Finite Group Databasebeta

← #30615938

Subgroups — #30615938

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462472C211yesmaximal, characteristic, complemented
2#21462439C211yesmaximal, characteristic
3C2×#16840752C211yesmaximal, characteristic
4C2×#16840728C211yesmaximal, characteristic, complemented
5#21461905C214yesmaximal, complemented
6#21461904C214yesmaximal, complemented
7#21461523C211yesmaximal, characteristic, complemented
8C2×#16840637C211yesmaximal, characteristic, complemented
9#413×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11C22×D61⋊C8C2211yescharacteristic
12C2×#14519856C2211yescharacteristic
13C2×#14519856C2211yescharacteristic
14#16840773—22no—
15#16840773C2214yes—
16#16840772—22no—
17#16840755C2212yes—
18#16840754C2212yes—
19#16840753—22no—
20#16840752—21no—
21#16840752—21no—
22#16840752—22no—
23#16840752—22no—
24C2×#14519840C2212yes—
25C2×#14519839C2212yes—
26#16840728—22no—
27#16840728C2214yes—
28C2×#14519831—21no—
29C2×C8×#6923—21no—
30#16840641C2214yes—
31#16840637C2214yes—
32C2×#41×D61C2211yescharacteristic
33C2×M16×D61C2211yescharacteristic
34C2×M16×D61C2211yescharacteristic
35#16839746C411yescharacteristic, complemented
36#16839745C412yescomplemented
37#16839744C411yescharacteristic, complemented
38#16839744C411yescharacteristic, complemented
39#16839743C412yescomplemented
40#101×D61C2212yes—
41#100×D61C2212yes—
42C61×#413C411yescharacteristic, complemented
43C2×#11758213—21no—
44D61⋊M16—21no—
45D61⋊M16—21no—
46D61⋊M16—21no—
47D61⋊M16C2312yes—
48C2×#11758217—21no—
49C2×D61⋊C8—21no—
50C2×D61⋊C8—21no—
51C2×D61⋊C8—21no—
52C2×D61⋊C8—21no—
53C2×D61⋊C8D411yescharacteristic
54C2×D61⋊C8D411yescharacteristic
55C2×D61⋊C8D412yes—
56C2×D61⋊C8C2312yes—
57C22×#92420—21no—
58#14519856—21no—
59#14519856—21no—
60#14519856—21no—
61#14519856—21no—
62#14519856—21no—
63#14519856—21no—
64#14519856D412yes—
65#14519856C2312yes—
66C2×C4⋊#6923D412yes—
67C2×C4⋊#6923C2312yes—
68C2×C22⋊#6923—21no—
69C2×C22⋊#6923—21no—
70C2×C4×#6923—21no—
71C2×C4×#6923—21no—
72#14519846—24no—
73#14519845—24no—
74#14519840—24no—
75#14519840—22no—
76#14519840—22no—
77#14519839—24no—
78#14519839—22no—
79#14519839—22no—
80#14519831—24no—
81C8×#6923—24no—
82C2×#11758226C2×C411yescharacteristic
83C2×#11758222—21no—
84C2×#11758222C2×C411yescharacteristic
85C2×#11758224—21no—
86C2×#11758224C2×C411yescharacteristic
87#41×D61—21no—
88#41×D61—21no—
89#41×D61—21no—
90#41×D61—21no—
91#41×D61—21no—
92#41×D61—21no—
93#41×D61—21no—
94#41×D61—21no—
95C2×Q8×D61C2311yescharacteristic
96C2×D4×D61—21no—
97C2×D4×D61C2311yescharacteristic
98C22×C4×D61—21no—
99C22×C4×D61C2311yescharacteristic
100C2×#11758194C2×C411yescharacteristic
101C2×#11758194C2×C411yescharacteristic
102C2×#11758189—21no—
103C2×#11758189—21no—
104M16×D61—21no—
