Finite Group Databasebeta

← #10816035

Subgroups — #10816035

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
1C2×#5202C211yesmaximal, characteristic
2#83965C211yesmaximal, characteristic, complemented
3#83797C211yesmaximal, characteristic
4#78882C211yesmaximal, characteristic
5#78774C211yesmaximal, characteristic, complemented
6C3×#1908C211yesmaximal, characteristic, complemented
7#72008C211yesmaximal, characteristic, complemented
8#70298C211yesmaximal, characteristic, complemented
9#70226C211yesmaximal, characteristic, complemented
10#70123C211yesmaximal, characteristic, complemented
11#70104C211yesmaximal, characteristic, complemented
12#69003C211yesmaximal, characteristic, complemented
13#68983C211yesmaximal, characteristic, complemented
14#68931C211yesmaximal, characteristic, complemented
15#68926C211yesmaximal, characteristic, complemented
16#22896—31nomaximal, Sylow
17C2×#1009C2211yescharacteristic
18C2×C3×#137C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20#5827C2211yescharacteristic
21C2×#947C2211yescharacteristic
22#5812C2211yescharacteristic, complemented
23C2×#942C2211yescharacteristic
24C2×#907C2211yescharacteristic
25#5504C2211yescharacteristic
26C3×#417C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic, complemented
28C2×C3×#101C2211yescharacteristic, complemented
29C2×#837C2211yescharacteristic
30#5233C2211yescharacteristic
31C2×#835C2211yescharacteristic, complemented
32#5202C418yescomplemented
33#5184—21no—
34Dic3×Q16—21no—
35#5164—21no—
36Dic3×SD16—21no—
37C2×#825C2211yescharacteristic
38#5133—21no—
39#5133C2211yescharacteristic
40#5131—21no—
41#5131C2211yescharacteristic
42#5131C2211yescharacteristic
43C2×#823C2211yescharacteristic
44#5122C2211yescharacteristic, complemented
45#5121C2211yescharacteristic, complemented
46#5113—21no—
47#5113C2211yescharacteristic
48C2×#821—21no—
49C2×#821C2211yescharacteristic
50#5101C2211yescharacteristic, complemented
51C2×C8×Dic3C2211yescharacteristic, complemented
52#4977C2211yescharacteristic, complemented
53C2×#809C2211yescharacteristic
54#4964C2211yescharacteristic, complemented
55#4817—21no—
56#4804—21no—
57#4803—21no—
58#4790—21no—
59#4789—21no—
60#4788—21no—
61#4770—21no—
62D3×#102—21no—
63C3×#343C2211yescharacteristic, complemented
64C3×#341C2211yescharacteristic, complemented
65#4558C2211yescharacteristic, complemented
66#4555C2211yescharacteristic, complemented
67#4555C2211yescharacteristic, complemented
68#4554C2211yescharacteristic, 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#1908D311yescharacteristic, complemented
79#1888—31no—
80#1860—31no—
81#1860—31no—
82#1764—31no—
83#1764—31no—
84C2×C3×#41C2311yescharacteristic
85C22×C3×Q8C2311yescharacteristic
86#1009—22no—
87#1009—22no—
88#1009—41no—
89#1009C2×C414yes—
90C2×#239—21no—
91C2×D3×D4—21no—
92C2×D3×D4—41no—
93C2×#235—21no—
94C2×#235C2311yescharacteristic
95C22×D12—21no—
96C22×D12C2311yescharacteristic
97C3×#137—22no—
98C3×#137C2×C414yescomplemented
99C2×C3×Q16—21no—
100C2×C3×SD16—21no—
101C3×#131—22no—
102C3×#131—41no—
103C2×C3×M16—21no—
104C2×C3×M16C2311yescharacteristic
105C2×C3×M16C2311yescharacteristic
106C2×C3×M16C2311yescharacteristic
107C22×C3×C8—21no—
108C22×C3×C8C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110#947—41no—
111#947C2×C414yescomplemented
112Q8×Dic3—21no—
113Q8×Dic3—21no—
114Q8×Dic3—21no—
115Q8×Dic3—21no—
116Q8×Dic3—21no—
117C2×#216—21no—
118#942—22no—
119#942C2×C414yescomplemented
120C2×#215—21no—
121D4×Dic3—21no—
122D4×Dic3—41no—
123C4×#67—41no—
124C2×(C2×D3)⋊C4—41no—
125#926—21no—
126#926—41no—
127#926C2311yescharacteristic
128#926C2311yescharacteristic
129C2×C4⋊Dic3—21no—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4×Dic3C2311yescharacteristic
132C2×#208C2311yescharacteristic
133C22×C3⋊C8C2311yescharacteristic
134#907—22no—
135#907—41no—
136C2×ΓL(1,𝔽25)—21no—
137C2×D3×C8—21no—
138#893—21no—
139#892—21no—
140D3×C4⋊C4—21no—
141#887—21no—
142C3×#103—21no—
143C3×#103—21no—
144C3×#103—21no—
145C3×#102—21no—
146C3×#101—41no—
