Finite Group Databasebeta

← #29682556

Subgroups — #29682556

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454406C211yesmaximal, characteristic, complemented
2#21454337C211yesmaximal, characteristic
3C2×#16839757C211yesmaximal, characteristic
4C2×#16839745C211yesmaximal, characteristic
5#405×D61C211yesmaximal, characteristic, complemented
6#21446482C211yesmaximal, characteristic, complemented
7#21446481C211yesmaximal, characteristic, complemented
8#21446480C211yesmaximal, characteristic, complemented
9#21446479C211yesmaximal, characteristic, complemented
10#21446477C211yesmaximal, characteristic, complemented
11#21446476C211yesmaximal, characteristic, complemented
12#21446471C211yesmaximal, characteristic, complemented
13#21445556C211yesmaximal, characteristic, complemented
14#341×D61C211yesmaximal, characteristic, complemented
15C61×#1930C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C2×M16×D61C2211yescharacteristic
18C2×M16×D61C2211yescharacteristic
19C2×M16×D61C2211yescharacteristic
20C22×C8×D61C2211yescharacteristic
21C2×#14519735C2211yescharacteristic
22#117×D61C2211yescharacteristic
23#16839805C2211yescharacteristic
24#16839804C2211yescharacteristic
25#16839803—21no—
26#16839802C2211yescharacteristic
27#16839801—21no—
28#16839799—21no—
29#16839797—21no—
30#16839771C2211yescharacteristic
31#16839768C2211yescharacteristic
32#16839766C2211yescharacteristic
33#16839762—21no—
34#16839762—21no—
35#16839757—21no—
36#16839757—21no—
37#16839757—21no—
38#16839757—21no—
39#16839757—21no—
40#16839757—21no—
41C2×#14519701C2211yescharacteristic
42C2×#14519699C2211yescharacteristic
43C2×#14519697C2211yescharacteristic
44#16839745C418yescomplemented
45C2×#14519696C2211yescharacteristic
46C2×#14519693C2211yescharacteristic
47C2×#14519693C2211yescharacteristic
48#16839720C2211yescharacteristic
49#16839719—21no—
50#16839718C2211yescharacteristic
51#16839717C2211yescharacteristic
52#16839716—21no—
53#97×D61—21no—
54C4×C8×D61—21no—
55#16839531C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839527C2211yescharacteristic
59#16839526C2211yescharacteristic
60#16839526C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C61×#419C2211yescharacteristic
63C2×#103×C61C2211yescharacteristic
64C2×#101×C61C2211yescharacteristic
65C61×#405C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#341C2211yescharacteristic
68C2×#11758226—22no—
69C2×#11758226—21no—
70C2×#11758226—21no—
71C2×#11758226—21no—
72C2×#11758226—21no—
73C2×#11758226—22no—
74C2×#11758226—22no—
75C2×#11758226C2×C414yes—
76C22×D244—21no—
77C22×D244C2311yescharacteristic
78C22×C4×D61—21no—
79C22×C4×D61C2311yescharacteristic
80C2×#11758194C2311yescharacteristic
81C2×#11758194C2311yescharacteristic
82C2×#11758194C2311yescharacteristic
83C2×#11758189—21no—
