Finite Group Databasebeta

← #10816033

Subgroups — #10816033

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#79202C211yesmaximal, characteristic, complemented
2#78925C211yesmaximal, characteristic
3#78742C211yesmaximal, characteristic
4C2×#4746C211yesmaximal, characteristic
5#78571C211yesmaximal, characteristic, complemented
6C3×#1931C211yesmaximal, characteristic, complemented
7#70681C211yesmaximal, characteristic, complemented
8#70325C211yesmaximal, characteristic, complemented
9#70158C211yesmaximal, characteristic, complemented
10#70072C211yesmaximal, characteristic, complemented
11#69887C211yesmaximal, characteristic, complemented
12#69007C211yesmaximal, characteristic, complemented
13#69003C211yesmaximal, characteristic, complemented
14#68949C211yesmaximal, characteristic, complemented
15#68926C211yesmaximal, characteristic, complemented
16#22896—31nomaximal, Sylow
17C2×#1004C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C2×#909C2211yescharacteristic
20C2×#909C2211yescharacteristic
21C2×#907C2211yescharacteristic
22C2×#901C2211yescharacteristic
23#5523C2211yescharacteristic
24#5516C2211yescharacteristic, complemented
25#5504C2211yescharacteristic
26C3×#428C2211yescharacteristic, complemented
27C3×#418C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#100C2211yescharacteristic
30C3×#405C2211yescharacteristic, complemented
31#5215C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#825C2211yescharacteristic
34C2×#825C2211yescharacteristic
35C2×#824C2211yescharacteristic, complemented
36#5133C2211yescharacteristic
37#5122C2211yescharacteristic, complemented
38#5121C2211yescharacteristic, complemented
39#5113C2211yescharacteristic
40C2×#816C2211yescharacteristic, complemented
41C2×C8×Dic3C2211yescharacteristic, complemented
42#5002C2211yescharacteristic
43#4977C2211yescharacteristic, complemented
44C2×#810C2211yescharacteristic, complemented
45#4966C2211yescharacteristic
46#4889—21no—
47#4873—21no—
48#4853—21no—
49#4852—21no—
50#4772—21no—
51#4772—21no—
52#4771—21no—
53#4771—21no—
54#4770—21no—
55#4770—21no—
56D3×#102—21no—
57D3×#102—21no—
58#4746C418yescomplemented
59#4716—21no—
60#4710—21no—
61C4×Dic12—21no—
62C4×#204—21no—
63C3×#343C2211yescharacteristic, complemented
64C3×#341C2211yescharacteristic, complemented
65#4558C2211yescharacteristic, complemented
66#4555C2211yescharacteristic, complemented
67#4554C2211yescharacteristic, complemented
68#4532C2211yescharacteristic, complemented
69C2×#455—31no—
70#2947—31no—
71#2947—31no—
72#2897—31no—
73#2897—31no—
74#1958—31no—
75#1931—31no—
76#1931D311yescharacteristic, complemented
77#1908—31no—
78#1908—31no—
79#1888—31no—
80#1860—31no—
81#1860—31no—
82#1764—31no—
83#1764—31no—
84C22×C3×D4C2311yescharacteristic
85C22×#67—41no—
86#1004—22no—
87#1004—22no—
88#1004—41no—
89#1004C2×C414yes—
90C2×#237—21no—
91C2×#237—21no—
92C2×D3×D4—21no—
93C2×D3×D4—21no—
94C2×#235C2311yescharacteristic
95C2×#235C2311yescharacteristic
96C22×Dic6C2311yescharacteristic
97C2×C3×M16C2311yescharacteristic
98C2×C3×M16C2311yescharacteristic
99C2×C3×M16C2311yescharacteristic
100C22×C3×C8C2311yescharacteristic
101C3×#117C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×#217—41no—
104#936—41no—
105#936C2×C414yescomplemented
106C4×#67—41no—
107#926C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#909—22no—
113#909—22no—
114#909C2×C414yescomplemented
115#909C2×C414yescomplemented
116#907—22no—
117#907—41no—
118C2×Dic12—21no—
119C2×Dic12—21no—
120C2×#204—21no—
121C2×#204—21no—
122#901—22no—
