Finite Group Databasebeta

← #10816031

Subgroups — #10816031

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#82571C211yesmaximal, characteristic, complemented
2#82563C211yesmaximal, characteristic
3#79202C211yesmaximal, characteristic, complemented
4C2×#4825C211yesmaximal, characteristic
5#78882C211yesmaximal, characteristic
6C3×#1860C211yesmaximal, characteristic, complemented
7#70673C211yesmaximal, characteristic, complemented
8#70369C211yesmaximal, characteristic, complemented
9#70304C211yesmaximal, characteristic, complemented
10#69562C211yesmaximal, characteristic, complemented
11#69551C211yesmaximal, characteristic, complemented
12#69007C211yesmaximal, characteristic, complemented
13#68983C211yesmaximal, characteristic, complemented
14#68766C211yesmaximal, characteristic, complemented
15#68739C211yesmaximal, characteristic, complemented
16#22896—31nomaximal, Sylow
17C2×#1009C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C2×#951C2211yescharacteristic
20C2×#948C2211yescharacteristic
21C2×#909C2211yescharacteristic
22C2×#907C2211yescharacteristic
23#5516C2211yescharacteristic, complemented
24#5504C2211yescharacteristic
25C2×#875C2211yescharacteristic
26C3×#428C2211yescharacteristic, complemented
27C2×C3×#104C2211yescharacteristic
28C3×#417C2211yescharacteristic
29C3×#382C2211yescharacteristic, complemented
30#5233C2211yescharacteristic
31C2×#824C2211yescharacteristic, complemented
32#5131C2211yescharacteristic
33C2×#823C2211yescharacteristic
34#5122C2211yescharacteristic, complemented
35#5030—21no—
36C4×#216—21no—
37#5018—21no—
38C4×#214—21no—
39#5002—21no—
40#5002C2211yescharacteristic
41#5001—21no—
42#5001C2211yescharacteristic
43#4997C2211yescharacteristic, complemented
44C2×#810C2211yescharacteristic, complemented
45C2×#810C2211yescharacteristic, complemented
46#4966—21no—
47#4966C2211yescharacteristic
48C2×#809—21no—
49C2×#809C2211yescharacteristic
50C2×#809C2211yescharacteristic
51#4964C2211yescharacteristic, complemented
52C2×#805C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic, complemented
54#4889—21no—
55#4873—21no—
56#4853—21no—
57#4852—21no—
58#4825C418yescomplemented
59#4804—21no—
60#4803—21no—
61#4770—21no—
62D3×#102—21no—
63C3×#343C2211yescharacteristic, complemented
64C3×#339C2211yescharacteristic, complemented
65#4558C2211yescharacteristic, complemented
66#4558C2211yescharacteristic, complemented
67#4530C2211yescharacteristic, 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#1860D311yescharacteristic, complemented
82#1764—31no—
83#1764—31no—
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
97C2×C3×M16C2311yescharacteristic
98C2×C3×M16C2311yescharacteristic
99C3×C4×Q8—21no—
100C3×C4×D4—21no—
101C3×C4×D4—41no—
102C3×#117—21no—
103C3×#117C2311yescharacteristic
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108#951—22no—
109#951C2×C414yescomplemented
110#948—22no—
111#948—41no—
112C2×#216—21no—
113C2×#214—21no—
114C4×#67—41no—
115C2×(C2×D3)⋊C4—41no—
116#926C2311yescharacteristic
117C2×#208—21no—
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C22×C3⋊C8—21no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#909—22no—