84C2×#11758189—21no—
85C2×#11758189—21no—
86M16×D61—22no—
87M16×D61—22no—
88M16×D61—22no—
89M16×D61—22no—
90M16×D61C2×C414yes—
91M16×D61C2×C414yes—
92C22×#92418C2311yescharacteristic
93C2×C8×D61—21no—
94C2×C8×D61—21no—
95C2×C8×D61—22no—
96C2×C8×D61—22no—
97C2×C8×D61—21no—
98C2×C8×D61D411yescharacteristic
99C2×C8×D61D411yescharacteristic
100C2×C8×D61D411yescharacteristic
101C2×C8×D61D411yescharacteristic
102#14519736C2311yescharacteristic
103#14519735—21no—
104#14519735—21no—
105#14519735—21no—
106#14519735—21no—
107#14519735D411yescharacteristic
108#14519735D411yescharacteristic
109#14519734—21no—
110#14519733—21no—
111C2×C4⋊Dic61C2311yescharacteristic
112C4⋊C4×D61D411yescharacteristic
113C4⋊C4×D61D411yescharacteristic
114C2×#11758198—21no—
115C22⋊C4×D61—21no—
116C2×C4×Dic61—21no—
117C42×D61—21no—
118#14519711—21no—
119#14519710—21no—
120#14519710—21no—
121#14519709—21no—
122#14519708—21no—
123#14519708—21no—
124#14519701—21no—
125#14519701—21no—
126#14519701—21no—
127#14519701—21no—
128#14519699—21no—
129#14519699—21no—
130#14519699—21no—
131#14519699—21no—
132#14519697—21no—
133#14519697—21no—
134#14519697—21no—
135#14519697—21no—
136#14519696C2×C414yescomplemented
137#14519693C2×C414yescomplemented
138#14519693C2×C414yescomplemented
139#14519688—21no—
140#14519687—21no—
141#14519687—21no—
142C4×#92418—21no—
143C8×Dic61—21no—
144C8×Dic61—21no—
145C22×Q8×C61C2311yescharacteristic
146C2×M16×C61C2311yescharacteristic
147C2×M16×C61C2311yescharacteristic
148C2×M16×C61C2311yescharacteristic
149C22×C8×C61C2311yescharacteristic
150#117×C61C2311yescharacteristic
151C2×C4⋊C4×C61C2311yescharacteristic
152#107×C61—21no—
153#106×C61—21no—
154#103×C61—21no—
155#103×C61—21no—
156#103×C61—21no—
157#103×C61—21no—
158#101×C61C2×C414yescomplemented
159#97×C61—21no—
160C4×C8×C61—21no—
161C23×D61—21no—
162C23×D61#4111yescharacteristic
163C2×Q8×C61—21no—
164C2×Q8×C61—21no—
165C2×Q8×C61—21no—
166C2×Q8×C61—21no—
167C2×Q8×C61—21no—
168C2×Q8×C61C22×C412yes—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61C2411yescharacteristic
172C22×Dic61—21no—
173C22×Dic61#4111yescharacteristic
174#11758226—24no—
175#11758226—24no—
176#11758226—24no—
177#11758226—24no—
178#11758226—42no—
179#11758226—42no—
180#11758226—42no—
181#11758226—42no—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244—21no—
185C2×D244—21no—
186C2×D244—24no—
187C2×D244—21no—
188C2×D244—41no—
189C2×D244—41no—
190C2×D244—42no—
191C2×D244C22×C412yes—
192C2×C4×D61—21no—
193C2×C4×D61—21no—
194C2×C4×D61—24no—
195C2×C4×D61—41no—
196C2×C4×D61—41no—
197C2×C4×D61—42no—
198C2×C4×D61C22×C412yes—
199C2×C4×D61C2×D411yescharacteristic
200C2×C4×D61C2×D411yescharacteristic
201C2×C4×D61C2×D411yescharacteristic
202C2×C4×D61C2×D411yescharacteristic
203C2×C4×D61#4112yes—