123#901—41no—
124C2×ΓL(1,𝔽25)—21no—
125C2×ΓL(1,𝔽25)—21no—
126C2×D3×C8—21no—
127C2×D3×C8—21no—
128#893—21no—
129#892—21no—
130#892—21no—
131D3×C4⋊C4—21no—
132D3×C4⋊C4—21no—
133#887—21no—
134#887—21no—
135#887—21no—
136#67⋊C4—41no—
137#877—41no—
138C4×D12—21no—
139C4×Dic6—21no—
140C4×Dic6—21no—
141C4×Dic6—21no—
142C3×#107—21no—
143C3×#106—21no—
144C3×#102—21no—
145C3×#102—21no—
146C3×#102—21no—
147C3×#102—21no—
148C3×#100C2×C414yescomplemented
149C3×#97—21no—
150C3×C4×C8—21no—
151#832—21no—
152#832—21no—
153#832—21no—
154#832—21no—
155C3⋊#95—41no—
156#825C2×C414yescomplemented
157#825C2×C414yescomplemented
158#824—41no—
159#821—21no—
160#821D411yescharacteristic
161#821D411yescharacteristic
162#818—21no—
163#817—21no—
164#816—21no—
165#816—21no—
166#816—21no—
167#816D411yescharacteristic
168#816D411yescharacteristic
169#815—21no—
170C8×Dic3—21no—
171C8×Dic3—41no—
172#810—21no—
173#810—21no—
174#810—21no—
175#810D411yescharacteristic
176#810D411yescharacteristic
177#809—21no—
178#809D411yescharacteristic
179#809D411yescharacteristic
180C2×H(5,𝔽2)—31no—
181C2×#137—31no—
182C2×#137—31no—
183C2×#131—31no—
184C2×#131—31no—
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#428C2×D311yescharacteristic, complemented
200C2×#104—31no—
201C2×#104—31no—
202#418—31no—
203#418—61no—
204#418—61no—
205#418C2×D311yescharacteristic
206#417—31no—
207#417—31no—
208#417—61no—
209#417—61no—
210#416—31no—
211#416—31no—
212#416—61no—
213#416—61no—
214C2×#103—31no—
215C2×#103—31no—
216C2×#102—31no—
217C2×#102—61no—
218C2×#102—61no—
219C2×#102C2×D311yescharacteristic
220C2×#101—31no—
221C2×#100C2×D311yescharacteristic
222#405—31no—
223#405C2×D311yescharacteristic, complemented
224C2×C4×C8—31no—
225#382—31no—
226#343—31no—
227#343C2×D311yescharacteristic, complemented
228#341—31no—
229#341C2×D311yescharacteristic, complemented
230#339—31no—
231#339—31no—
232C24×C3—21no—
233C23×D3—41no—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4—21no—
239C2×C3×D4C22×C412yes—
240C22×C3×C4—21no—
241C22×C3×C4C2411yescharacteristic
242C2×#67—21no—
243C2×#67—21no—
244C2×#67—41no—
245C2×#67—41no—
246C2×#67—44no—
247C2×#67—44no—
248C22×Dic3—21no—
249C22×Dic3—21no—
250C22×Dic3—21no—
251C22×Dic3—41no—
252#237—24no—
253#237—24no—
254#237—42no—
255#237—42no—
256D3×D4—24no—
257D3×D4—24no—
258D3×D4—42no—
259D3×D4—42no—
260#235—21no—
261#235—21no—
262#235—21no—
263#235—21no—
264#235C22×C412yes—
265#235C22×C412yes—
266#235#4112yes—
267#235#4112yes—
268C2×D12C2×D411yescharacteristic
269C2×D12C2×D411yescharacteristic
270C2×C4×D3—21no—
271C2×C4×D3—21no—
272C2×C4×D3—41no—
273C2×C4×D3—41no—
274C2×Dic6—21no—
275C2×Dic6—21no—
276C2×Dic6—21no—
277C2×Dic6—21no—
278C2×Dic6—21no—
279C2×Dic6C22×C412yes—
280C2×Dic6C2×D411yescharacteristic
281C2×Dic6C2×D411yescharacteristic
282C3×M16—21no—
283C3×M16—21no—
284C3×M16—21no—
285C3×M16—21no—
286C3×M16C22×C412yes—
287C3×M16C22×C412yes—
288C2×C3×C8—21no—
289C2×C3×C8—21no—
290C2×C3×C8—21no—
291C2×C3×C8—21no—
292C2×C3×C8—21no—
293C2×C3×C8C2×D411yescharacteristic
294C2×C3×C8C2×D411yescharacteristic
295C3×C4⋊C4—21no—
296C3×C4⋊C4—21no—
297C3×C4⋊C4C2×D411yescharacteristic
298C3×C4⋊C4C2×D411yescharacteristic
299C3×C22⋊C4—21no—
300C3×C42—21no—
301#217—21no—
302#217—44no—
303(C2×D3)⋊C4—41no—
304(C2×D3)⋊C4—41no—
305C4⋊Dic3—21no—
306C4⋊Dic3—21no—
307C4⋊Dic3C2×D411yescharacteristic
308C4⋊Dic3C2×D411yescharacteristic
309Dic3⋊C4—41no—
310Dic3⋊C4—41no—
311Dic3⋊C4—41no—
312Dic3⋊C4—41no—
313C4×Dic3—21no—
314C4×Dic3—21no—
315C4×Dic3—41no—
316C4×Dic3—41no—
317C4×Dic3—41no—