105M16×D61—21no—
106M16×D61—21no—
107M16×D61—21no—
108M16×D61C2×C412yes—
109M16×D61C2×C412yes—
110M16×D61C2312yes—
111M16×D61C2312yes—
112C2×C8×D61—21no—
113C2×C8×D61—21no—
114#14519696C2×C412yescomplemented
115#14519695C2×C412yescomplemented
116#14519695C2×C412yescomplemented
117#14519694C2×C412yescomplemented
118#14519693C2×C412yescomplemented
119#14519693C2×C412yescomplemented
120C2×#41×C61C2×C411yescharacteristic
121C2×M16×C61C2×C411yescharacteristic
122C2×M16×C61C2×C411yescharacteristic
123#101×C61C2×C412yescomplemented
124#100×C61C2×C412yescomplemented
125C23×D61—21no—
126C23×D61#4111yescharacteristic
127C22×#6923—22no—
128C22×#6923—22no—
129#41×C61—21no—
130#41×C61—21no—
131#41×C61—21no—
132#41×C61—21no—
133C2×Q8×C61C22×C411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61C22×C411yescharacteristic
136C22×C4×C61—21no—
137C22×C4×C61C22×C411yescharacteristic
138C2×C22⋊D61—21no—
139C2×C22⋊D61—21no—
140C2×C22⋊D61—21no—
141C22×Dic61—21no—
142C22×Dic61#4111yescharacteristic
143#11758226—21no—
144#11758226—21no—
145#11758226—21no—
146#11758226—21no—
147Q8×D61—21no—
148Q8×D61—21no—
149Q8×D61—21no—
150Q8×D61—21no—
151#11758224—21no—
152#11758224—21no—
153#11758224—21no—
154#11758224—21no—
155#11758224—41no—
156#11758224—41no—
157#11758224—41no—
158#11758224—41no—
159D4×D61—21no—
160D4×D61—21no—
161D4×D61—21no—
162D4×D61—21no—
163D4×D61—41no—
164D4×D61—41no—
165D4×D61—41no—
166D4×D61—41no—
167#11758222—21no—
168#11758222—21no—
169#11758222—21no—
170#11758222—21no—
171#11758222—41no—
172#11758222—41no—
173#11758222—41no—
174#11758222—41no—
175C2×D244—21no—
176C2×D244C22×C411yescharacteristic
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×C4×D61—41no—
181C2×C4×D61—41no—
182C2×C4×D61—41no—
183C2×C4×D61—41no—
184C2×C4×D61C22×C411yescharacteristic
185C2×C4×D61C2×D411yescharacteristic
186C2×C4×D61C2×D411yescharacteristic
187C2×C4×D61C2×D411yescharacteristic
188C2×C4×D61C2×D411yescharacteristic
189C2×C4×D61#4111yescharacteristic
190C2×C4×D61#4111yescharacteristic
191C2×C4×D61#4111yescharacteristic
192C2×C4×D61#4111yescharacteristic
193C2×C4×D61#4111yescharacteristic
194C2×C4×D61#4111yescharacteristic
195C2×C4×D61C2411yescharacteristic
196C2×Dic122—21no—
197C2×Dic122C22×C411yescharacteristic
198C22⋊#6923—22no—
199C22⋊#6923—22no—
200C22⋊#6923—41no—
201C22⋊#6923—41no—
202#11758217—22no—
203#11758217—41no—
204C2×#92420—21no—
205C2×#92420—21no—
206C2×#92420—22no—
207C2×#92420—22no—
208C2×#92420—41no—
209C4⋊#6923—22no—
210C4⋊#6923—22no—
211C4⋊#6923—22no—
212C4⋊#6923C2×D414yes—
213C4×#6923—22no—
214C4×#6923—22no—
215C4×#6923—41no—
216C4×#6923—41no—
217#11758213—22no—
218#11758213—41no—
219D61⋊C8—21no—
220D61⋊C8—21no—
221D61⋊C8—22no—
222D61⋊C8—22no—
223D61⋊C8—22no—
224D61⋊C8—41no—
225D61⋊C8C2×D412yes—
