Finite Group Databasebeta

← #29665186

Subgroups — #29665186

Order 15616.

792 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456733C211yesmaximal, characteristic, complemented
2#21455278C211yesmaximal, characteristic, complemented
3#459×D61C211yesmaximal, characteristic, complemented
4#21454840C211yesmaximal, characteristic, complemented
5#21445257C211yesmaximal, characteristic, complemented
6#21445256C211yesmaximal, characteristic, complemented
7#21445255C211yesmaximal, characteristic, complemented
8#21445254C211yesmaximal, characteristic, complemented
9#21445173C211yesmaximal, characteristic, complemented
10#21445172C211yesmaximal, characteristic, complemented
11#21445171C211yesmaximal, characteristic, complemented
12#21444592C211yesmaximal, characteristic, complemented
13#21444485C211yesmaximal, characteristic, complemented
14#21444465C211yesmaximal, characteristic, complemented
15C61×#1667C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17C2×D8×D61—21no—
18#16840310C2211yescharacteristic
19#127×D61C2211yescharacteristic
20C4×D4×D61C2211yescharacteristic
21#16840138C2211yescharacteristic, complemented
22#16840109C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic, complemented
24#16839976—21no—
25#16839975—21no—
26#16839968C2211yescharacteristic, complemented
27#16839967C2211yescharacteristic, complemented
28#16839958C2211yescharacteristic
29#16839957C2211yescharacteristic
30#16839956—21no—
31#16839955—21no—
32#16839955—21no—
33#16839955—21no—
34#16839954C2211yescharacteristic
35#16839954C2211yescharacteristic, complemented
36#16839894—21no—
37#16839894—21no—
38C8×D244C2211yescharacteristic, complemented
39D4×#92418C2211yescharacteristic, complemented
40#16839827C2211yescharacteristic, complemented
41C8×C22⋊D61—21no—
42#16839825—21no—
43#16839814C2211yescharacteristic
44#16839813C2211yescharacteristic
45#16839784C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839760—21no—
48#102×D61—21no—
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839514C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839507C2211yescharacteristic, complemented
56#16839506C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic, complemented
59#16839505C2211yescharacteristic, complemented
60C61×#493C2211yescharacteristic, complemented
61C61×#459C2211yescharacteristic
62C61×#459C2211yescharacteristic, complemented
63C8×D4×C61C2211yescharacteristic, complemented
64C61×#335C2211yescharacteristic, complemented
65C61×#331C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61D411yescharacteristic
72C22×C4×D61—21no—
73#14519801—44no—
74C2×#11758190—21no—
75C2×#11758200—21no—
76C2×D488—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758199—21no—
81C2×#11758199—21no—
82D8×D61—44no—
83C2×#11758189—21no—
84C22×#92418—21no—
85C2×C8×D61—21no—
86C2×C8×D61—21no—
87C2×C8×D61—21no—
88#14519777C2311yescharacteristic
89#14519774C2311yescharacteristic
90#14519773—21no—
91#14519773—21no—
92#14519773—21no—
93#14519764—21no—
94#14519748—21no—
95#14519748—21no—
96C4×D244C2311yescharacteristic
97D4×Dic61—21no—
98D4×Dic61—21no—
99D4×Dic61D411yescharacteristic
100#14519739D411yescharacteristic
101C4×C22⋊D61—21no—
102#14519737—21no—
103C4⋊C4×D61D411yescharacteristic
104C22⋊C4×D61—21no—
105C42×D61C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519706—21no—
111#14519706D411yescharacteristic
112#14519706D411yescharacteristic
113#14519702—21no—
114#14519700—21no—
115#14519699—21no—
116#14519699—21no—
117#14519699—21no—
118#14519698—21no—
119#14519698—21no—
120#14519698—21no—
121#14519698—21no—
122#14519697—21no—
123C22⋊#92418—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689—21no—
127#14519689D411yescharacteristic
128#14519689D411yescharacteristic
129C4×#92418C2311yescharacteristic
130C8×Dic61—21no—
131C2×D8×C61—21no—
132C2×D8×C61—21no—
133C2×D8×C61—21no—
134C2×D8×C61—21no—
135C22×C8×C61—21no—
136#127×C61C2311yescharacteristic
137#127×C61C2311yescharacteristic
138C4×D4×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#102×C61—21no—
143#102×C61—21no—
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—21no—
147C23×D61—41no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C2×C22⋊D61—21no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—41no—
159C2×C22⋊D61—41no—
160C2×C22⋊D61—41no—
161C22×Dic61—21no—
162C22×Dic61—41no—
163#11758224—42no—
164#11758224—42no—
165D4×D61—22no—
166D4×D61—42no—
167D4×D61—42no—
168D4×D61—81no—
169D4×D61—81no—