147#837C2×C414yescomplemented
148#835—21no—
149#835—21no—
150#835D411yescharacteristic
151#835D411yescharacteristic
152#832D411yescharacteristic
153#832D411yescharacteristic
154C3⋊#95—41no—
155#825C2×C414yescomplemented
156#823C2×C414yescomplemented
157#821—21no—
158#821—21no—
159#821—21no—
160#821—21no—
161#821—21no—
162#821—21no—
163#821—21no—
164#821D411yescharacteristic
165#821D411yescharacteristic
166#818—21no—
167#817—21no—
168#816—21no—
169#816D411yescharacteristic
170#816D411yescharacteristic
171#815—21no—
172#815—21no—
173C8×Dic3—21no—
174C8×Dic3—21no—
175C8×Dic3—41no—
176#809—21no—
177#809—21no—
178#808—21no—
179#807—21no—
180C2×H(5,𝔽2)—31no—
181C2×#137—31no—
182C2×#137C2×D311yescharacteristic
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×#104—31no—
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×#103C2×D311yescharacteristic, complemented
216C2×#102—31no—
217C2×#102—31no—
218C2×#102—61no—
219C2×#102—61no—
220C2×#101C2×D311yescharacteristic, complemented
221C2×#100—31no—
222#405—31no—
223#405—31no—
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—
232C23×D3—21no—
233C23×D3—41no—
234C3×#41—21no—
235C3×#41—21no—
236C3×#41C22×C412yes—
237C3×#41#4112yes—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—21no—
241C2×C3×Q8—21no—
242C2×C3×Q8—21no—
243C2×C3×Q8C22×C412yes—
244C2×C3×Q8C2×D411yescharacteristic
245C2×C3×D4—21no—
246C2×C3×D4C2×D411yescharacteristic
247C22×C3×C4—21no—
248C22×C3×C4—21no—
249C22×C3×C4—21no—
250C22×C3×C4C2411yescharacteristic
251C2×#67—21no—
252C2×#67—41no—
253C22×Dic3—21no—
254C22×Dic3—21no—
255C22×Dic3—41no—
256#239—24no—
257#239—42no—
258D3×D4—24no—
259D3×D4—42no—
260D3×D4—44no—
261D3×D4—44no—
262#235—21no—
263#235—21no—
264#235—24no—
265#235—42no—
266#235C22×C412yes—
267#235#4112yes—
268C2×D12—21no—
269C2×D12—21no—
270C2×D12—21no—
271C2×D12—21no—
272C2×D12—24no—
273C2×D12—21no—
274C2×D12—42no—
275C2×D12—41no—
276C2×D12C22×C412yes—
277C2×D12C2×D411yescharacteristic
278C2×C4×D3—21no—
279C2×C4×D3—41no—
280C2×C4×D3—41no—
281C2×Dic6C2×D411yescharacteristic
282C3×Q16—24no—
283C3×SD16—24no—
284C3×M16—21no—
285C3×M16—21no—
286C3×M16—21no—
287C3×M16—21no—
288C3×M16—21no—
289C3×M16—42no—
290C3×M16C22×C412yes—
291C2×C3×C8—21no—
292C2×C3×C8—21no—
293C2×C3×C8—21no—
294C2×C3×C8—21no—
295C2×C3×C8—21no—
296C2×C3×C8—42no—
297C2×C3×C8C2×D411yescharacteristic
298C2×C3×C8C2×D411yescharacteristic
299C3×C4⋊C4—21no—
300C3×C4⋊C4—21no—
301#217—21no—
302#217—21no—
303#217—41no—
304#216—24no—
305#215—24no—
306(C2×D3)⋊C4—44no—
307(C2×D3)⋊C4—41no—
308C4⋊Dic3—21no—
309C4⋊Dic3—21no—
310C4⋊Dic3—41no—
311C4⋊Dic3—41no—
312C4⋊Dic3—41no—
313C4⋊Dic3C2×D411yescharacteristic
314C4⋊Dic3C2×D411yescharacteristic
315C4⋊Dic3C2×D411yescharacteristic
316C4⋊Dic3C2×D411yescharacteristic
317Dic3⋊C4—41no—
318C4×Dic3—21no—
319C4×Dic3—21no—
320C4×Dic3—21no—
321C4×Dic3—41no—
322C4×Dic3—41no—
323C4×Dic3—41no—
324C4×Dic3—41no—
325C4×Dic3C22×C412yes—
326#208—21no—
327#208C22×C412yes—
328C2×C3⋊C8—21no—
329C2×C3⋊C8—21no—
330C2×C3⋊C8—21no—
331C2×C3⋊C8C2×D411yescharacteristic
332C2×C3⋊C8C2×D411yescharacteristic
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×Q8C22×D311yescharacteristic
344C22×D4—31no—
345C22×D4—61no—
346C22×D4—61no—
347C22×D4—121no—
348#137—22no—
349#137—34no—
350#137—62no—
351#137C4×D314yescomplemented
352C2×Q16—21no—
353C2×Q16—61no—
354C2×SD16—21no—
355C2×SD16—61no—
356#131—22no—
357#131—41no—
358#131—62no—
359#131—121no—
360C2×M16—21no—
361C2×M16—31no—
362C2×M16—61no—
363C2×M16C22×D311yescharacteristic
364C2×M16C22×D311yescharacteristic
365C2×M16C22×D311yescharacteristic
366C22×C8—21no—
367C22×C8—31no—
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—31no—
384#117—61no—
385#117—61no—
386#117—121no—
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—61no—
394#107—61no—
395#106—61no—
396#106—61no—
397#104—34no—
398#104—34no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—31no—
403#103—31no—
404#103—31no—
405#103—31no—
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—41no—
424#100—34no—