226D61⋊C8C2×D412yes—
227M16×C61—21no—
228M16×C61—21no—
229M16×C61C22×C412yes—
230M16×C61C22×C412yes—
231C2×C8×C61—21no—
232C2×C8×C61—21no—
233#11758194—21no—
234#11758194—21no—
235#11758194C22×C412yes—
236#11758194C22×C412yes—
237C2×#92418—21no—
238C2×#92418—21no—
239#11758189—22no—
240#11758189—22no—
241#11758189—22no—
242#11758189—22no—
243C8×D61—22no—
244C8×D61—22no—
245C8×D61—22no—
246C8×D61—22no—
247C23×C61—21no—
248C23×C61#11711yescharacteristic
249C22×D61—21no—
250C22×D61—42no—
251C22×D61—41no—
252C22×D61—41no—
253C22×D61—82no—
254C22×D61#11711yescharacteristic
255C22×D61#12111yescharacteristic
256C22×D61#12111yescharacteristic
257C22×D61#12611yescharacteristic
258C22×D61#12611yescharacteristic
259C22×D61C2×#4111yescharacteristic
260C2×#6923—24no—
261C2×#6923—44no—
262C2×#6923—42no—
263C2×#6923—82no—
264Q8×C61—21no—
265Q8×C61—21no—
266D4×C61—21no—
267D4×C61—21no—
268D4×C61—41no—
269D4×C61—41no—
270C2×C4×C61—21no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61#11711yescharacteristic
274C2×C4×C61C4×D411yescharacteristic
275C2×C4×C61C4×D411yescharacteristic
276C2×C4×C61C4×D411yescharacteristic
277C2×C4×C61C4×D411yescharacteristic
278C2×C4×C61C23×C411yescharacteristic
279C22⋊D61—41no—
280C22⋊D61—41no—
281C22⋊D61—41no—
282C22⋊D61—41no—
283C22⋊D61—41no—
284C22⋊D61—41no—
285C2×Dic61—21no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61#11711yescharacteristic
289C2×Dic61#12311yescharacteristic
290C2×Dic61#12311yescharacteristic
291C2×Dic61#12411yescharacteristic
292C2×Dic61#12411yescharacteristic
293C2×Dic61C2×#4111yescharacteristic
294D244—21no—
295D244—21no—
296D244—41no—
297D244—41no—
298C4×D61—41no—
299C4×D61—41no—
300C4×D61—41no—
301C4×D61—41no—
302C4×D61—41no—
303C4×D61—41no—
304C4×D61C4×D411yescharacteristic
305C4×D61C4×D411yescharacteristic
306C4×D61C4×D411yescharacteristic
307C4×D61C4×D411yescharacteristic
308C4×D61#12111yescharacteristic
309C4×D61#12111yescharacteristic
310C4×D61#12311yescharacteristic
311C4×D61#12311yescharacteristic
312C4×D61#12411yescharacteristic
313C4×D61#12411yescharacteristic
314C4×D61#12611yescharacteristic
315C4×D61#12611yescharacteristic
316C4×D61C22×D411yescharacteristic
317C4×D61C22×D411yescharacteristic
318C4×D61C2×#4111yescharacteristic
319C4×D61C2×#4111yescharacteristic
320Dic122—21no—
321Dic122—21no—
322Dic122—41no—
323Dic122—41no—
324#92420—22no—
325#92420—22no—
326#92420—42no—
327C8×C61—22no—
328C8×C61—22no—
329#92418—22no—
330#92418—22no—
331#21075—611nomaximal, Sylow
332C22×C61—41no—
333C22×C61—41no—
334C22×C61—81no—
335C22×C61#38811yescharacteristic
336C22×C61#38811yescharacteristic
337C22×C61C2×#11711yescharacteristic
338C2×D61—22no—
339C2×D61—22no—
340C2×D61—22no—
341C2×D61—41no—
342C2×D61—41no—
343C2×D61—41no—
344C2×D61—41no—