170D4×D61—82no—
171D4×D61D814yescomplemented
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244C2×D411yescharacteristic
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—81no—
180C2×C4×D61—81no—
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61#4111yescharacteristic
184C2×Dic122—21no—
185D8×C61—42no—
186D8×C61—42no—
187D8×C61—81no—
188D8×C61—81no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—81no—
193C2×C8×C61C2×D411yescharacteristic
194C2×C8×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D411yescharacteristic
196C22⋊C4×C61—21no—
197C42×C61C2411yescharacteristic
198C22⋊Dic61—21no—
199C22⋊Dic61—41no—
200C22⋊Dic61—41no—
201#11758200—42no—
202#11758199—42no—
203#11758199—42no—
204#11758199—42no—
205#11758199—81no—
206#11758199—81no—
207#11758198—21no—
208#11758198—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61C2×D411yescharacteristic
212#11758196—21no—
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61#4111yescharacteristic
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—81no—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222D488—42no—
223#11758190—42no—
224#11758189—42no—
225C8×D61—42no—
226C8×D61—81no—
227C8×D61—81no—
228C23×C61—21no—
229C23×C61—41no—
230C23×C61—41no—
231C22×D61—21no—
232C22×D61—24no—
233C22×D61—41no—
234C22×D61—81no—
235C22×D61—81no—
236C22×D61—82no—
237C22×D61—82no—
238C22×D61—82no—
239C22×D61#12111yescharacteristic
240D4×C61—21no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61—81no—
250D4×C61—81no—
251D4×C61C2×D812yes—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—24no—
259C22⋊D61—82no—
260C22⋊D61—82no—
261C22⋊D61—82no—
262C22⋊D61—82no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265C2×Dic61—81no—
266C2×Dic61—82no—
267C2×Dic61#12111yescharacteristic
268C2×Dic61#12111yescharacteristic
269C2×Dic61#12111yescharacteristic
270D244—21no—
271D244—41no—
272D244—41no—
273D244—41no—
274D244—41no—
275D244—81no—
276D244C2×D812yes—
277C4×D61—21no—
278C4×D61—21no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—81no—
283C4×D61—81no—
284C4×D61C2×D812yes—
285C4×D61#13512yes—
286Dic122—41no—
287Dic122—41no—
288C8×C61—21no—
289C8×C61—21no—
290C8×C61—41no—
291C8×C61—41no—
292C8×C61—41no—
293#92418—21no—
294#92418—21no—
295#92418—41no—
296#92418—41no—
297#92418—41no—
298#19386—611nomaximal, Sylow
299C22×C61—22no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61C2×#12111yescharacteristic
309C2×D61—21no—
310C2×D61—21no—
311C2×D61—22no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—84no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—164no—
319C2×D61#46612yes—
320C4×C61—21no—
321C4×C61—41no—
322C4×C61C22×D811yescharacteristic
323C4×C61C2×#13511yescharacteristic
324C4×C61#57611yescharacteristic
325C4×C61#57611yescharacteristic
326Dic61—21no—
327Dic61—21no—
328Dic61—41no—
329Dic61—81no—
330Dic61#46612yes—
331#3205—611no—
332#3036—611no—
333C2×#459—611no—
334#2972—611no—
335#1681—611no—
336#1681—611no—
337#1681—611no—
338#1681—611no—
339#1667—611no—
340#1667—611no—
341#1667—611no—
342#1667D6111yescharacteristic, complemented
343#1571—611no—
344#1545—611no—
345#1540—611no—
346C2×C61—42no—
347C2×C61—161no—
348C2×C61—161no—
349C2×C61#304111yescharacteristic
350C2×C61C2×#46611yescharacteristic
351C2×C61#306811yescharacteristic
352D61—22no—
353D61—42no—
354D61—161no—
355C2×AGL(1,ℤ/8)—1221no—
356C22×D8—1221no—
357#532—611no—
358C2×#127—611no—
359C2×C4×D4—611no—
360#493—611no—
361#493C2×D6111yescharacteristic, complemented
362#486—611no—
363#486—611no—
364#463—1221no—
365#463—1221no—
366#461—611no—
367#461—611no—
368#459—611no—
369#459—611no—
370#459—611no—
371#459—611no—
372#459—1221no—
373#459—1221no—
374#459—1221no—
375#459—1221no—
376#459C2×D6111yescharacteristic
377#459C2×D6111yescharacteristic, complemented
378#447—1221no—
379#447—1221no—
380C8×D4—611no—
381C8×D4—611no—
382C8×D4—611no—
383C8×D4—1221no—
384C8×D4—1221no—
385C8×D4C2×D6111yescharacteristic, complemented
386#431—611no—
387#431—611no—
388#423—611no—
389C2×C4⋊C8—611no—
390#417—1221no—
391C2×#102—1221no—
392C2×C22⋊C8—1221no—
393C2×C22⋊C8—1221no—
394#335—611no—
395#335—611no—
396#335—611no—
397#335C2×D6111yescharacteristic, complemented
398#332—611no—
399#332—611no—
400#331—611no—
401#331—611no—
402#331—611no—
403#331—611no—
404#331C2×D6111yescharacteristic, complemented
405#331C2×D6111yescharacteristic, complemented
406C61#1938611yescharacteristic, complemented, Sylow
407C2×#41—1221no—