318C4×Dic3—41no—
319C4×Dic3C22×C412yes—
320#208—21no—
321#208—21no—
322C2×C3⋊C8—21no—
323C2×C3⋊C8—21no—
324C2×C3⋊C8—21no—
325C2×C3⋊C8C2×D411yescharacteristic
326C2×C3⋊C8C2×D411yescharacteristic
327Dic12—24no—
328Dic12—24no—
329#204—24no—
330#204—24no—
331ΓL(1,𝔽25)—42no—
332ΓL(1,𝔽25)—42no—
333D3×C8—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—31no—
341C2×#41—61no—
342C2×#41—61no—
343C22×Q8—31no—
344C22×D4—61no—
345C22×D4—61no—
346C22×D4—121no—
347C22×D4C22×D311yescharacteristic
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—62no—
357#131—62no—
358#131—121no—
359#131—121no—
360C2×M16—31no—
361C2×M16—61no—
362C2×M16—61no—
363C2×M16C22×D311yescharacteristic
364C2×M16C22×D311yescharacteristic
365C2×M16C22×D311yescharacteristic
366C22×C8—31no—
367C22×C8—61no—
368C22×C8—61no—
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—61no—
389C2×C4⋊C4—61no—
390C2×C4⋊C4C22×D311yescharacteristic
391C2×C22⋊C4—121no—
392C2×C42—31no—
393#107—21no—
394#107—61no—
395#106—21no—
396#106—61no—
397#104—34no—
398#104—34no—
399#103—31no—
400#103—31no—
401#103—31no—
402#103—31no—
403#103—61no—
404#103—61no—
405#103—61no—
406#103—61no—
407#103—61no—
408#103—61no—
409#102—21no—
410#102—21no—
411#102—21no—
412#102—21no—
413#102—31no—
414#102—31no—
415#102—31no—
416#102—31no—
417#102—61no—
418#102—61no—
419#102—61no—
420#102—61no—
421#102—61no—
422#102—61no—
423#101—121no—
424#100C4×D314yescomplemented
425#97—21no—
426#97—61no—
427C4×C8—21no—
428C4×C8—61no—
429C4×C8—121no—
430#95—121no—
431C23×C3—22no—
432C23×C3—41no—
433C23×C3—41no—
434C23×C3—81no—
435C23×C3—81no—
436C23×C3H(5,𝔽2)11yescharacteristic
437C22×D3—21no—
438C22×D3—21no—
439C22×D3—44no—
440C22×D3—161no—
441C22×D3—161no—
442C3×D4—22no—
443C3×D4—22no—
444C3×D4—41no—
445C3×D4—41no—
446C2×C3×C4—21no—
447C2×C3×C4—21no—
448C2×C3×C4—41no—
449C2×C3×C4—81no—
450C2×C3×C4C23×C411yescharacteristic
451C2×C3×C4C22×D411yescharacteristic
452C2×C3×C4C22×D411yescharacteristic
453C2×C3×C4C2×#4111yescharacteristic
454C2×C3×C4C2×#4111yescharacteristic
455C2×C3×C4H(5,𝔽2)11yescharacteristic
456#67—22no—
457#67—22no—
458#67—22no—
459#67—22no—
460#67—44no—
461#67—44no—
462C2×Dic3—21no—
463C2×Dic3—21no—
464C2×Dic3—22no—
465C2×Dic3—22no—
466C2×Dic3—21no—
467C2×Dic3—21no—
468C2×Dic3—44no—
469C2×Dic3—41no—
470C2×Dic3—41no—
471C2×Dic3—41no—
472C2×Dic3—41no—
473C2×Dic3—41no—
474C2×Dic3—81no—
475D12—21no—
476D12—21no—
477D12C4×D412yes—
478D12C4×D412yes—
479C4×D3—22no—
480C4×D3—22no—
481C4×D3—22no—
482C4×D3—22no—
483C4×D3—161no—
484C4×D3—161no—
485Dic6—21no—
486Dic6—21no—
487Dic6—22no—
488Dic6—22no—
489Dic6—41no—
490Dic6—41no—
491Dic6C4×D412yes—
492Dic6C4×D412yes—
493C3×C8—22no—
494C3×C8—22no—
495C3×C8—21no—
496C3×C8—21no—
497C3×C8—41no—
498C3⋊C8—21no—
499C3⋊C8—21no—
500C3⋊C8—41no—
501C24—21no—
502C24—121no—
503#41—32no—
504#41—32no—
505#41—32no—
506#41—32no—
507#41—61no—
508#41—61no—
509#41—61no—
510#41—61no—
511#41—64no—
512#41—64no—
513#41—122no—
514#41—122no—
515C2×Q8—32no—
516C2×Q8—31no—
517C2×Q8—31no—
518C2×Q8—61no—
519C2×Q8—61no—
520C2×Q8—61no—
521C2×Q8—61no—
522C2×Q8—61no—
523C2×D4—21no—
524C2×D4—21no—
525C2×D4—21no—
526C2×D4—21no—
527C2×D4—21no—
528C2×D4—31no—
529C2×D4—31no—
530C2×D4—64no—
531C2×D4—64no—
532C2×D4—61no—
533C2×D4—61no—
534C2×D4—122no—
535C2×D4—122no—
536C2×D4—121no—
537C2×D4—121no—
538C2×D4—124no—
539C2×D4—124no—
540C2×D4C2×C4×D312yes—
541C22×C4—21no—
542C22×C4—61no—
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—21no—
559M16—61no—
560M16—61no—
561M16—122no—
562M16—122no—
563M16C2×C4×D312yes—
