Finite Group Databasebeta

← #10839409

Subgroups — #10839409

Order 768.

904 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82563C211yesmaximal, characteristic, complemented
2#82557C211yesmaximal, characteristic, complemented
3#79221C211yesmaximal, characteristic, complemented
4#78886C211yesmaximal, characteristic
5C3×#1859C211yesmaximal, characteristic, complemented
6#70673C211yesmaximal, characteristic, complemented
7#70384C211yesmaximal, characteristic, complemented
8#70377C211yesmaximal, characteristic, complemented
9#70300C211yesmaximal, characteristic, complemented
10#69593C211yesmaximal, characteristic, complemented
11#69586C211yesmaximal, characteristic, complemented
12#69010C211yesmaximal, characteristic, complemented
13#68986C211yesmaximal, characteristic, complemented
14#68776C211yesmaximal, characteristic, complemented
15#68744C211yesmaximal, characteristic, complemented
16#24925—31nomaximal, Sylow
17C2×#1009C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C2×#950C2211yescharacteristic
20C2×#948C2211yescharacteristic
21C2×#908C2211yescharacteristic
22#5523C2211yescharacteristic, complemented
23#5510C2211yescharacteristic
24C3×#428C2211yescharacteristic, complemented
25C2×C3×#104C2211yescharacteristic
26C3×#417C2211yescharacteristic
27C3×#384C2211yescharacteristic, complemented
28#5232C2211yescharacteristic
29#5133C2211yescharacteristic
30C2×#823C2211yescharacteristic
31#5129C2211yescharacteristic
32#5129C2211yescharacteristic, complemented
33C2×#822C2211yescharacteristic, complemented
34C4×#216—21no—
35#5026—21no—
36#5018—21no—
37C4×#213—21no—
38#5002—21no—
39#5002C2211yescharacteristic
40#5001—21no—
41#5001C2211yescharacteristic
42#4997C2211yescharacteristic, complemented
43#4966—21no—
44#4966C2211yescharacteristic
45#4966C2211yescharacteristic, complemented
46C2×#809—21no—
47C2×#809C2211yescharacteristic
48C2×#809C2211yescharacteristic
49C2×#807C2211yescharacteristic, complemented
50C2×#807C2211yescharacteristic, complemented
51#4939C2211yescharacteristic
52C2×#805C2211yescharacteristic
53C2×#805C2211yescharacteristic, complemented
54#4884—21no—
55#4882—21no—
56#4868—21no—
57#4866—21no—
58#4810—21no—
59#4808—21no—
60#4776—21no—
61#4773—21no—
62C3×#343C2211yescharacteristic, complemented
63C3×#339C2211yescharacteristic, complemented
64#4558C2211yescharacteristic, complemented
65#4554C2211yescharacteristic, complemented
66#4530C2211yescharacteristic, complemented
67#4524C2211yescharacteristic, complemented
68#3098—31no—
69#3012—31no—
70#2951—31no—
71#2897—31no—
72#2093—31no—
73#2078—31no—
74#2051—31no—
75#2039—31no—
76#1931—31no—
77#1929—31no—
78#1883—31no—
79#1876—31no—