345C2×D61—82no—
346C2×D61—82no—
347C2×D61—81no—
348C2×D61#38811yescharacteristic
349C2×D61#38811yescharacteristic
350C2×D61C2×#12111yescharacteristic
351C2×D61C2×#12611yescharacteristic
352#6923—44no—
353#6923—82no—
354C4×C61—41no—
355C4×C61#38811yescharacteristic
356C4×C61#38811yescharacteristic
357C4×C61C2×C4×D411yescharacteristic
358C4×C61C2×C4×D411yescharacteristic
359Dic61—41no—
360Dic61#38811yescharacteristic
361Dic61#38811yescharacteristic
362Dic61C2×#12311yescharacteristic
363Dic61C2×#12411yescharacteristic
364#2911—611no—
365#2896—611no—
366C2×#416—611no—
367C2×#413—611no—
368C2×#405—611no—
369#1931—614no—
370#1930—614no—
371#1742—611no—
372C2×#341—611no—
373C2×C61—21no—
374C2×C61—21no—
375C2×C61—21no—
376C2×C61—81no—
377C2×C61C2×#38811yescharacteristic
378D61—21no—
379D61—21no—
380D61—21no—
381D61—81no—
382D61#306712yes—
383C22×#41—611no—
384C22×M16—611no—
385C22×M16—611no—
386C22×M16—611no—
387C23×C8—611no—
388C2×#117—611no—
389C2×#117—611no—
390#428—614no—
391#428—1222no—
392#427—1222no—
393#419—612no—
394#418—612no—
395#417—1222no—
396#416—1222no—
397#416—1221no—
398#416—1221no—
399#416—1222no—
400C2×#103—612no—
401C2×#102—612no—
402#413—611no—
403#413—611no—
404#413—611no—
405#413—612no—
406#413—612no—
407#413#692311yescharacteristic, complemented
408C2×#101—612no—
409C2×#100—612no—
410#405—614no—
411#405—1222no—
412C2×#97—1221no—
413C2×C4×C8—1221no—
414#343—614no—
415#341—614no—
416C61#2107511yescharacteristic, complemented, Sylow
417C2×#41—611no—
418C2×#41—611no—
419C2×#41—611no—
420C2×#41—1221no—
421C2×#41—1221no—
422C2×#41—1221no—
423C2×#41—1221no—
424C2×#41—1221no—
425C2×#41—1221no—
426C2×#41—1221no—
427C2×#41—1221no—
428C2×#41—1221no—
429C2×#41—1221no—
430C2×#41C2×#692311yescharacteristic
431C22×Q8—611no—
432C22×D4—611no—
433C22×D4—1221no—
434C23×C4—611no—
435C23×C4—1221no—
436C2×M16—611no—
437C2×M16—611no—
438C2×M16—612no—
439C2×M16—612no—
440C2×M16—612no—
441C2×M16—612no—
442C2×M16—612no—
443C2×M16—1221no—
444C2×M16—1221no—
445C2×M16—1221no—
446C2×M16—1221no—
447C2×M16—1221no—
448C2×M16—1221no—
449C2×M16—1221no—
450C2×M16—1221no—
451C2×M16—1221no—
452C2×M16—1221no—
453C2×M16—1221no—
454C2×M16C2×#692311yescharacteristic
455C2×M16C2×#692311yescharacteristic
456C22×C8—611no—
457C22×C8—611no—
458C22×C8—612no—
459C22×C8—612no—
460C22×C8—1221no—
461C22×C8—1221no—
462C22×C8—1221no—
463C22×C8—1221no—
464C22×C8—1221no—
465C22×C8—1221no—
466C22×C8—1221no—
467#117—612no—
468#117—612no—
469#117—1221no—
470#117—1221no—
471#117—1221no—
472#117—1221no—
473#117—1221no—
474#117—1221no—
475C2×C4⋊C4—612no—
476C2×C4⋊C4—612no—
477C2×C22⋊C4—1221no—
478C2×C22⋊C4—1221no—
479C2×C42—1221no—
480C2×C42—1221no—
481#107—1224no—
482#106—1224no—
483#103—1224no—
484#103—1222no—
485#103—1222no—
486#102—1224no—