425#97—61no—
426#97—61no—
427C4×C8—61no—
428C4×C8—61no—
429C4×C8—121no—
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×Q8—22no—
444C3×Q8—22no—
445C3×Q8—41no—
446C3×Q8—41no—
447C3×Q8C4×D412yes—
448C3×D4—22no—
449C3×D4—22no—
450C3×D4—21no—
451C3×D4C4×D412yes—
452C2×C3×C4—21no—
453C2×C3×C4—22no—
454C2×C3×C4—22no—
455C2×C3×C4—22no—
456C2×C3×C4—41no—
457C2×C3×C4—41no—
458C2×C3×C4—41no—
459C2×C3×C4—81no—
460C2×C3×C4C23×C411yescharacteristic
461C2×C3×C4C22×D411yescharacteristic
462C2×C3×C4C22×D411yescharacteristic
463C2×C3×C4C2×#4111yescharacteristic
464C2×C3×C4C2×#4111yescharacteristic
465C2×C3×C4H(5,𝔽2)11yescharacteristic
466#67—22no—
467#67—22no—
468#67—44no—
469C2×Dic3—21no—
470C2×Dic3—22no—
471C2×Dic3—21no—
472C2×Dic3—21no—
473C2×Dic3—21no—
474C2×Dic3—21no—
475C2×Dic3—41no—
476C2×Dic3—41no—
477C2×Dic3—41no—
478C2×Dic3—81no—
479C2×Dic3—161no—
480D12—22no—
481D12—22no—
482D12—21no—
483D12—44no—
484D12—41no—
485D12—41no—
486D12C4×D412yes—
487C4×D3—22no—
488C4×D3—22no—
489C4×D3—44no—
490C4×D3—161no—
491Dic6—21no—
492Dic6C4×D412yes—
493C3×C8—22no—
494C3×C8—21no—
495C3×C8—21no—
496C3×C8—41no—
497C3×C8—41no—
498C3⋊C8—22no—
499C3⋊C8—21no—
500C3⋊C8—21no—
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#23712yes—
515C2×Q8—21no—
516C2×Q8—21no—
517C2×Q8—21no—
518C2×Q8—21no—
519C2×Q8—21no—
520C2×Q8—31no—
521C2×Q8C2×C4×D312yes—
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—61no—
545C22×C4—61no—
546C22×C4—61no—
547C22×C4—121no—
548C22×C4—121no—
549C22×C4—121no—
550C22×C4C23×D311yescharacteristic
551Q16—24no—
552Q16—64no—
553SD16—24no—
554SD16—64no—
555M16—21no—
556M16—21no—
557M16—21no—
558M16—21no—
559M16—21no—
560M16—32no—
561M16—42no—
562M16—61no—
563M16—122no—
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—42no—
573C2×C8—61no—
574C2×C8—61no—
575C2×C8—61no—
576C2×C8—122no—
577C2×C8D3×D411yescharacteristic
578C2×C8D3×D411yescharacteristic
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—31no—
582C4⋊C4—31no—
583C4⋊C4—31no—
584C4⋊C4—31no—
585C4⋊C4—61no—
586C4⋊C4—61no—
587C4⋊C4—121no—
588C4⋊C4—121no—
589C4⋊C4—121no—
590C4⋊C4—121no—
591C22⋊C4—61no—
592C22⋊C4—61no—
593C22⋊C4—124no—
594C22⋊C4—121no—
595C22⋊C4—121no—
596C42—32no—
597C42—61no—
598C42—61no—
599C42—61no—
600C42—121no—
601C42—121no—
602C42—121no—
603C42—121no—
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—81no—
624C3×C4C2×C4×D411yescharacteristic
625C3×C4C2×C4×D411yescharacteristic
626C3×C4#54611yescharacteristic
627C3×C4#54611yescharacteristic
628Dic3—22no—
629Dic3—42no—
630Dic3—41no—
631Dic3—41no—
632Dic3—41no—
633Dic3—41no—
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—22no—
648Q8—22no—
649Q8—32no—
650Q8—41no—
651Q8—41no—
652Q8—61no—
653Q8#67⋊C412yes—
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#67⋊C412yes—
668C2×C4—21no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—61no—
676C2×C4—62no—
677C2×C4—62no—
678C2×C4—62no—
679C2×C4—61no—
680C2×C4—61no—
681C2×C4—61no—
682C2×C4—61no—
683C2×C4—81no—
684C2×C4—124no—
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×C4C2×#23711yescharacteristic
695C2×C4#100411yescharacteristic
696C2×C4D3×#4111yescharacteristic
697C8—22no—
698C8—21no—
699C8—21no—
700C8—41no—
701C8—41no—
702C8—62no—
703C8—61no—
704C8—61no—
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#560911yescharacteristic
729C22#565811yescharacteristic
730C4—22no—
731C4—42no—
732C4—62no—
733C4—81no—
734C4—122no—
735C4—121no—
736C4—121no—
737C4—121no—
738C4—121no—
739C4—241no—
740C4C2×#67⋊C411yescharacteristic
741C4C4×D3×D411yescharacteristic
742C4#559111yescharacteristic
743C4#561011yescharacteristic
744C3#2289611yescharacteristic, complemented, Sylow
745C2—21no—
746C2—21no—
747C2—42no—
748C2—122no—
749C2—242no—
750C2#7487911yescharacteristic
751C2#8093411yescharacteristic
752C2#8094611yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1908C2
socleC22×C3D42
prime radicalC2×#1009C22
cosocleC22×C3×C4C24
hypercenter#1908D3
hypocenterC3#22896
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1908