80#1859D311yescharacteristic, complemented
81#1816—31no—
82#1792—31no—
83C2×C3×#41C2311yescharacteristic
84#1009—22no—
85#1009—22no—
86#1009—22no—
87#1009—41no—
88C2×#239—21no—
89C2×D3×D4—21no—
90C2×D3×D4—41no—
91C2×#235—21no—
92C2×#235C2311yescharacteristic
93C2×#235C2311yescharacteristic
94C22×D12—21no—
95C22×D12C2311yescharacteristic
96C2×C3×M16C2311yescharacteristic
97C2×C3×M16C2311yescharacteristic
98C3×C4×Q8—21no—
99C3×C4×D4—21no—
100C3×C4×D4—41no—
101C3×#117—21no—
102C3×#117C2311yescharacteristic
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—41no—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108#950—22no—
109#950—22no—
110#948—22no—
111#948—41no—
112C2×#216D411yescharacteristic
113C2×#215D411yescharacteristic
114C2×#214D411yescharacteristic
115C2×#213D411yescharacteristic
116#930—41no—
117C2×(C2×D3)⋊C4—41no—
118C2×C4⋊Dic3C2311yescharacteristic
119C2×#208—21no—
120C2×#208C2311yescharacteristic
121C2×#208C2311yescharacteristic
122C22×C3⋊C8—21no—
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#908—22no—
126#908—41no—
127C2×D24—21no—
128C2×#204—21no—
129#897—21no—
130#897—21no—
131#895—21no—
132#895—21no—
133#892—21no—
134D3×C4⋊C4—21no—
135#67⋊C4—41no—
136C4×D12—21no—
137C4×Dic6—21no—
138C3×#107—21no—
139C3×#106—21no—
140C3×#104—22no—
141C3×#103—21no—
142C3×#102—21no—
143#835—21no—
144#832—21no—
145#823—22no—
146#822—41no—
147#821—21no—
148(C2×D3)⋊C8—21no—
149(C2×D3)⋊C8—21no—
150(C2×D3)⋊C8—21no—
151(C2×D3)⋊C8—21no—
152#816—21no—
153#810—21no—
154#810—21no—
155#810D411yescharacteristic
156#810D411yescharacteristic
157#809—21no—
158#809—21no—
159#809—21no—
160#809—21no—
161#809—21no—
162#809—21no—
163#809—21no—
164#809—21no—
165#809D411yescharacteristic
166#809D411yescharacteristic
167#807—21no—
168#807—21no—
169#807—21no—
170#807—21no—
171#805—41no—
172#805—41no—
173#803—21no—
174C4×C3⋊C8—21no—
175C2×H(5,𝔽2)—31no—
176C2×AGL(1,ℤ/8)—31no—
177C2×#135—31no—
178C2×#131—31no—
179#537—31no—
180#520—31no—
181#520C2×D311yescharacteristic
182#485—61no—
183#484—61no—
184#482—61no—
185#480—61no—
186#452—61no—
187#450—61no—
188#449—61no—
189#447—61no—
190#440—61no—
191#440—61no—
192C4×Q16—61no—
193C4×D8—61no—
194C2×#108—31no—
195#428C2×D311yescharacteristic, complemented
196C2×#107—31no—
197C2×#107—31no—
198C2×#104—31no—
199C2×#104—31no—
200C2×#104C2×D311yescharacteristic
201#418—31no—
202#418—61no—
203#417—31no—
204#417—31no—
205#417—31no—
206#417—31no—
207#417—61no—
208#417C2×D311yescharacteristic
209#416—31no—
210#416—61no—
211C2×#102—31no—
212C2×#102—31no—
213C2×#102—61no—
214C2×#100—31no—
215#408—31no—
216#408—31no—
217#405—31no—
218#390—31no—
219#384C2×D311yescharacteristic, complemented
