Finite Group Databasebeta

← #30558415

Subgroups — #30558415

Order 15616.

2020 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459739C211yesmaximal, characteristic, complemented
2#21457990C214yesmaximal, complemented
3#21457915C214yesmaximal, complemented
4#21457870C214yesmaximal, complemented
5#21457795C214yesmaximal, complemented
6C2×#16840130C211yesmaximal, characteristic, complemented
7#21455782C212yesmaximal, complemented
8#21455641C211yesmaximal, characteristic, complemented
9#21455506C212yesmaximal, complemented
10#21455302C211yesmaximal, characteristic, complemented
11#21455134C212yesmaximal, complemented
12#21454617C211yesmaximal, characteristic, complemented
13#21454537C211yesmaximal, characteristic, complemented
14#21454352C212yesmaximal, complemented
15C61×#3499C211yesmaximal, characteristic, complemented
16C2×#14519808C2212yes—
17C2×#14519797C2211yescharacteristic
18C2×#14519784C2212yes—
19#16840458C2214yes—
20#16840455—24no—
21#16840452C2214yes—
22#16840373—21no—
23#16840366—21no—
24#16840340C2212yes—
25#16840291—22no—
26C2×#14519769C2211yescharacteristic
27#16840245C2212yes—
28C4×#11758222C2211yescharacteristic
29#16840130C2214yes—
30#16840130C2214yes—
31#16840100C2214yes—
32#16840100C2214yes—
33#16840095C2214yes—
34#16840090C2214yes—
35#16840069C2214yes—
36#16840062C2214yes—
37#16840018C2214yes—
38#16840010C2214yes—
39#16840001C2214yes—
40#16839993C2214yes—
41#16839969C2214yes—
42#16839964C2214yes—
43#16839924C2214yes—
44#16839918C2214yes—
45#16839907C2214yes—
46#16839901C2214yes—
47#16839872C2212yes—
48#16839867C2212yes—
49#16839862C2212yes—
50#16839860C2212yes—
51#16839834C2214yes—
52#16839834C2214yes—
53#16839805C2212yes—
54#16839798C2211yescharacteristic
55#16839786C2211yescharacteristic
56#16839771C2214yes—
57#16839771C2212yes—
58#16839763C2212yes—
59#16839763C2214yes—
60#16839758C2211yescharacteristic
61#16839758C2214yes—
62#16839756C2211yescharacteristic
63C2×#14519701C2212yes—
64C2×#14519701C2214yes—
65C2×#14519701C2212yes—
66#16839729C2212yes—
67C2×#14519688C2211yescharacteristic
68C2×2−1+4×C61C2212yes—
69D4×Q8×C61C2214yes—
70C61×#547—24no—
71C61×#547C2214yes—
72C61×#536—22no—
73C61×#536C2212yes—
74C61×#532—21no—
75C61×#532C2211yescharacteristic
76C2×C4⋊Q8×C61—21no—
77C2×C4⋊Q8×C61C2211yescharacteristic
78C4×#41×C61C2211yescharacteristic
79C2×#11758222C2311yescharacteristic
80#14519808—44no—
81#14519797—24no—
82C2×#11758202C2311yescharacteristic
83C2×#11758202C2314yes—
84C2×#11758200C2311yescharacteristic
85C2×#11758201C2311yescharacteristic
86C2×#11758201C2314yes—