487#102—1222no—
488#102—1222no—
489#101—612no—
490#101—612no—
491#101—612no—
492#101C2×#692312yescomplemented
493#100—612no—
494#100—612no—
495#100—612no—
496#100C2×#692312yescomplemented
497#97—1224no—
498C4×C8—1224no—
499C24—611no—
500C24—1221no—
501#41—21no—
502#41—21no—
503#41—21no—
504#41—21no—
505#41—1221no—
506#41—1221no—
507#41—1221no—
508#41—1221no—
509#41—1221no—
510#41—1221no—
511#41—1221no—
512#41—1221no—
513#41—1221no—
514#41—1221no—
515#41—1221no—
516#41—1221no—
517#41—2441no—
518#41—2441no—
519#41—2441no—
520#41—2441no—
521#41—2441no—
522#41—2441no—
523#41—2441no—
524#41—2441no—
525C2×Q8—611no—
526C2×Q8—1221no—
527C2×Q8—1221no—
528C2×Q8—1221no—
529C2×Q8—1221no—
530C2×Q8—1221no—
531C2×Q8C22×#692311yescharacteristic
532C2×D4—21no—
533C2×D4—611no—
534C2×D4—1221no—
535C2×D4—1221no—
536C2×D4—1221no—
537C2×D4—1221no—
538C2×D4—1221no—
539C2×D4—1221no—
540C2×D4—1221no—
541C2×D4—1221no—
542C2×D4—2441no—
543C2×D4—2441no—
544C2×D4—2441no—
545C2×D4—2441no—
546C2×D4C22×#692311yescharacteristic
547C22×C4—21no—
548C22×C4—611no—
549C22×C4—611no—
550C22×C4—611no—
551C22×C4—611no—
552C22×C4—611no—
553C22×C4—611no—
554C22×C4—611no—
555C22×C4—611no—
556C22×C4—611no—
557C22×C4—611no—
558C22×C4—611no—
559C22×C4—611no—
560C22×C4—611no—
561C22×C4—1221no—
562C22×C4—1221no—
563C22×C4—1221no—
564C22×C4—1222no—
565C22×C4—1221no—
566C22×C4—1222no—
567C22×C4—2441no—
568C22×C4—2441no—
569C22×C4—2441no—
570C22×C4—2441no—
571C22×C4C22×#692311yescharacteristic
572M16—21no—
573M16—21no—
574M16—612no—
575M16—612no—
576M16—1222no—
577M16—1222no—
578M16—1222no—
579M16—1222no—
580M16—1221no—
581M16—1221no—
582M16—1222no—
583M16—1222no—
584M16—2441no—
585M16—2441no—
586M16C22×#692312yes—
587M16C22×#692312yes—
588C2×C8—21no—
589C2×C8—21no—
590C2×C8—612no—
591C2×C8—612no—
592C2×C8—1222no—
593C2×C8—1222no—
594C2×C8—1222no—
595C2×C8—1222no—
596C2×C8—1222no—
597C2×C8—1221no—
598C2×C8—1221no—
599C2×C8—1222no—
600C2×C8—1221no—
601C2×C8—1221no—
602C2×C8—1222no—
603C2×C8—1222no—
604C2×C8—1221no—
605C2×C8—1221no—
606C2×C8—1222no—
607C2×C8—2441no—
608C2×C8—2441no—
609C4⋊C4—614no—
610C4⋊C4—1222no—
611C4⋊C4—1222no—
612C4⋊C4—1222no—
613C22⋊C4—1222no—
614C22⋊C4—1222no—
615C22⋊C4—2441no—
616C22⋊C4—2441no—
617C42—1222no—
618C42—1222no—
619C42—2441no—
620C42—2441no—
621C23—21no—
622C23—611no—
623C23—611no—
624C23—611no—
625C23—611no—
626C23—611no—
627C23—611no—
628C23—1221no—
629C23—4882no—
630C23—2441no—
631C23—2441no—
632C23—2442no—
633C23#1451985611yescharacteristic
634Q8—21no—
635Q8—21no—
636Q8—1221no—
637Q8—1221no—
638Q8—2441no—
639Q8—2441no—
640D4—21no—
641D4—21no—
642D4—41no—
643D4—41no—
644D4—1221no—
645D4—1221no—
646D4—2441no—
647D4—2441no—
648D4—2441no—
649D4—2441no—
650D4—2441no—
651D4—2441no—
652D4—2441no—
653D4—2441no—