408C22×D4—611no—
409C22×D4—1221no—
410C22×D4—1221no—
411C22×D4—1221no—
412C23×C4—1221no—
413AGL(1,ℤ/8)—2444no—
414C2×SD16—1221no—
415C2×SD16—1221no—
416C2×D8—21no—
417C2×D8—21no—
418C2×D8—21no—
419C2×D8—21no—
420C2×D8—1221no—
421C2×D8—1221no—
422C2×D8—1221no—
423C2×D8—1221no—
424C2×D8—1221no—
425C2×D8—1221no—
426C2×D8—2444no—
427C2×M16—1221no—
428C22×C8—21no—
429C22×C8—1221no—
430C22×C8—1221no—
431C22×C8—1221no—
432C22×C8—1221no—
433C4⋊Q8—611no—
434#127—611no—
435#127—1221no—
436#127—1221no—
437#127—1221no—
438#127C22×D6111yescharacteristic
439#127C22×D6111yescharacteristic
440#124—1221no—
441#121—1221no—
442#121—1221no—
443C4×D4—611no—
444C4×D4—611no—
445C4×D4—611no—
446C4×D4—1221no—
447C4×D4—1221no—
448C4×D4—1221no—
449C4×D4—1221no—
450C4×D4C22×D6111yescharacteristic
451C2×C4⋊C4—611no—
452C2×C22⋊C4—1221no—
453C2×C42—611no—
454C4⋊C8—611no—
455C4⋊C8—611no—
456C4⋊C8—611no—
457C4⋊C8—611no—
458C4⋊C8—611no—
459C4⋊C8—1221no—
460C4⋊C8—1221no—
461C4⋊C8C22×D6111yescharacteristic
462C4⋊C8C22×D6111yescharacteristic
463C4⋊C8C22×D6111yescharacteristic
464#103—1221no—
465#103—1221no—
466#102—21no—
467#102—21no—
468#102—1221no—
469#102—1221no—
470#102—1221no—
471#102—1221no—
472#102—1221no—
473#102—1221no—
474#102—1221no—
475#102—1221no—
476C22⋊C8—21no—
477C22⋊C8—611no—
478C22⋊C8—611no—
479C22⋊C8—1221no—
480C22⋊C8—1221no—
481C22⋊C8—1221no—
482C22⋊C8—1221no—
483C4×C8—611no—
484C4×C8—1221no—
485C4×C8C22×D6111yescharacteristic
486C24—1221no—
487C24—2441no—
488#41—2442no—
489#41—2442no—
490C2×Q8—1221no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—614no—
498C2×D4—611no—
499C2×D4—1222no—
500C2×D4—1221no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4—4881no—
505C2×D4—4881no—
506C2×D4—4882no—
507C2×D4—2442no—
508C2×D4—2442no—
509C2×D4—2441no—
510C2×D4—2441no—
511C2×D4—2441no—
512C2×D4—2441no—
513C2×D4D4×D6111yescharacteristic
514C22×C4—21no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—4881no—
523C22×C4—4881no—
524C22×C4—2441no—
525SD16—2442no—
526SD16—2442no—
527D8—42no—
528D8—42no—
529D8—81no—
530D8—81no—
531D8—4881no—
532D8—4881no—
533D8—2442no—
534D8—2442no—
535D8—2442no—
536D8—2442no—
537M16—2442no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—81no—
542C2×C8—611no—
543C2×C8—611no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—4881no—
548C2×C8—4881no—
549C2×C8—4881no—
550C2×C8—2442no—
551C2×C8D4×D6111yescharacteristic
552C2×C8D4×D6111yescharacteristic
553C4⋊C4—611no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4D4×D6111yescharacteristic
558C22⋊C4—21no—
559C22⋊C4—1221no—
560C22⋊C4—1221no—
561C22⋊C4—1221no—
562C22⋊C4—2441no—
563C22⋊C4—2441no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42C23×D6111yescharacteristic
568C23—21no—
569C23—41no—
570C23—41no—
571C23—611no—
572C23—1224no—
573C23—1221no—
574C23—4881no—
575C23—4881no—
576C23—4882no—
577C23—4882no—
578C23—4882no—
579C23—2441no—
580Q8—2441no—
581Q8—2441no—
582D4—21no—
583D4—41no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—81no—
592D4—81no—
593D4—612no—
594D4—1224no—
595D4—1221no—
596D4—4881no—
597D4—4882no—
598D4—4882no—
599D4—4882no—
600D4—4882no—
601D4—2441no—
602D4—2441no—
603D4—2441no—
604D4—2441no—
605D4D8×D6112yes—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—611no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—4881no—
618C2×C4—4881no—
619C2×C4—4882no—
620C2×C4—4881no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4C2×D4×D6111yescharacteristic
626C2×C4C2×D4×D6111yescharacteristic
627C2×C4#41×D6111yescharacteristic
628C8—21no—
629C8—21no—
630C8—41no—
631C8—41no—
632C8—41no—
633C8—1221no—
634C8—1221no—
635C8—2441no—
636C8—2441no—
637C8—2441no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—612no—
648C22—1221no—
649C22—1221no—
650C22—1222no—
651C22—9764no—
652C22—4884no—
653C22—4881no—
654C22—4881no—
655C22—4881no—
656C22—2441no—
657C22—2441no—
658C22#121×D6111yescharacteristic
659C4—21no—
660C4—41no—
661C4—612no—
662C4—1221no—
663C4—1221no—
664C4—4881no—
665C4—2441no—
666C4C2×D8×D6111yescharacteristic
667C4#135×D6111yescharacteristic
668C4#1684056311yescharacteristic
669C4#1684056611yescharacteristic
670C2—42no—
671C2—161no—
672C2—161no—
673C2—1222no—
674C2—9761no—
675C2—2442no—
676C2#2145531711yescharacteristic
677C2#466×D6111yescharacteristic
678C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1667C2
socleC22×C61C2×#121
hypercenter#1667D61
hypocenterC61#19386
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1667