87C2×#11758199C2311yescharacteristic
88#14519784—44no—
89C2×#11758194C2312yes—
90C2×#11758194C2314yes—
91C22×#92418C2311yescharacteristic
92C22×#92418C2311yescharacteristic
93C22×#92418C2314yes—
94#14519772—24no—
95#14519770—24no—
96#14519769—21no—
97#14519769—21no—
98#14519769C2314yes—
99#14519767—24no—
100#14519759—24no—
101#14519758—24no—
102#14519756C2314yes—
103#14519753—22no—
104#14519751C2314yes—
105#14519749—24no—
106#14519749—24no—
107#14519748—22no—
108Q8×Dic61—22no—
109C4×Dic122C2312yes—
110C4×D244C2312yes—
111D4×Dic61—24no—
112D4×Dic61—22no—
113C4×C22⋊D61—22no—
114#14519737—24no—
115#14519736C2314yes—
116#14519736C2311yescharacteristic
117#14519734—21no—
118#14519733—24no—
119C2×C4⋊Dic61C2314yes—
120C2×C4⋊Dic61C2312yes—
121C2×#11758196—22no—
122C42×D61—21no—
123#14519710C2314yes—
124#14519710C2312yes—
125#14519708C2314yes—
126#14519708C2312yes—
127C4⋊#92418C2314yes—
128#14519701C2314yes—
129#14519701C2314yes—
130#14519701C2314yes—
131#14519701C2314yes—
132#14519701C2312yes—
133#14519701C2314yes—
134#14519701C2314yes—
135#14519701C2314yes—
136#14519700C2312yes—
137#14519698C2312yes—
138#14519698C2314yes—
139#14519697C2312yes—
140C22⋊#92418C2314yes—
141C22⋊#92418C2314yes—
142#14519688C2312yes—
143#14519688C2312yes—
1442−1+4×C61—28no—
1452−1+4×C61—44no—
146C2×#41×C61—22no—
147C2×#41×C61—24no—
148C2×#41×C61C2314yes—
149C2×#41×C61C2311yescharacteristic
150C2×#41×C61C2312yes—
151C22×Q8×C61—22no—
152C22×Q8×C61C2312yes—
153C22×Q8×C61C2314yes—
154C4⋊Q8×C61—22no—
155C4⋊Q8×C61—24no—
156C4⋊Q8×C61—21no—
157C4⋊Q8×C61—21no—
158C4⋊Q8×C61D411yescharacteristic
159C4⋊Q8×C61C2314yes—
160#127×C61D411yescharacteristic
161#124×C61—22no—
162#124×C61D412yes—
163#123×C61—24no—
164#123×C61—24no—
165#123×C61—24no—
166C22⋊Q8×C61—24no—
167C22⋊Q8×C61—24no—
168C22⋊Q8×C61C2314yes—
169C22⋊Q8×C61C2314yes—
170#121×C61—24no—
171#121×C61D414yes—
172C4×Q8×C61C2312yes—
173C4×D4×C61—24no—
174C4×D4×C61—22no—
175C4×D4×C61C2312yes—
176C4×D4×C61C2314yes—
177#117×C61—21no—
178#117×C61C2311yescharacteristic
179C2×C4⋊C4×C61—22no—
180C2×C4⋊C4×C61—24no—
181C2×C42×C61—21no—
182C2×C42×C61C2311yescharacteristic
183#41×C61—24no—
184#41×C61—22no—
185#41×C61—24no—
186#41×C61—28no—
187#41×C61—44no—
188#41×C61—48no—
189#41×C61—416no—
190#41×C61#4114yes—
191C2×Q8×C61—24no—
192C2×Q8×C61—22no—
193C2×Q8×C61—28no—
194C2×Q8×C61—24no—
195C2×Q8×C61—24no—
196C2×Q8×C61—44no—
197C2×Q8×C61—48no—
198C2×Q8×C61C2×D412yes—
199C2×Q8×C61C2414yes—
200C2×Q8×C61C2411yescharacteristic
201C2×D4×C61—21no—
202C2×D4×C61—24no—
203C2×D4×C61C2×D414yes—
204C2×D4×C61C2×D412yes—
205C2×D4×C61C2×D414yes—