564M16C2×C4×D312yes—
565C2×C8—21no—
566C2×C8—21no—
567C2×C8—21no—
568C2×C8—21no—
569C2×C8—21no—
570C2×C8—31no—
571C2×C8—31no—
572C2×C8—61no—
573C2×C8—61no—
574C2×C8—61no—
575C2×C8—122no—
576C2×C8—122no—
577C2×C8D3×D411yescharacteristic
578C2×C8D3×D411yescharacteristic
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—31no—
582C4⋊C4—31no—
583C4⋊C4—61no—
584C4⋊C4—61no—
585C4⋊C4—121no—
586C4⋊C4—121no—
587C4⋊C4—121no—
588C4⋊C4—121no—
589C4⋊C4D3×D411yescharacteristic
590C4⋊C4D3×D411yescharacteristic
591C22⋊C4—21no—
592C22⋊C4—61no—
593C22⋊C4—121no—
594C22⋊C4—121no—
595C22⋊C4—124no—
596C42—21no—
597C42—32no—
598C42—61no—
599C42—61no—
600C42—121no—
601C42—121no—
602C42—121no—
603C42—121no—
604C22×C3—21no—
605C22×C3—21no—
606C22×C3—42no—
607C22×C3—82no—
608C22×C3—81no—
609C22×C3—81no—
610C22×C3—161no—
611C22×C3#51811yescharacteristic
612C22×C3D4211yescharacteristic
613C22×C3#55111yescharacteristic
614C2×D3—22no—
615C2×D3—22no—
616C2×D3—41no—
617C2×D3—41no—
618C2×D3—41no—
619C2×D3—41no—
620C2×D3—164no—
621C3×C4—41no—
622C3×C4—41no—
623C3×C4—81no—
624C3×C4C2×C4×D411yescharacteristic
625C3×C4C2×C4×D411yescharacteristic
626C3×C4#54611yescharacteristic
627C3×C4#54611yescharacteristic
628Dic3—22no—
629Dic3—22no—
630Dic3—42no—
631Dic3—42no—
632Dic3—41no—
633Dic3—41no—
634Dic3—81no—
635C23—22no—
636C23—41no—
637C23—41no—
638C23—61no—
639C23—61no—
640C23—81no—
641C23—81no—
642C23—124no—
643C23—481no—
644C23—481no—
645C23#100911yescharacteristic
646Q8—32no—
647Q8—32no—
648Q8—61no—
649Q8—61no—
650Q8—62no—
651Q8—62no—
652Q8—121no—
653Q8—121no—
654D4—22no—
655D4—22no—
656D4—32no—
657D4—32no—
658D4—41no—
659D4—41no—
660D4—62no—
661D4—62no—
662D4—62no—
663D4—62no—
664D4—61no—
665D4—61no—
666D4—124no—
667D4—124no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—41no—
671C2×C4—61no—
672C2×C4—61no—
673C2×C4—62no—
674C2×C4—62no—
675C2×C4—62no—
676C2×C4—62no—
677C2×C4—62no—
678C2×C4—62no—
679C2×C4—61no—
680C2×C4—61no—
681C2×C4—81no—
682C2×C4—124no—
683C2×C4—121no—
684C2×C4—121no—
685C2×C4—121no—
686C2×C4—121no—
687C2×C4—121no—
688C2×C4—241no—
689C2×C4—481no—
690C2×C4—481no—
691C2×C4C22×C4×D311yescharacteristic
692C2×C4C2×D3×D411yescharacteristic
693C2×C4C2×D3×D411yescharacteristic
694C2×C4D3×#4111yescharacteristic
695C2×C4D3×#4111yescharacteristic
696C2×C4#100911yescharacteristic
697C8—21no—
698C8—21no—
699C8—22no—
700C8—22no—
701C8—41no—
702C8—61no—
703C8—61no—
704C8—121no—
705C2×C3—21no—
706C2×C3—21no—
707C2×C3—82no—
708C2×C3#235811yescharacteristic
709C2×C3#329011yescharacteristic
710C2×C3#329011yescharacteristic
711D3—42no—
712D3—42no—
713C22—21no—
714C22—21no—
715C22—42no—
716C22—62no—
717C22—62no—
718C22—82no—
719C22—81no—
720C22—81no—
721C22—121no—
722C22—121no—
723C22—121no—
724C22—121no—
725C22—161no—
726C22—484no—
727C22#552111yescharacteristic
728C22#560911yescharacteristic
729C22#570511yescharacteristic
730C4—41no—
731C4—41no—
732C4—62no—
733C4—62no—
734C4—81no—
735C4—122no—
736C4—121no—
737C4—121no—
738C4—122no—
739C4—241no—
740C4C4×D3×D411yescharacteristic
741C4C4×D3×D411yescharacteristic
742C4#565411yescharacteristic
743C4#567611yescharacteristic
744C3#2289611yescharacteristic, complemented, Sylow
745C2—21no—
746C2—21no—
747C2—82no—
748C2—122no—
749C2—122no—
750C2#7583611yescharacteristic
751C2#8093411yescharacteristic
752C2#8094011yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1931C2
socleC22×C3D42
prime radicalC2×#1004C22
cosocleC22×C3×C4C24
hypercenter#1931D3
hypocenterC3#22896
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1931