124#909C2×C414yescomplemented
125#907—22no—
126#907—41no—
127C2×Dic12—21no—
128C2×#204—21no—
129C2×ΓL(1,𝔽25)—21no—
130C2×D3×C8—21no—
131#893—21no—
132#892—21no—
133D3×C4⋊C4—21no—
134#887—21no—
135#887—21no—
136#887—21no—
137#877—41no—
138#875—41no—
139#875C2×C414yescomplemented
140C4×Dic6—21no—
141C4×Dic6—21no—
142C3×#107—21no—
143C3×#106—21no—
144C3×#104C2×C414yescomplemented
145C3×#103—21no—
146C3×#102—21no—
147C3×#95—41no—
148#835D411yescharacteristic
149#835D411yescharacteristic
150#832D411yescharacteristic
151#832D411yescharacteristic
152#824—41no—
153#823C2×C414yescomplemented
154#821—21no—
155#821—21no—
156#815—21no—
157C8×Dic3—21no—
158#810—21no—
159#810—21no—
160#810—21no—
161#810—21no—
162#810—21no—
163#810D411yescharacteristic
164#810D411yescharacteristic
165#809—21no—
166#809—21no—
167#809—21no—
168#809—21no—
169#809—21no—
170#809—21no—
171#809—21no—
172#809D411yescharacteristic
173#809D411yescharacteristic
174#808—21no—
175#807—21no—
176#805C2×C414yescomplemented
177#803—21no—
178C4×C3⋊C8—21no—
179C4×C3⋊C8—41no—
180C2×H(5,𝔽2)—31no—
181C2×#137—31no—
182C2×#137—31no—
183C2×#131—31no—
184C2×#131—31no—
185C2×#124—31no—
186#520—31no—
187#520C2×D311yescharacteristic
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×#104C2×D311yescharacteristic
202#418—31no—
203#418—31no—
204#418—61no—
205#418—61no—
206#417—31no—
207#417—61no—
208#417—61no—
209#417C2×D311yescharacteristic
210#416—31no—
211#416—31no—
212#416—61no—
213#416—61no—
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#405—31no—
224C2×C4×C8—31no—
225#382C2×D311yescharacteristic, complemented
226#343—31no—
227#343C2×D311yescharacteristic, complemented
228#341—31no—
229#341—31no—
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×C4—21no—
244C22×C3×C4—21no—
245C22×C3×C4—41no—
246C22×C3×C4C2411yescharacteristic
247C2×#67—21no—
248C2×#67—41no—
249C22×Dic3—21no—
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×M16C22×C412yes—
286C2×C3×C8—21no—
287C2×C3×C8—21no—
288C3×C4⋊C4—21no—
289C3×C4⋊C4—21no—
290C3×C4⋊C4—21no—
291C3×C4⋊C4—21no—
292C3×C4⋊C4—41no—
293C3×C4⋊C4C2×D411yescharacteristic
294C3×C4⋊C4C2×D411yescharacteristic
295C3×C22⋊C4—21no—
296C3×C22⋊C4—41no—
297C3×C42—21no—
298C3×C42—21no—
299C3×C42—41no—
300C3×C42C22×C412yes—
301#217—21no—
302#216—24no—
303#214—24no—
304(C2×D3)⋊C4—44no—
305(C2×D3)⋊C4—41no—
306C4⋊Dic3C2×D411yescharacteristic
307C4⋊Dic3C2×D411yescharacteristic
308Dic3⋊C4—41no—
309Dic3⋊C4—41no—
310Dic3⋊C4—41no—
311C4×Dic3—21no—
312C4×Dic3—41no—
313C4×Dic3—41no—
314C4×Dic3—41no—
315#208—21no—
316#208—21no—
317#208—21no—
318#208—42no—
319#208C22×C412yes—
320C2×C3⋊C8—21no—
321C2×C3⋊C8—21no—
322C2×C3⋊C8—21no—
323C2×C3⋊C8—21no—
324C2×C3⋊C8—21no—
325C2×C3⋊C8—21no—
326C2×C3⋊C8—42no—
327C2×C3⋊C8C2×D411yescharacteristic
328C2×C3⋊C8C2×D411yescharacteristic
329C2×C3⋊C8C2×D411yescharacteristic
330C2×C3⋊C8C2×D411yescharacteristic
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—62no—
357#131—62no—
358#131—121no—
359#131—121no—
360C2×M16—31no—
361C2×M16—31no—
362C2×M16—61no—