206C2×D4×C61C2411yescharacteristic
207C22×C4×C61—21no—
208C22×C4×C61—22no—
209C22×C4×C61—24no—
210C22×C4×C61—22no—
211C22×C4×C61—24no—
212C22×C4×C61C2×D412yes—
213C22×C4×C61C2414yes—
214C22×C4×C61C2411yescharacteristic
215C2×C22⋊D61—21no—
216C2×C22⋊D61—24no—
217C22×Dic61—22no—
218C22×Dic61—24no—
219#11758222—22no—
220#11758222—22no—
221C2×D244C2411yescharacteristic
222C2×C4×D61—21no—
223C2×C4×D61—22no—
224C2×Dic122C2411yescharacteristic
225C4⋊C4×C61—24no—
226C4⋊C4×C61—24no—
227C4⋊C4×C61—24no—
228C4⋊C4×C61—24no—
229C4⋊C4×C61—22no—
230C4⋊C4×C61C2×D414yes—
231C4⋊C4×C61C2412yes—
232C22⋊C4×C61—21no—
233C22⋊C4×C61—24no—
234C22⋊C4×C61—24no—
235C22⋊C4×C61—22no—
236C22⋊C4×C61C2×D414yes—
237C22⋊C4×C61C2×D414yes—
238C42×C61—21no—
239C42×C61—22no—
240C42×C61C2×D411yescharacteristic
241C42×C61C2×D411yescharacteristic
242C42×C61C2412yes—
243C22⋊Dic61—24no—
244C22⋊Dic61—21no—
245C22⋊Dic61—24no—
246#11758202—22no—
247#11758202—44no—
248#11758201—22no—
249#11758201—44no—
250#11758200—22no—
251#11758199—22no—
252#11758198—22no—
253#11758198—24no—
254C4⋊Dic61—22no—
255C4⋊Dic61—24no—
256C4⋊Dic61C2412yes—
257C4⋊Dic61C2414yes—
258#11758196—24no—
259#11758196—24no—
260#11758196—24no—
261#11758196—22no—
262C4×Dic61—22no—
263C4×Dic61—21no—
264C4×Dic61—24no—
265#11758194—24no—
266#11758194—44no—
267C2×#92418—22no—
268C2×#92418—22no—
269C2×#92418—44no—
270C2×#92418C2411yescharacteristic
271C2×#92418C2411yescharacteristic
272C2×#92418C2412yes—
273C2×#92418C2414yes—
274C23×C61—21no—
275C23×C61C22×D414yes—
276C23×C61C22×D411yescharacteristic
277C22×D61—21no—
278Q8×C61—24no—
279Q8×C61—21no—
280Q8×C61—24no—
281Q8×C61—24no—
282Q8×C61—48no—
283Q8×C61—44no—
284Q8×C61C2×#4112yes—
285D4×C61—24no—
286D4×C61—28no—
287D4×C61—24no—
288D4×C61—21no—
289D4×C61—44no—
290D4×C61—48no—
291D4×C61C2×#4112yes—
292C2×C4×C61—21no—
293C2×C4×C61—24no—
294C2×C4×C61—24no—
295C2×C4×C61—24no—
296C2×C4×C61—22no—
297C2×C4×C61—24no—
298C2×C4×C61—42no—
299C2×C4×C61—48no—
300C2×C4×C61—48no—
301C2×C4×C61—44no—
302C2×C4×C61—44no—
303C2×C4×C61C22×D412yes—
304C2×C4×C61C22×D414yes—
305C2×C4×C61C22×D412yes—
306C2×C4×C61C22×D411yescharacteristic
307C2×C4×C61C2×#4112yes—
308C2×C4×C61H(5,𝔽2)12yes—
309C2×C4×C61C2511yescharacteristic
310C22⋊D61—42no—
311C22⋊D61—84no—
312C2×Dic61—22no—
313C2×Dic61—21no—
314C2×Dic61—24no—
315C2×Dic61—84no—
316C2×Dic61—82no—
317D244—21no—
318D244—21no—
319C4×D61—42no—
320C4×D61—82no—
321Dic122—21no—
322Dic122—21no—
323#92418—21no—
324#92418—21no—
325#92418—22no—
326#92418—24no—
327#61224—611nomaximal, Sylow
328C22×C61—22no—
329C22×C61—24no—
330C22×C61—24no—