Complements

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

NH ≅ G/Ntype
1. #21456733C2semidirect
2. #21455278C2semidirect
3. #459×D61C2semidirect
4. #21454840C2semidirect
5. #21445257C2semidirect
6. #21445256C2semidirect
7. #21445255C2semidirect
8. #21445254C2semidirect
9. #21445173C2semidirect
10. #21445172C2semidirect
11. #21445171C2semidirect
12. #21444592C2semidirect
13. #21444485C2semidirect
14. #21444465C2semidirect
15. C61×#1667C2semidirect
21. #16840138C22semidirect
22. #16840109C22semidirect
23. #16840108C22semidirect
26. #16839968C22semidirect
27. #16839967C22semidirect
35. #16839954C22semidirect
38. C8×D244C22semidirect
39. D4×#92418C22semidirect
40. #16839827C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839514C22semidirect
54. #16839509C22semidirect
55. #16839507C22semidirect
56. #16839506C22semidirect
57. #16839506C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#493C22semidirect
62. C61×#459C22semidirect
63. C8×D4×C61C22semidirect
64. C61×#335C22semidirect
65. C61×#331C22semidirect
66. C61×#331C22semidirect
171. D4×D61D8semidirect
342. #1667D61semidirect
361. #493C2×D61semidirect
377. #459C2×D61semidirect
385. C8×D4C2×D61semidirect
397. #335C2×D61semidirect
404. #331C2×D61semidirect
405. #331C2×D61semidirect
406. C61#19386semidirect

Maximal subgroups

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

subgroupindex
#214567332
#214552782
#459×D612
#214548402
#214452572
#214452562
#214452552
#214452542
#214451732
#214451722
#214451712
#214445922
#214444852
#214444652
C61×#16672
#1938661