363C2×M16—61no—
364C2×M16C22×D311yescharacteristic
365C2×M16C22×D311yescharacteristic
366C22×C8—31no—
367C22×C8—31no—
368C22×C8—61no—
369C22×C8—61no—
370#124—34no—
371#124—121no—
372C4×Q8—21no—
373C4×Q8—61no—
374C4×Q8—61no—
375C4×Q8—61no—
376C4×Q8—61no—
377C4×Q8—61no—
378C4×D4—21no—
379C4×D4—41no—
380C4×D4—61no—
381C4×D4—121no—
382#117—21no—
383#117—31no—
384#117—61no—
385#117—121no—
386#117C22×D311yescharacteristic
387C2×C4⋊C4—21no—
388C2×C4⋊C4—61no—
389C2×C4⋊C4C22×D311yescharacteristic
390C2×C4⋊C4C22×D311yescharacteristic
391C2×C22⋊C4—121no—
392C2×C42C22×D311yescharacteristic
393#107—21no—
394#107—61no—
395#106—21no—
396#106—61no—
397#104—34no—
398#104C4×D314yescomplemented
399#103—21no—
400#103—31no—
401#103—31no—
402#103—31no—
403#103—31no—
404#103—61no—
405#103—61no—
406#103—61no—
407#103—61no—
408#103—61no—
409#102—21no—
410#102—31no—
411#102—31no—
412#102—31no—
413#102—31no—
414#102—61no—
415#102—61no—
416#102—61no—
417#102—61no—
418#102—61no—
419#102—61no—
420#102—61no—
421#102—61no—
422#102—61no—
423#101—121no—
424#100—34no—
425#97—61no—
426#97—61no—
427C4×C8—61no—
428C4×C8—61no—
429C4×C8—121no—
430#95—41no—
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×C4—41no—
457C2×C3×C4—41no—
458C2×C3×C4—81no—
459C2×C3×C4—81no—
460C2×C3×C4—161no—
461C2×C3×C4C23×C411yescharacteristic
462C2×C3×C4C22×D411yescharacteristic
463C2×C3×C4C22×D411yescharacteristic
464C2×C3×C4C2×#4111yescharacteristic
465C2×C3×C4C2×#4111yescharacteristic
466C2×C3×C4H(5,𝔽2)11yescharacteristic
467#67—22no—
468#67—22no—
469#67—44no—
470C2×Dic3—21no—
471C2×Dic3—22no—
472C2×Dic3—21no—
473C2×Dic3—21no—
474C2×Dic3—41no—
475C2×Dic3—41no—
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—41no—
495C3⋊C8—22no—
496C3⋊C8—21no—
497C3⋊C8—21no—
498C3⋊C8—21no—
499C3⋊C8—21no—
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#23912yes—
515C2×Q8—32no—
516C2×Q8—31no—
517C2×Q8—61no—
518C2×Q8—61no—
519C2×Q8—61no—
520C2×Q8—61no—
521C2×Q8—61no—
522C2×Q8D3×D411yescharacteristic
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×D4D3×D411yescharacteristic
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—21no—
544C22×C4—21no—
545C22×C4—41no—
546C22×C4—61no—
547C22×C4—61no—
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—61no—
560M16—61no—
561M16—61no—
562M16—122no—
563M16—122no—
564M16C2×C4×D312yes—
565C2×C8—21no—
566C2×C8—21no—
567C2×C8—31no—
568C2×C8—31no—
569C2×C8—31no—
570C2×C8—31no—
571C2×C8—61no—
572C2×C8—61no—
573C2×C8—61no—
574C2×C8—61no—
575C2×C8—61no—
576C2×C8—61no—
577C2×C8—122no—
578C2×C8—122no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—31no—
584C4⋊C4—31no—
585C4⋊C4—41no—
586C4⋊C4—121no—
587C4⋊C4—121no—
588C4⋊C4—121no—
589C4⋊C4D3×D411yescharacteristic
590C4⋊C4D3×D411yescharacteristic
591C22⋊C4—21no—
592C22⋊C4—41no—
593C22⋊C4—61no—
594C22⋊C4—124no—
595C22⋊C4—121no—
596C42—21no—
597C42—21no—
598C42—41no—
599C42—61no—
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×C4—81no—
626C3×C4—81no—
627C3×C4C2×C4×D411yescharacteristic
628C3×C4C2×C4×D411yescharacteristic
629C3×C4#54611yescharacteristic
630C3×C4#54611yescharacteristic