331C22×C61—22no—
332C22×C61—24no—
333C22×C61—42no—
334C22×C61#54612yes—
335C22×C61C23×D411yescharacteristic
336C2×D61—42no—
337C2×D61—41no—
338C4×C61—22no—
339C4×C61—22no—
340C4×C61—22no—
341C4×C61—24no—
342C4×C61—42no—
343C4×C61—44no—
344C4×C61#54612yes—
345C4×C61C22×#4111yescharacteristic
346C4×C61C2×H(5,𝔽2)11yescharacteristic
347Dic61—42no—
348Dic61—44no—
349Dic61—41no—
350#3512—611no—
351#3499D6111yescharacteristic, complemented
352#3320—614no—
353#3315—614no—
354#3312—614no—
355#3307—614no—
356C2×#491—611no—
357#3097—612no—
358#3080—611no—
359#3063—612no—
360#3039—611no—
361#3013—612no—
362#2943—611no—
363#2932—611no—
364#2898—612no—
365C2×C61—22no—
366C2×C61—44no—
367C2×C61—42no—
368C2×C61C2×#54611yescharacteristic
369C2×C61#358612yes—
370D61—81no—
371C2×2−1+4C2×D6112yes—
372C2×#137—612no—
373C2×#135—611no—
374C2×#131—612no—
375#553—614no—
376#553—1224no—
377#552—614no—
378D4×Q8C2×D6114yes—
379#547—24no—
380#547C2×D6114yes—
381#541—1221no—
382#540—1221no—
383#537—612no—
384#536—22no—
385#536C2×D6112yes—
386#532—21no—
387#532C2×D6111yescharacteristic
388C2×C4⋊Q8—21no—
389C2×C4⋊Q8C2×D6111yescharacteristic
390#529—1222no—
391C2×#125—611no—
392#520—612no—
393C4×#41—611no—
394C4×#41C2×D6111yescharacteristic
395#491—614no—
396#491—614no—
397#485—614no—
398#485—614no—
399#484—614no—
400#483—614no—
401#479—614no—
402#478—614no—
403#470—614no—
404#468—614no—
405#467—614no—
406#465—614no—
407#462—614no—
408#461—614no—
409#452—614no—
410#451—614no—
411#449—614no—
412#448—614no—
413#442—612no—
414#441—612no—
415#440—612no—
416#440—612no—
417#435—614no—
418#435—614no—
419#428—612no—
420#427—611no—
421#423—611no—
422#419—612no—
423#419—614no—
424#417—612no—
425#417—614no—
426#416—611no—
427#416—614no—
428#416—611no—
429C2×#103—612no—
430C2×#103—612no—
431C2×#103—614no—
432#408—612no—
433C2×#97—611no—
434C61#6122411yescharacteristic, complemented, Sylow
4352−1+4—28no—
4362−1+4—44no—
437C2×#41—22no—
438C2×#41—24no—
439C2×#41—611no—
440C2×#41C22×D6114yes—
441C2×#41C22×D6111yescharacteristic
442C2×#41C22×D6112yes—
443C22×Q8—22no—
444C22×Q8C22×D6112yes—
445C22×Q8C22×D6114yes—
446#137—2444no—
447#135—1224no—
448C2×Q16—611no—
449C2×Q16—614no—
450C2×SD16—611no—
451C2×SD16—611no—
452C2×SD16—614no—
453C2×D8—611no—
454#131—2444no—
455C2×M16—612no—
456C2×M16—614no—
457C22×C8—611no—
458C22×C8—611no—
459C22×C8—614no—
460C4⋊Q8—22no—
461C4⋊Q8—24no—
462C4⋊Q8—21no—
463C4⋊Q8—21no—
464C4⋊Q8C22⋊D6111yescharacteristic
465C4⋊Q8C22×D6114yes—
466#127C22⋊D6111yescharacteristic
467#126—1224no—
468#126—1224no—
469#125—614no—
470#125—1224no—