204C2×C4×D61#4112yes—
205C2×C4×D61#4112yes—
206M16×C61—21no—
207M16×C61—21no—
208M16×C61—21no—
209M16×C61—21no—
210M16×C61C22×C412yes—
211M16×C61C22×C412yes—
212C2×C8×C61—21no—
213C2×C8×C61—21no—
214C2×C8×C61—21no—
215C2×C8×C61—21no—
216C2×C8×C61—21no—
217C2×C8×C61C2×D411yescharacteristic
218C2×C8×C61C2×D411yescharacteristic
219C4⋊C4×C61—21no—
220C4⋊C4×C61—21no—
221C4⋊C4×C61C2×D411yescharacteristic
222C4⋊C4×C61C2×D411yescharacteristic
223C22⋊C4×C61—21no—
224C42×C61—21no—
225C22⋊Dic61—21no—
226#11758198—21no—
227#11758198—21no—
228#11758198—41no—
229#11758198—41no—
230C4⋊Dic61—21no—
231C4⋊Dic61—21no—
232C4⋊Dic61C2×D411yescharacteristic
233C4⋊Dic61C2×D411yescharacteristic
234#11758196—21no—
235#11758196—21no—
236C4×Dic61—21no—
237C4×Dic61—21no—
238C4×Dic61—21no—
239C4×Dic61—41no—
240C4×Dic61—41no—
241#11758194—21no—
242#11758194—21no—
243#11758194—21no—
244#11758194—21no—
245#11758194C22×C412yes—
246#11758194C22×C412yes—
247C2×#92418—21no—
248C2×#92418—21no—
249C2×#92418—21no—
250C2×#92418—21no—
251C2×#92418—21no—
252C2×#92418C2×D411yescharacteristic
253C2×#92418C2×D411yescharacteristic
254#11758189—24no—
255#11758189—24no—
256#11758189—42no—
257C8×D61—24no—
258C8×D61—24no—
259C8×D61—22no—
260C8×D61—22no—
261C8×D61—22no—
262C8×D61—22no—
263C8×D61—42no—
264C23×C61C2×#4111yescharacteristic
265C22×D61—22no—
266C22×D61—41no—
267C22×D61—41no—
268C22×D61—41no—
269C22×D61—41no—
270C22×D61—81no—
271C22×D61—81no—
272C22×D61#11712yes—
273C22×D61#12111yescharacteristic
274C22×D61#12111yescharacteristic
275C22×D61#12612yes—
276Q8×C61—22no—
277Q8×C61—22no—
278Q8×C61—41no—
279Q8×C61—41no—
280C2×C4×C61—22no—
281C2×C4×C61—21no—
282C2×C4×C61—21no—
283C2×C4×C61—41no—
284C2×C4×C61—41no—
285C2×C4×C61—41no—
286C2×C4×C61—81no—
287C2×C4×C61C23×C411yescharacteristic
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61C2×#4111yescharacteristic
291C2×C4×C61C2×#4111yescharacteristic
292C2×C4×C61C2×#4111yescharacteristic
293C2×Dic61—21no—
294C2×Dic61—21no—
295C2×Dic61—41no—
296C2×Dic61—81no—
297C2×Dic61#11712yes—
298C2×Dic61#12111yescharacteristic
299C2×Dic61#12111yescharacteristic
300C2×Dic61#12612yes—
301D244—22no—
302D244—22no—
303D244—44no—
304D244—44no—
305D244—41no—
306D244—41no—
307C4×D61—42no—
308C4×D61—42no—
309C4×D61—44no—
310C4×D61—44no—
311C4×D61C4×D412yes—
312C4×D61C4×D412yes—
313C4×D61C4×D412yes—
314C4×D61C4×D412yes—
315C4×D61#12312yes—
316C4×D61#12312yes—
317C4×D61#12412yes—
318C4×D61#12412yes—
319C8×C61—22no—
320C8×C61—22no—
321C8×C61—21no—
322C8×C61—21no—
323C8×C61—41no—
324#92418—22no—
325#92418—22no—
326#92418—21no—
327#92418—21no—
328#92418—41no—
329#21075—611nomaximal, Sylow
330C22×C61—21no—
331C22×C61—21no—
332C22×C61C2×#11711yescharacteristic
333C22×C61C2×#12111yescharacteristic
334C22×C61C2×#12611yescharacteristic
