—
597C42—21no—
598C42—61no—
599C42—121no—
600C42—121no—
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—164no—
620C2×D3—161no—
621C3×C4—22no—
622C3×C4—42no—
623C3×C4—41no—
624C3×C4—41no—
625C3×C4C2×C4×D411yescharacteristic
626C3×C4C2×C4×D411yescharacteristic
627C3×C4#54611yescharacteristic
628C3×C4#54611yescharacteristic
629Dic3—22no—
630Dic3—42no—
631Dic3—41no—
632Dic3—41no—
633Dic3—81no—
634Dic3—81no—
635C23—21no—
636C23—61no—
637C23—62no—
638C23—124no—
639C23—121no—
640C23—121no—
641C23—241no—
642C23—241no—
643C23—481no—
644C23—481no—
645C23#100911yescharacteristic
646Q8—21no—
647Q8—32no—
648Q8—61no—
649Q8—62no—
650Q8—62no—
651Q8—121no—
652Q8—121no—
653Q8C4×D1212yes—
654D4—22no—
655D4—22no—
656D4—21no—
657D4—32no—
658D4—62no—
659D4—62no—
660D4—62no—
661D4—61no—
662D4—62no—
663D4—124no—
664D4—124no—
665D4—121no—
666D4—121no—
667D4C4×D1212yes—
668C2×C4—21no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—61no—
677C2×C4—62no—
678C2×C4—62no—
679C2×C4—62no—
680C2×C4—61no—
681C2×C4—61no—
682C2×C4—124no—
683C2×C4—121no—
684C2×C4—121no—
685C2×C4—121no—
686C2×C4—121no—
687C2×C4—241no—
688C2×C4—241no—
689C2×C4—481no—
690C2×C4—481no—
691C2×C4C22×C4×D311yescharacteristic
692C2×C4C22×D1211yescharacteristic
693C2×C4C2×#23511yescharacteristic
694C2×C4C2×D3×D411yescharacteristic
695C2×C4#100411yescharacteristic
696C2×C4D3×#4111yescharacteristic
697C8—21no—
698C8—21no—
699C8—22no—
700C8—21no—
701C8—21no—
702C8—41no—
703C8—62no—
704C8—121no—
705C2×C3—21no—
706C2×C3—21no—
707C2×C3—42no—
708C2×C3#235811yescharacteristic
709C2×C3#329011yescharacteristic
710C2×C3#329011yescharacteristic
711D3—42no—
712D3—82no—
713C22—21no—
714C22—21no—
715C22—22no—
716C22—41no—
717C22—41no—
718C22—62no—
719C22—121no—
720C22—121no—
721C22—122no—
722C22—242no—
723C22—241no—
724C22—241no—
725C22—481no—
726C22—484no—
727C22#552711yescharacteristic
728C22#554511yescharacteristic
729C22D4×D1211yescharacteristic
730C4—22no—
731C4—42no—
732C4—41no—
733C4—41no—
734C4—62no—
735C4—121no—
736C4—121no—
737C4—122no—
738C4—241no—
739C4—241no—
740C4C2×C4×D1211yescharacteristic
741C4#549011yescharacteristic
742C4C4×D3×D411yescharacteristic
743C4#555411yescharacteristic
744C3#2289611yescharacteristic, complemented, Sylow
745C2—21no—
746C2—21no—
747C2—42no—
748C2—122no—
749C2—242no—
750C2#7357711yescharacteristic
751C2#7989111yescharacteristic
752C2#7989211yescharacteristic
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1764C2
socleC22×C3D42
prime radicalC2×#1009C22
cosocleC22×C3×C4C24
hypercenter#1764D3
hypocenterC3#22896
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
4. #78821C2semidirect
5. #78571C2semidirect
6. C3×#1764C2semidirect
7. #72019C2semidirect
8. #70364C2semidirect
9. #69892C2semidirect
10. #69789C2semidirect
11. #69773C2semidirect
12. #68949C2semidirect
13. #68931C2semidirect
14. #68772C2semidirect
15. #68766C2semidirect
24. #5516C22semidirect
28. C3×#405C22semidirect
29. C2×C3×C4×C8C22semidirect
31. C2×#824C22semidirect
37. #5121C22semidirect
43. C2×#816C22semidirect
44. C2×#816C22semidirect
45. #5101C22semidirect
47. #4995C22semidirect
48. #4936C22semidirect
50. #4825C4semidirect
63. C3×#341C22semidirect
64. C3×#339C22semidirect
65. #4554C22semidirect
66. #4554C22semidirect
67. #4552C22semidirect
68. #4523C22semidirect
83. #1764D3semidirect
108. #951C2×C4semidirect
123. #909C2×C4semidirect
137. #875C2×C4semidirect
142. C3×#104C2×C4semidirect
154. #823C2×C4semidirect
179. #805C2×C4semidirect
223. #405C2×D3semidirect
224. C2×C4×C8C2×D3semidirect
229. #341C2×D3semidirect
231. #339C2×D3semidirect
398. #104C4×D3semidirect
744. C3#22896semidirect

Maximal subgroups

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

subgroupindex
#837972
C2×#48252
#788312
#788212
#785712
C3×#17642
#720192
#703642
#698922
#697892
#697732
#689492
#689312
#687722
#687662
#228963