471#125—1221no—
472#125—1221no—
473#124—22no—
474#124C22⋊D6112yes—
475#123—24no—
476#123—24no—
477#123—24no—
478#123—1224no—
479#123—1224no—
480C22⋊Q8—24no—
481C22⋊Q8—24no—
482C22⋊Q8—614no—
483C22⋊Q8—1222no—
484C22⋊Q8C22×D6114yes—
485C22⋊Q8C22×D6114yes—
486#121—24no—
487#121—614no—
488#121—1222no—
489#121—1224no—
490#121—1224no—
491#121C22⋊D6114yes—
492C4×Q8—612no—
493C4×Q8—1222no—
494C4×Q8C22×D6112yes—
495C4×D4—24no—
496C4×D4—22no—
497C4×D4—612no—
498C4×D4—1222no—
499C4×D4—1224no—
500C4×D4—1222no—
501C4×D4—1224no—
502C4×D4C22×D6112yes—
503C4×D4C22×D6114yes—
504#117—21no—
505#117—611no—
506#117—614no—
507#117—1224no—
508#117—1221no—
509#117C22×D6111yescharacteristic
510C2×C4⋊C4—22no—
511C2×C4⋊C4—24no—
512C2×C4⋊C4—612no—
513C2×C4⋊C4—614no—
514C2×C4⋊C4—1222no—
515C2×C42—21no—
516C2×C42—1221no—
517C2×C42C22×D6111yescharacteristic
518#107—612no—
519#107—614no—
520#106—612no—
521#106—614no—
522C4⋊C8—614no—
523#103—612no—
524#103—614no—
525#103—614no—
526#103—614no—
527#103—614no—
528#103—614no—
529#103—614no—
530#103—614no—
531#103—612no—
532#102—612no—
533#102—614no—
534#102—612no—
535C22⋊C8—614no—
536C22⋊C8—614no—
537#97—612no—
538#97—612no—
539#41—24no—
540#41—24no—
541#41—22no—
542#41—28no—
543#41—44no—
544#41—48no—
545#41—416no—
546#41—1222no—
547#41—1222no—
548#41#1175822414yes—
549C2×Q8—24no—
550C2×Q8—22no—
551C2×Q8—28no—
552C2×Q8—24no—
553C2×Q8—24no—
554C2×Q8—44no—
555C2×Q8—48no—
556C2×Q8—611no—
557C2×Q8C2×C22⋊D6112yes—
558C2×Q8C23×D6114yes—
559C2×Q8C23×D6111yescharacteristic
560C2×D4—21no—
561C2×D4—24no—
562C2×D4—611no—
563C2×D4—1224no—
564C2×D4—1221no—
565C2×D4C2×C22⋊D6114yes—
566C2×D4C2×C22⋊D6112yes—
567C2×D4C2×C22⋊D6114yes—
568C2×D4C23×D6111yescharacteristic
569C22×C4—21no—
570C22×C4—22no—
571C22×C4—24no—
572C22×C4—22no—
573C22×C4—24no—
574C22×C4—1222no—
575C22×C4—1224no—
576C22×C4—1222no—
577C22×C4—1221no—
578C22×C4C2×C22⋊D6112yes—
579C22×C4C23×D6114yes—
580C22×C4C23×D6111yescharacteristic
581Q16—1222no—
582Q16—2444no—
583SD16—1222no—
584SD16—1222no—
585SD16—2444no—
586D8—1222no—
587M16—1224no—
588M16—2444no—
589C2×C8—611no—
590C2×C8—611no—
591C2×C8—612no—
592C2×C8—614no—
593C2×C8—1222no—
594C2×C8—1222no—
595C2×C8—2444no—
596C4⋊C4—24no—
597C4⋊C4—24no—
598C4⋊C4—24no—
599C4⋊C4—24no—
600C4⋊C4—22no—
601C4⋊C4—612no—
602C4⋊C4—614no—
603C4⋊C4—1222no—
604C4⋊C4—1224no—
605C4⋊C4—1224no—
606C4⋊C4—1224no—
607C4⋊C4—1222no—
608C4⋊C4—1224no—
609C4⋊C4C2×C22⋊D6114yes—
610C4⋊C4C23×D6112yes—
611C22⋊C4—21no—
612C22⋊C4—24no—
613C22⋊C4—24no—
614C22⋊C4—22no—
615C22⋊C4—1221no—
616C22⋊C4—1224no—
617C22⋊C4—1224no—
618C22⋊C4—1222no—