335C2×D61—21no—
336C2×D61—21no—
337C2×D61—21no—
338C2×D61—21no—
339C2×D61—42no—
340C2×D61—42no—
341C2×D61—42no—
342C2×D61—82no—
343C2×D61—81no—
344C2×D61—81no—
345C2×D61—82no—
346C2×D61#38812yes—
347C2×D61#38812yes—
348C4×C61—42no—
349C4×C61—41no—
350C4×C61—41no—
351C4×C61—81no—
352C4×C61C2×C4×D411yescharacteristic
353C4×C61C2×C4×D411yescharacteristic
354C4×C61C2×#12311yescharacteristic
355C4×C61C2×#12411yescharacteristic
356Dic61—41no—
357Dic61—41no—
358Dic61—81no—
359Dic61#38812yes—
360Dic61#38812yes—
361#2911—611no—
362#2896—611no—
363C2×#416—611no—
364C2×#413—611no—
365C2×#405—611no—
366#1931—611no—
367#1931—611no—
368#1931—611no—
369#1931—611no—
370#1930—611no—
371#1930—611no—
372#1930—611no—
373#1930D6111yescharacteristic, complemented
374#1742—611no—
375C2×#341—611no—
376C2×C61—21no—
377C2×C61—21no—
378C2×C61C2×#38811yescharacteristic
379C2×C61#306711yescharacteristic
380C2×C61#306711yescharacteristic
381D61—22no—
382D61—22no—
383D61—82no—
384C22×#41—611no—
385C22×M16—611no—
386C22×M16—611no—
387C22×M16—611no—
388C23×C8—611no—
389C2×#117—611no—
390C2×#117—611no—
391#428—611no—
392#428—611no—
393#428—611no—
394#428—1221no—
395#428—1221no—
396#428C2×D6111yescharacteristic
397#427—1221no—
398#427—1221no—
399#419—611no—
400#419C2×D6111yescharacteristic
401#418—611no—
402#418—611no—
403#417—1221no—
404#417—1221no—
405#416—1221no—
406#416—1221no—
407#416—1221no—
408#416—1221no—
409#416—1221no—
410#416—1221no—
411C2×#103—611no—
412C2×#103C2×D6111yescharacteristic
413C2×#102—611no—
414C2×#102—611no—
415#413—618no—
416C2×#101—611no—
417C2×#101C2×D6111yescharacteristic
418C2×#100—611no—
419C2×#100—611no—
420#405—611no—
421#405—611no—
422#405—611no—
423#405—1221no—
424#405—1221no—
425#405C2×D6111yescharacteristic
426C2×#97—1221no—
427C2×C4×C8—1221no—
428#343—611no—
429#343—611no—
430#343—611no—
431#343C2×D6111yescharacteristic
432#341—611no—
433#341—611no—
434#341—611no—
435#341C2×D6111yescharacteristic
436C61#2107511yescharacteristic, complemented, Sylow
437C2×#41—614no—
438C2×#41—1222no—
439C2×#41—1222no—
440C2×#41—1222no—
441C2×#41—1221no—
442C2×#41—1221no—
443C2×#41—1221no—
444C2×#41—1221no—
445C22×Q8C22×D6111yescharacteristic
446C22×D4—611no—
447C22×D4—1221no—
448C23×C4—611no—
449C23×C4—1221no—
450C2×M16—611no—
451C2×M16—611no—
452C2×M16—614no—
453C2×M16—614no—
454C2×M16—611no—
455C2×M16—1221no—
456C2×M16—1221no—
457C2×M16—1221no—
458C2×M16—1222no—
459C2×M16—1222no—
460C2×M16—1222no—
461C2×M16—1222no—
462C2×M16C22×D6111yescharacteristic
463C2×M16C22×D6111yescharacteristic
464C2×M16C22×D6111yescharacteristic
465C22×C8—611no—
466C22×C8—611no—
467C22×C8—611no—
468C22×C8—611no—
469C22×C8—611no—
470C22×C8—1222no—
471C22×C8—1222no—
472C22×C8—1221no—
473C22×C8—1221no—
474C22×C8—1221no—