654C2×C4—21no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—611no—
658C2×C4—611no—
659C2×C4—611no—
660C2×C4—611no—
661C2×C4—611no—
662C2×C4—611no—
663C2×C4—611no—
664C2×C4—611no—
665C2×C4—611no—
666C2×C4—611no—
667C2×C4—611no—
668C2×C4—611no—
669C2×C4—611no—
670C2×C4—611no—
671C2×C4—611no—
672C2×C4—611no—
673C2×C4—611no—
674C2×C4—611no—
675C2×C4—611no—
676C2×C4—611no—
677C2×C4—611no—
678C2×C4—611no—
679C2×C4—1224no—
680C2×C4—1221no—
681C2×C4—4882no—
682C2×C4—2441no—
683C2×C4—2441no—
684C2×C4—2441no—
685C2×C4—2441no—
686C2×C4—2441no—
687C2×C4—2441no—
688C2×C4—2442no—
689C2×C4—2444no—
690C2×C4—2441no—
691C2×C4—2441no—
692C2×C4#1451985611yescharacteristic
693C2×C4D4×#692311yescharacteristic
694C2×C4D4×#692311yescharacteristic
695C2×C4D4×#692311yescharacteristic
696C2×C4D4×#692311yescharacteristic
697C2×C4C23×#692311yescharacteristic
698C8—22no—
699C8—22no—
700C8—1222no—
701C8—1222no—
702C8—1222no—
703C8—1222no—
704C8—2442no—
705C22—41no—
706C22—41no—
707C22—81no—
708C22—611no—
709C22—611no—
710C22—611no—
711C22—611no—
712C22—1222no—
713C22—1222no—
714C22—1222no—
715C22—4881no—
716C22—4882no—
717C22—4882no—
718C22—2441no—
719C22—2441no—
720C22—2441no—
721C22—2441no—
722C22#1684071211yescharacteristic
723C22#1684071211yescharacteristic
724C22C2×#1451985611yescharacteristic
725C4—41no—
726C4—611no—
727C4—611no—
728C4—611no—
729C4—611no—
730C4—4882no—
731C4—2444no—
732C4—2441no—
733C4#1684071211yescharacteristic
734C4#1684071211yescharacteristic
735C4C2×D4×#692311yescharacteristic
736C4C2×D4×#692311yescharacteristic
737C2—21no—
738C2—21no—
739C2—21no—
740C2—81no—
741C2—612no—
742C2—1221no—
743C2—1221no—
744C2—1221no—
745C2—4881no—
746C2C2×#1684071211yescharacteristic
subgroupG/H
centerC4#16840712
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#413C4
socleC2×C61C2×#388
prime radicalC2×#41×D61C22
cosocleC2×C4×D61C24
hypercenter#413#6923
hypocenterC61#21075
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #21462472C2semidirect
4. C2×#16840728C2semidirect
5. #21461905C2semidirect
6. #21461904C2semidirect
7. #21461523C2semidirect
8. C2×#16840637C2semidirect
35. #16839746C4semidirect
36. #16839745C4semidirect
37. #16839744C4semidirect
38. #16839744C4semidirect
39. #16839743C4semidirect
42. C61×#413C4semidirect
114. #14519696C2×C4semidirect
115. #14519695C2×C4semidirect
116. #14519695C2×C4semidirect
117. #14519694C2×C4semidirect
118. #14519693C2×C4semidirect
119. #14519693C2×C4semidirect
123. #101×C61C2×C4semidirect
124. #100×C61C2×C4semidirect
407. #413#6923semidirect
416. C61#21075semidirect
492. #101C2×#6923semidirect
496. #100C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624722
#214624392
C2×#168407522
C2×#168407282
#214619052
#214619042
#214615232
C2×#168406372
#413×D612
#2107561