619C22⋊C4—1224no—
620C22⋊C4C2×C22⋊D6114yes—
621C22⋊C4C2×C22⋊D6114yes—
622C42—21no—
623C42—22no—
624C42—1221no—
625C42—1222no—
626C42—1224no—
627C42C2×C22⋊D6111yescharacteristic
628C42C2×C22⋊D6111yescharacteristic
629C42C23×D6112yes—
630C23—21no—
631C23—1221no—
632C23C22×C22⋊D6114yes—
633C23C22×C22⋊D6111yescharacteristic
634Q8—21no—
635Q8—24no—
636Q8—24no—
637Q8—24no—
638Q8—48no—
639Q8—44no—
640Q8—1221no—
641Q8—1221no—
642Q8C2×#1175822412yes—
643D4—24no—
644D4—28no—
645D4—21no—
646D4—24no—
647D4—44no—
648D4—48no—
649D4—1221no—
650D4—1221no—
651D4—4884no—
652D4—2442no—
653D4C2×#1175822412yes—
654C2×C4—21no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—24no—
660C2×C4—42no—
661C2×C4—48no—
662C2×C4—48no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—1222no—
666C2×C4—1221no—
667C2×C4—1224no—
668C2×C4—4882no—
669C2×C4—4884no—
670C2×C4—4882no—
671C2×C4—2442no—
672C2×C4C22×C22⋊D6112yes—
673C2×C4C22×C22⋊D6112yes—
674C2×C4C22×C22⋊D6114yes—
675C2×C4C22×C22⋊D6111yescharacteristic
676C2×C4C2×#1175822412yes—
677C2×C4#1451982212yes—
678C2×C4C24×D6111yescharacteristic
679C8—1221no—
680C8—1221no—
681C8—1222no—
682C8—1224no—
683C22—22no—
684C22—22no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—42no—
689C22—2441no—
690C22—2442no—
691C22#1684040612yes—
692C22C23×C22⋊D6111yescharacteristic
693C4—22no—
694C4—24no—
695C4—22no—
696C4—22no—
697C4—44no—
698C4—42no—
699C4—2441no—
700C4—2442no—
701C4—2444no—
702C4#1684040612yes—
703C4C22×#1175822411yescharacteristic
704C4C2×#1451982211yescharacteristic
705C2—22no—
706C2—44no—
707C2—42no—
708C2—4881no—
709C2C2×#1684040611yescharacteristic
710C2#2146060612yes—
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3499C2
socleC22×C61C23×D4
hypercenter#3499D61
hypocenterC61#61224
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #532 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #21459739C2semidirect
2. #21457990C2semidirect
3. #21457915C2semidirect
4. #21457870C2semidirect
5. #21457795C2semidirect
6. C2×#16840130C2semidirect
7. #21455782C2semidirect
8. #21455641C2semidirect
9. #21455506C2semidirect
10. #21455302C2semidirect
11. #21455134C2semidirect
12. #21454617C2semidirect
13. #21454537C2semidirect
14. #21454352C2semidirect
15. C61×#3499C2semidirect
351. #3499D61semidirect
434. C61#61224semidirect

Maximal subgroups

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

subgroupindex
#214597392
#214579902
#214579152
#214578702
#214577952
C2×#168401302
#214557822
#214556412
#214555062
#214553022
#214551342
#214546172
#214545372
#214543522
C61×#34992
#6122461