220#343—31no—
221#343—31no—
222#343C2×D311yescharacteristic, complemented
223#341—31no—
224#339C2×D311yescharacteristic, complemented
225#336—31no—
226C23×D3—21no—
227C23×D3—41no—
228C3×#41—21no—
229C3×#41—21no—
230C3×#41#4112yes—
231C3×#41#4112yes—
232C2×C3×Q8C2×D411yescharacteristic
233C2×C3×D4—21no—
234C2×C3×D4C2×D411yescharacteristic
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—41no—
240C22×C3×C4C2411yescharacteristic
241C2×#67—21no—
242C2×#67—21no—
243C2×#67—41no—
244C22×Dic3—21no—
245#239—42no—
246#239—42no—
247D3×D4—42no—
248D3×D4—44no—
249D3×D4—44no—
250D3×D4—42no—
251#235—24no—
252#235—21no—
253#235—21no—
254#235—21no—
255#235—21no—
256#235—21no—
257#235—21no—
258#235—21no—
259#235—21no—
260#235—42no—
261C2×D12—24no—
262C2×D12—21no—
263C2×D12—21no—
264C2×D12—21no—
265C2×D12—21no—
266C2×D12—21no—
267C2×D12—42no—
268C2×D12—41no—
269C2×D12—41no—
270C2×D12C2×D411yescharacteristic
271C2×D12C2×D411yescharacteristic
272C2×D12#4112yes—
273C2×C4×D3—21no—
274C2×C4×D3—21no—
275C2×C4×D3—41no—
276C2×C4×D3—41no—
277C2×Dic6C2×D411yescharacteristic
278C2×Dic6C2×D411yescharacteristic
279C3×M16—21no—
280C3×M16—21no—
281C3×M16—21no—
282C3×M16—21no—
283C2×C3×C8—21no—
284C2×C3×C8—21no—
285C3×C4⋊C4—21no—
286C3×C4⋊C4—21no—
287C3×C4⋊C4—21no—
288C3×C4⋊C4—21no—
289C3×C4⋊C4—41no—
290C3×C4⋊C4—82no—
291C3×C4⋊C4C2×D411yescharacteristic
292C3×C4⋊C4C2×D411yescharacteristic
293C3×C22⋊C4—21no—
294C3×C22⋊C4—41no—
295C3×C42—21no—
296C3×C42—21no—
297C3×C42—21no—
298C3×C42—41no—
299#216—22no—
300#215—22no—
301#214—22no—
302#213—22no—
303(C2×D3)⋊C4—41no—
304(C2×D3)⋊C4—41no—
305(C2×D3)⋊C4—41no—
306(C2×D3)⋊C4—82no—
307C4⋊Dic3—21no—
308C4⋊Dic3—21no—
309C4⋊Dic3—41no—
310Dic3⋊C4—41no—
311C4×Dic3—41no—
312#208—21no—
313#208—21no—
314#208—21no—
315#208—21no—
316#208—42no—
317C2×C3⋊C8—21no—
318C2×C3⋊C8—21no—
319C2×C3⋊C8—21no—
320C2×C3⋊C8—21no—
321C2×C3⋊C8—21no—
322C2×C3⋊C8—21no—
323C2×C3⋊C8—42no—
324C2×C3⋊C8C2×D411yescharacteristic
325C2×C3⋊C8C2×D411yescharacteristic
326C2×C3⋊C8C2×D411yescharacteristic
327C2×C3⋊C8C2×D411yescharacteristic
328D24—42no—
329#204—42no—
330H(5,𝔽2)—62no—
331H(5,𝔽2)—62no—
332H(5,𝔽2)—62no—
333H(5,𝔽2)—121no—
334C2×#41—31no—
335C2×#41—31no—
336C2×#41—61no—
337C2×#41—61no—
338C2×#41C22×D311yescharacteristic
339C22×D4—31no—
340C22×D4—61no—
341C22×D4—61no—
342C22×D4—121no—
343AGL(1,ℤ/8)—62no—
344AGL(1,ℤ/8)—121no—
345#135—62no—
346#135—62no—
347C2×Q16—31no—
348C2×SD16—31no—
349C2×SD16—31no—
350C2×SD16—61no—
351C2×D8—31no—
352C2×D8—61no—
353#131—62no—
354#131—121no—
355C2×M16—31no—
356C2×M16—31no—
357C2×M16—61no—
358C2×M16C22×D311yescharacteristic
359C2×M16C22×D311yescharacteristic
360C22×C8—31no—