Complements

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

NH ≅ G/Ntype
1. #79202C2semidirect
5. #78571C2semidirect
6. C3×#1931C2semidirect
7. #70681C2semidirect
8. #70325C2semidirect
9. #70158C2semidirect
10. #70072C2semidirect
11. #69887C2semidirect
12. #69007C2semidirect
13. #69003C2semidirect
14. #68949C2semidirect
15. #68926C2semidirect
24. #5516C22semidirect
26. C3×#428C22semidirect
30. C3×#405C22semidirect
35. C2×#824C22semidirect
37. #5122C22semidirect
38. #5121C22semidirect
40. C2×#816C22semidirect
41. C2×C8×Dic3C22semidirect
43. #4977C22semidirect
44. C2×#810C22semidirect
58. #4746C4semidirect
63. C3×#343C22semidirect
64. C3×#341C22semidirect
65. #4558C22semidirect
66. #4555C22semidirect
67. #4554C22semidirect
68. #4532C22semidirect
76. #1931D3semidirect
105. #936C2×C4semidirect
114. #909C2×C4semidirect
115. #909C2×C4semidirect
148. C3×#100C2×C4semidirect
156. #825C2×C4semidirect
157. #825C2×C4semidirect
199. #428C2×D3semidirect
223. #405C2×D3semidirect
227. #343C2×D3semidirect
229. #341C2×D3semidirect
424. #100C4×D3semidirect
744. C3#22896semidirect

Maximal subgroups

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

subgroupindex
#792022
#789252
#787422
C2×#47462
#785712
C3×#19312
#706812
#703252
#701582
#700722
#698872
#690072
#690032
#689492
#689262
#228963