631Dic3—22no—
632Dic3—42no—
633Dic3—41no—
634Dic3—41no—
635C23—21no—
636C23—61no—
637C23—62no—
638C23—124no—
639C23—121no—
640C23—121no—
641C23—241no—
642C23—241no—
643C23—481no—
644C23—481no—
645C23#100411yescharacteristic
646Q8—21no—
647Q8—32no—
648Q8—61no—
649Q8—62no—
650Q8—62no—
651Q8—121no—
652Q8—121no—
653Q8#89312yes—
654D4—22no—
655D4—22no—
656D4—21no—
657D4—32no—
658D4—62no—
659D4—62no—
660D4—62no—
661D4—62no—
662D4—61no—
663D4—124no—
664D4—124no—
665D4—121no—
666D4—121no—
667D4#89312yes—
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—41no—
677C2×C4—41no—
678C2×C4—61no—
679C2×C4—62no—
680C2×C4—62no—
681C2×C4—62no—
682C2×C4—61no—
683C2×C4—61no—
684C2×C4—81no—
685C2×C4—81no—
686C2×C4—124no—
687C2×C4—121no—
688C2×C4—121no—
689C2×C4—161no—
690C2×C4—481no—
691C2×C4C22×C4×D311yescharacteristic
692C2×C4C2×D3×D411yescharacteristic
693C2×C4C2×D3×D411yescharacteristic
694C2×C4C2×#23911yescharacteristic
695C2×C4D3×#4111yescharacteristic
696C2×C4#100911yescharacteristic
697C8—22no—
698C8—41no—
699C8—61no—
700C8—61no—
701C8—62no—
702C8—61no—
703C8—61no—
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#554511yescharacteristic
728C22#559011yescharacteristic
729C22#563711yescharacteristic
730C4—22no—
731C4—42no—
732C4—41no—
733C4—41no—
734C4—62no—
735C4—81no—
736C4—81no—
737C4—122no—
738C4—121no—
739C4—121no—
740C4C2×#89311yescharacteristic
741C4C4×D3×D411yescharacteristic
742C4#563011yescharacteristic
743C4#565411yescharacteristic
744C3#2289611yescharacteristic, complemented, Sylow
745C2—21no—
746C2—21no—
747C2—42no—
748C2—122no—
749C2—242no—
750C2#7459611yescharacteristic
751C2#8028911yescharacteristic
752C2#8029611yescharacteristic
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1860C2
socleC22×C3D42
prime radicalC2×#1009C22
cosocleC22×C3×C4C24
hypercenter#1860D3
hypocenterC3#22896
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1860

Complements

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

NH ≅ G/Ntype
1. #82571C2semidirect
3. #79202C2semidirect
6. C3×#1860C2semidirect
7. #70673C2semidirect
8. #70369C2semidirect
9. #70304C2semidirect
10. #69562C2semidirect
11. #69551C2semidirect
12. #69007C2semidirect
13. #68983C2semidirect
14. #68766C2semidirect
15. #68739C2semidirect
23. #5516C22semidirect
26. C3×#428C22semidirect
29. C3×#382C22semidirect
31. C2×#824C22semidirect
34. #5122C22semidirect
43. #4997C22semidirect
44. C2×#810C22semidirect
45. C2×#810C22semidirect
51. #4964C22semidirect
53. C2×C4×C3⋊C8C22semidirect
58. #4825C4semidirect
63. C3×#343C22semidirect
64. C3×#339C22semidirect
65. #4558C22semidirect
66. #4558C22semidirect
67. #4530C22semidirect
68. #4523C22semidirect
81. #1860D3semidirect
109. #951C2×C4semidirect
124. #909C2×C4semidirect
139. #875C2×C4semidirect
144. C3×#104C2×C4semidirect
153. #823C2×C4semidirect
176. #805C2×C4semidirect
199. #428C2×D3semidirect
225. #382C2×D3semidirect
227. #343C2×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
#825712
#825632
#792022
C2×#48252
#788822
C3×#18602
#706732
#703692
#703042
#695622
#695512
#690072
#689832
#687662
#687392
#228963