361C22×C8—31no—
362C22×C8—61no—
363C22⋊Q8—61no—
364C22⋊Q8—61no—
365#121—61no—
366#121—61no—
367#121—121no—
368C4×Q8—21no—
369C4×Q8—61no—
370C4×D4—21no—
371C4×D4—41no—
372C4×D4—61no—
373C4×D4—121no—
374#117—21no—
375#117—61no—
376#117C22×D311yescharacteristic
377C2×C4⋊C4—21no—
378C2×C4⋊C4—31no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4—61no—
381C2×C4⋊C4C22×D311yescharacteristic
382C2×C4⋊C4C22×D311yescharacteristic
383C2×C22⋊C4—121no—
384C2×C42C22×D311yescharacteristic
385#108—121no—
386#107—21no—
387#107—61no—
388#107—61no—
389#107—61no—
390#107—61no—
391#106—21no—
392#104—22no—
393#104—121no—
394#104—121no—
395#103—21no—
396#103—31no—
397#103—31no—
398#103—61no—
399#103—61no—
400#103—61no—
401#103—61no—
402#102—21no—
403#102—31no—
404#102—31no—
405#102—61no—
406#102—61no—
407#102—61no—
408#102—61no—
409#102—61no—
410#102—61no—
411#102—61no—
412#102—61no—
413#102—61no—
414#102—61no—
415#100—62no—
416C22⋊C8—61no—
417C22⋊C8—61no—
418C22⋊C8—61no—
419C22⋊C8—61no—
420#97—61no—
421C4×C8—61no—
422C23×C3—21no—
423C23×C3H(5,𝔽2)11yescharacteristic
424C22×D3—21no—
425C22×D3—21no—
426C22×D3—22no—
427C22×D3—41no—
428C22×D3—44no—
429C22×D3—41no—
430C22×D3—81no—
431C22×D3—81no—
432C22×D3—161no—
433C22×D3—161no—
434C3×Q8—21no—
435C3×Q8#12312yes—
436C3×D4—22no—
437C3×D4—22no—
438C3×D4—21no—
439C3×D4#12312yes—
440C2×C3×C4—21no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—21no—
445C2×C3×C4—21no—
446C2×C3×C4—41no—
447C2×C3×C4—41no—
448C2×C3×C4—41no—
449C2×C3×C4—41no—
450C2×C3×C4—81no—
451C2×C3×C4—81no—
452C2×C3×C4—161no—
453C2×C3×C4C22×D411yescharacteristic
454C2×C3×C4C22×D411yescharacteristic
455C2×C3×C4C2×#4111yescharacteristic
456C2×C3×C4C2×#4111yescharacteristic
457C2×C3×C4C2×#4111yescharacteristic
458C2×C3×C4H(5,𝔽2)11yescharacteristic
459#67—44no—
460#67—41no—
461#67—41no—
462#67—41no—
463#67—41no—
464C2×Dic3—21no—
465C2×Dic3—21no—
466C2×Dic3—41no—
467C2×Dic3—81no—
468D12—21no—
469D12—21no—
470D12—22no—
471D12—22no—
472D12—21no—
473D12—21no—
474D12—44no—
475D12—41no—
476D12—41no—
477D12—161no—
478C4×D3—44no—
479C4×D3—41no—
480C4×D3—41no—
481C4×D3—41no—
482C4×D3—41no—
483C4×D3—161no—
484Dic6—21no—
485Dic6—21no—
486Dic6—21no—
487Dic6—21no—
488C3×C8—41no—
489C3×C8—41no—
490C3⋊C8—21no—
491C3⋊C8—21no—
492C3⋊C8—21no—
493C3⋊C8—21no—
494C3⋊C8—41no—
495C3⋊C8—41no—
496C24—61no—
497C24—121no—
498#41—21no—
499#41—21no—
500#41—64no—
501#41—61no—
502#41—61no—
503#41—61no—
504#41—61no—
505#41—61no—
506#41—61no—
507#41—61no—
508#41—61no—
509#41—122no—
510#41—122no—
511#41—122no—
512#41#23512yes—
513#41#23912yes—
514C2×Q8—31no—
515C2×Q8—31no—
516C2×Q8D3×D411yescharacteristic
517C2×D4—21no—
518C2×D4—31no—