Complements

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

NH ≅ G/Ntype
2. #83965C2semidirect
5. #78774C2semidirect
6. C3×#1908C2semidirect
7. #72008C2semidirect
8. #70298C2semidirect
9. #70226C2semidirect
10. #70123C2semidirect
11. #70104C2semidirect
12. #69003C2semidirect
13. #68983C2semidirect
14. #68931C2semidirect
15. #68926C2semidirect
22. #5812C22semidirect
27. C2×C3×#103C22semidirect
28. C2×C3×#101C22semidirect
31. C2×#835C22semidirect
32. #5202C4semidirect
44. #5122C22semidirect
45. #5121C22semidirect
50. #5101C22semidirect
51. C2×C8×Dic3C22semidirect
52. #4977C22semidirect
54. #4964C22semidirect
63. C3×#343C22semidirect
64. C3×#341C22semidirect
65. #4558C22semidirect
66. #4555C22semidirect
67. #4555C22semidirect
68. #4554C22semidirect
78. #1908D3semidirect
98. C3×#137C2×C4semidirect
111. #947C2×C4semidirect
119. #942C2×C4semidirect
147. #837C2×C4semidirect
155. #825C2×C4semidirect
156. #823C2×C4semidirect
215. C2×#103C2×D3semidirect
220. C2×#101C2×D3semidirect
227. #343C2×D3semidirect
229. #341C2×D3semidirect
351. #137C4×D3semidirect
744. C3#22896semidirect

Maximal subgroups

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

subgroupindex
C2×#52022
#839652
#837972
#788822
#787742
C3×#19082
#720082
#702982
#702262
#701232
#701042
#690032
#689832
#689312
#689262
#228963