475C22×C8C22×D6111yescharacteristic
476#117—611no—
477#117—611no—
478#117—611no—
479#117—1221no—
480#117—1221no—
481#117—1221no—
482#117—1221no—
483#117—1221no—
484#117—1221no—
485#117C22×D6111yescharacteristic
486C2×C4⋊C4—611no—
487C2×C4⋊C4—611no—
488C2×C4⋊C4—611no—
489C2×C4⋊C4C22×D6111yescharacteristic
490C2×C22⋊C4—1221no—
491C2×C22⋊C4—1221no—
492C2×C42—1221no—
493C2×C42—1221no—
494#107—21no—
495#107—1221no—
496#107—1221no—
497#107—1221no—
498#106—21no—
499#106—1221no—
500#106—1221no—
501#106—1221no—
502#103—21no—
503#103—21no—
504#103—21no—
505#103—21no—
506#103—1221no—
507#103—1221no—
508#103—1221no—
509#103—1221no—
510#102—1221no—
511#102—1221no—
512#102—1221no—
513#102—1221no—
514#102—1221no—
515#102—1221no—
516#102—1221no—
517#102—1221no—
518#101—614no—
519#101C4×D6114yescomplemented
520#100—614no—
521#100—614no—
522#97—21no—
523#97—1221no—
524#97—1221no—
525#97—1221no—
526C4×C8—21no—
527C4×C8—1221no—
528C4×C8—1221no—
529C4×C8—1221no—
530C24—611no—
531C24—1221no—
532#41—1224no—
533#41—1224no—
534#41—1224no—
535#41—1224no—
536#41—2442no—
537#41—2442no—
538#41—2442no—
539#41—2442no—
540C2×Q8—21no—
541C2×Q8—21no—
542C2×Q8—21no—
543C2×Q8—21no—
544C2×Q8—21no—
545C2×Q8C2×C4×D6112yes—
546C2×D4—612no—
547C2×D4—1224no—
548C2×D4—1221no—
549C2×D4—1221no—
550C2×D4—1221no—
551C2×D4—1221no—
552C2×D4—1221no—
553C2×D4—2441no—
554C2×D4—2441no—
555C2×D4—2442no—
556C22×C4—21no—
557C22×C4—21no—
558C22×C4—611no—
559C22×C4—612no—
560C22×C4—611no—
561C22×C4—611no—
562C22×C4—611no—
563C22×C4—611no—
564C22×C4—612no—
565C22×C4—612no—
566C22×C4—612no—
567C22×C4—1224no—
568C22×C4—1221no—
569C22×C4—1221no—
570C22×C4—1221no—
571C22×C4—2441no—
572C22×C4—2441no—
573C22×C4—2442no—
574C22×C4C23×D6111yescharacteristic
575M16—21no—
576M16—21no—
577M16—21no—
578M16—21no—
579M16—612no—
580M16—612no—
581M16—1224no—
582M16—1224no—
583M16—1221no—
584M16—1221no—
585M16—1221no—
586M16—1221no—
587M16—2442no—
588M16C2×C4×D6112yes—
589M16C2×C4×D6112yes—
590C2×C8—21no—
591C2×C8—21no—
592C2×C8—21no—
593C2×C8—21no—
594C2×C8—21no—
595C2×C8—611no—
596C2×C8—611no—
597C2×C8—1222no—
598C2×C8—1222no—
599C2×C8—1222no—
600C2×C8—1222no—
601C2×C8—1224no—
602C2×C8—1224no—
603C2×C8—1221no—
604C2×C8—1221no—
605C2×C8—1221no—
606C2×C8—1221no—
607C2×C8—1221no—
608C2×C8—2442no—
609C2×C8D4×D6111yescharacteristic
610C2×C8D4×D6111yescharacteristic
611C4⋊C4—21no—
612C4⋊C4—21no—
613C4⋊C4—611no—
614C4⋊C4—611no—
615C4⋊C4—1221no—
616C4⋊C4—1221no—
617C4⋊C4—1221no—
618C4⋊C4—1221no—
619C4⋊C4D4×D6111yescharacteristic
620C4⋊C4D4×D6111yescharacteristic
621C22⋊C4—21no—
622C22⋊C4—1221no—
623C22⋊C4—1221no—
624C22⋊C4—1221no—
625C22⋊C4—2441no—
626C22⋊C4—2441no—
627C42—21no—
628C42—1221no—
629C42—1221no—
630C42—1221no—
631C42—2441no—
632C42—2441no—
633C23—611no—
634C23—611no—
635C23—612no—