519C2×D4—31no—
520C2×D4—32no—
521C2×D4—64no—
522C2×D4—61no—
523C2×D4—61no—
524C2×D4—61no—
525C2×D4—61no—
526C2×D4—61no—
527C2×D4—61no—
528C2×D4—61no—
529C2×D4—122no—
530C2×D4—121no—
531C2×D4—121no—
532C2×D4—121no—
533C2×D4—122no—
534C2×D4—124no—
535C2×D4—124no—
536C2×D4—122no—
537C2×D4D3×D411yescharacteristic
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—41no—
543C22×C4—61no—
544C22×C4—61no—
545C22×C4—61no—
546C22×C4—121no—
547C22×C4—121no—
548C22×C4C23×D311yescharacteristic
549Q16—62no—
550SD16—62no—
551SD16—62no—
552SD16—122no—
553D8—62no—
554D8—122no—
555M16—21no—
556M16—21no—
557M16—21no—
558M16—21no—
559M16—61no—
560M16—61no—
561M16—61no—
562M16—61no—
563M16—122no—
564C2×C8—21no—
565C2×C8—21no—
566C2×C8—31no—
567C2×C8—31no—
568C2×C8—31no—
569C2×C8—31no—
570C2×C8—61no—
571C2×C8—61no—
572C2×C8—61no—
573C2×C8—61no—
574C2×C8—61no—
575C2×C8—61no—
576C2×C8—122no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—41no—
582C4⋊C4—61no—
583C4⋊C4—61no—
584C4⋊C4—82no—
585C4⋊C4—121no—
586C4⋊C4—121no—
587C4⋊C4D3×D411yescharacteristic
588C4⋊C4D3×D411yescharacteristic
589C22⋊C4—21no—
590C22⋊C4—41no—
591C22⋊C4—121no—
592C22⋊C4—121no—
593C22⋊C4—121no—
594C22⋊C4—242no—
595C42—21no—
596C42—21no—
597C42—21no—
598C42—41no—
599C42—121no—
600C22×C3—21no—
601C22×C3—21no—
602C22×C3—22no—
603C22×C3—41no—
604C22×C3—41no—
605C22×C3#53811yescharacteristic
606C22×C3D4211yescharacteristic
607C22×C3#55111yescharacteristic
608C2×D3—41no—
609C2×D3—42no—
610C2×D3—41no—
611C2×D3—41no—
612C2×D3—41no—
613C2×D3—41no—
614C2×D3—41no—
615C2×D3—81no—
616C2×D3—81no—
617C2×D3—82no—
618C2×D3—161no—
619C2×D3—164no—
620C3×C4—22no—
621C3×C4—42no—
622C3×C4—41no—
623C3×C4—41no—
624C3×C4—81no—
625C3×C4—81no—
626C3×C4C2×#12311yescharacteristic
627C3×C4#52511yescharacteristic
628C3×C4#54611yescharacteristic
629C3×C4#54611yescharacteristic
630Dic3—41no—
631Dic3—41no—
632Dic3—81no—
633C23—21no—
634C23—61no—
635C23—61no—
636C23—62no—
637C23—124no—
638C23—121no—
639C23—121no—
640C23—241no—
641C23—241no—
642C23—481no—
643C23—481no—
644C23#100411yescharacteristic
645Q8—21no—
646Q8—61no—
647Q8—61no—
648Q8—61no—
649Q8—61no—
650Q8#89412yes—
651D4—22no—
652D4—22no—
653D4—21no—
654D4—61no—
655D4—62no—
656D4—61no—
657D4—61no—
658D4—62no—
659D4—61no—
660D4—124no—
661D4—124no—
662D4—121no—
663D4—121no—
664D4—121no—
665D4—121no—
666D4—121no—
667D4—121no—
668D4—481no—
669D4#89412yes—
670C2×C4—21no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—21no—
675C2×C4—21no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—61no—
681C2×C4—61no—
682C2×C4—81no—
683C2×C4—81no—
684C2×C4—124no—
685C2×C4—121no—
686C2×C4—121no—
687C2×C4—121no—
688C2×C4—121no—
689C2×C4—121no—