636C23—612no—
637C23—1222no—
638C23—4881no—
639C23—4881no—
640C23—2441no—
641C23—2441no—
642C23—2441no—
643C23—2441no—
644C23#41×D6111yescharacteristic
645Q8—22no—
646Q8—22no—
647Q8—41no—
648Q8—41no—
649D4—1222no—
650D4—1222no—
651D4—2444no—
652D4—2444no—
653D4—2441no—
654D4—2441no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—81no—
662C2×C4—612no—
663C2×C4—611no—
664C2×C4—611no—
665C2×C4—612no—
666C2×C4—612no—
667C2×C4—612no—
668C2×C4—612no—
669C2×C4—612no—
670C2×C4—612no—
671C2×C4—612no—
672C2×C4—612no—
673C2×C4—612no—
674C2×C4—1221no—
675C2×C4—1221no—
676C2×C4—4881no—
677C2×C4—2444no—
678C2×C4—2444no—
679C2×C4—2442no—
680C2×C4—2442no—
681C2×C4—2441no—
682C2×C4C22×C4×D6111yescharacteristic
683C2×C4C2×D4×D6111yescharacteristic
684C2×C4C2×D4×D6111yescharacteristic
685C2×C4#41×D6111yescharacteristic
686C2×C4#41×D6111yescharacteristic
687C2×C4#41×D6111yescharacteristic
688C8—22no—
689C8—22no—
690C8—21no—
691C8—21no—
692C8—41no—
693C8—1222no—
694C8—1222no—
695C8—1221no—
696C8—1221no—
697C8—2441no—
698C22—21no—
699C22—21no—
700C22—612no—
701C22—612no—
702C22—1221no—
703C22—1221no—
704C22—1221no—
705C22—1221no—
706C22—4882no—
707C22—4882no—
708C22—4881no—
709C22—4881no—
710C22—2442no—
711C22—2442no—
712C22—2442no—
713C22#117×D6111yescharacteristic
714C22#121×D6111yescharacteristic
715C22#126×D6111yescharacteristic
716C4—42no—
717C4—41no—
718C4—41no—
719C4—81no—
720C4—612no—
721C4—612no—
722C4—4881no—
723C4—2441no—
724C4—2441no—
725C4C4×D4×D6111yescharacteristic
726C4C4×D4×D6111yescharacteristic
727C4#123×D6111yescharacteristic
728C4#124×D6111yescharacteristic
729C2—21no—
730C2—21no—
731C2—1222no—
732C2—1222no—
733C2—4882no—
734C2#388×D6111yescharacteristic
735C2#2145553411yescharacteristic
736C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1930C2
socleC22×C61C2×#121
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1930D61
hypocenterC61#21075
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1930

Complements

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

NH ≅ G/Ntype
1. #21454406C2semidirect
5. #405×D61C2semidirect
6. #21446482C2semidirect
7. #21446481C2semidirect
8. #21446480C2semidirect
9. #21446479C2semidirect
10. #21446477C2semidirect
11. #21446476C2semidirect
12. #21446471C2semidirect
13. #21445556C2semidirect
14. #341×D61C2semidirect
15. C61×#1930C2semidirect
44. #16839745C4semidirect
136. #14519696C2×C4semidirect
137. #14519693C2×C4semidirect
138. #14519693C2×C4semidirect
158. #101×C61C2×C4semidirect
373. #1930D61semidirect
436. C61#21075semidirect
519. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214544062
#214543372
C2×#168397572
C2×#168397452
#405×D612
#214464822
#214464812
#214464802
#214464792
#214464772
#214464762
#214464712
#214455562
#341×D612
C61×#19302
#2107561