690C2×C4—161no—
691C2×C4—241no—
692C2×C4—481no—
693C2×C4C2×#23511yescharacteristic
694C2×C4C2×D3×D411yescharacteristic
695C2×C4C2×D3×D411yescharacteristic
696C2×C4C2×#23911yescharacteristic
697C2×C4D3×#4111yescharacteristic
698C2×C4#100911yescharacteristic
699C8—41no—
700C8—41no—
701C8—61no—
702C8—61no—
703C8—61no—
704C8—61no—
705C8—121no—
706C8—121no—
707C2×C3—21no—
708C2×C3—21no—
709C2×C3—42no—
710C2×C3#241811yescharacteristic
711C2×C3#328911yescharacteristic
712C2×C3#329211yescharacteristic
713D3—81no—
714D3—81no—
715D3—82no—
716C22—21no—
717C22—21no—
718C22—22no—
719C22—41no—
720C22—41no—
721C22—121no—
722C22—122no—
723C22—121no—
724C22—121no—
725C22—121no—
726C22—121no—
727C22—121no—
728C22—242no—
729C22—241no—
730C22—241no—
731C22—481no—
732C22—484no—
733C22#556011yescharacteristic
734C22#559011yescharacteristic
735C22#563711yescharacteristic
736C4—22no—
737C4—42no—
738C4—41no—
739C4—41no—
740C4—81no—
741C4—81no—
742C4—121no—
743C4—121no—
744C4—241no—
745C4C2×#89411yescharacteristic
746C4#554711yescharacteristic
747C4#563011yescharacteristic
748C4#565411yescharacteristic
749C3#2492511yescharacteristic, complemented, Sylow
750C2—21no—
751C2—21no—
752C2—42no—
753C2—241no—
754C2—241no—
755C2—242no—
756C2#7460311yescharacteristic
757C2#8028211yescharacteristic
758C2#8030111yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1859C2
socleC22×C3D42
prime radical#78886C2
hypercenter#1859D3
hypocenterC3#24925
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1859

Complements

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

NH ≅ G/Ntype
1. #82563C2semidirect
2. #82557C2semidirect
3. #79221C2semidirect
5. C3×#1859C2semidirect
6. #70673C2semidirect
7. #70384C2semidirect
8. #70377C2semidirect
9. #70300C2semidirect
10. #69593C2semidirect
11. #69586C2semidirect
12. #69010C2semidirect
13. #68986C2semidirect
14. #68776C2semidirect
15. #68744C2semidirect
22. #5523C22semidirect
24. C3×#428C22semidirect
27. C3×#384C22semidirect
32. #5129C22semidirect
33. C2×#822C22semidirect
42. #4997C22semidirect
45. #4966C22semidirect
49. C2×#807C22semidirect
50. C2×#807C22semidirect
53. C2×#805C22semidirect
62. C3×#343C22semidirect
63. C3×#339C22semidirect
64. #4558C22semidirect
65. #4554C22semidirect
66. #4530C22semidirect
67. #4524C22semidirect
80. #1859D3semidirect
195. #428C2×D3semidirect
219. #384C2×D3semidirect
222. #343C2×D3semidirect
224. #339C2×D3semidirect
749. C3#24925semidirect

Maximal subgroups

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

subgroupindex
#825632
#825572
#792212
#788862
C3×#18592
#706732
#703842
#703772
#703002
#695932
#695862
#690102
#689862
#687762
#687442
#249253