Finite Group Databasebeta

← #30617225

Subgroups — #30617225

Order 15616.

768 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462539C211yesmaximal, characteristic
2#21462384C211yesmaximal, characteristic, complemented
3#21461903C211yesmaximal, characteristic, complemented
4#21461903C211yesmaximal, characteristic, complemented
5#21461886C211yesmaximal, characteristic, complemented
6#21461884C211yesmaximal, characteristic, complemented
7#21461800C211yesmaximal, characteristic
8#21461771C211yesmaximal, characteristic, complemented
9#21461727C211yesmaximal, characteristic, complemented
10#21461627C211yesmaximal, characteristic, complemented
11#21461623C211yesmaximal, characteristic, complemented
12#21461600C211yesmaximal, characteristic, complemented
13#21461596C211yesmaximal, characteristic, complemented
14#21461562C211yesmaximal, characteristic, complemented
15#21455737C211yesmaximal, characteristic
16C2×#14519865C2211yescharacteristic
17C22×#11758213C2211yescharacteristic
18C2×D61⋊M16C2211yescharacteristic
19#16840796—21no—
20#16840795—21no—
21#16840794—21no—
22#16840792—21no—
23C2×#14519848C2211yescharacteristic
24C2×#14519848C2211yescharacteristic
25#16840769C2211yescharacteristic
26C2×#14519842C2211yescharacteristic
27C2×#14519842C2211yescharacteristic
28#16840737C2211yescharacteristic
29#16840733C2211yescharacteristic
30#16840727C2211yescharacteristic
31#16840727C2211yescharacteristic
32#16840727C2211yescharacteristic
33#16840727C2211yescharacteristic
34#16840727C2211yescharacteristic
35#16840726—21no—
36#16840726—21no—
37C4×#11758213—21no—
38C4×#11758213—21no—
39C2×#14519831C2211yescharacteristic
40C2×C8×#6923C2211yescharacteristic
41C4×C22⋊#6923C2211yescharacteristic
42#16840638C2211yescharacteristic
43#16840638C2211yescharacteristic
44#16840632C2211yescharacteristic
45#16840632C2211yescharacteristic
46#16840632C2211yescharacteristic
47#16840632C2211yescharacteristic
48#16840632C2211yescharacteristic
49#16840632C2211yescharacteristic
50(C2×#6923)⋊C8C2211yescharacteristic
51(C2×#6923)⋊C8C2211yescharacteristic
52#16840275C2211yescharacteristic
53#117×D61C2211yescharacteristic
54#16840100C411yescharacteristic, complemented
55#16840096C411yescharacteristic, complemented
56#16840091C411yescharacteristic, complemented
57#16840087C411yescharacteristic, complemented
58#16840086C411yescharacteristic, complemented
59#16840082C411yescharacteristic, complemented
60#16840077C411yescharacteristic, complemented
61#16839799C2211yescharacteristic
62#16839797C2211yescharacteristic
63#16839757C2211yescharacteristic
64#16839757C2211yescharacteristic
65C22⋊C8×D61C2211yescharacteristic
66C61×#485C411yescharacteristic, complemented
67#14519865—22no—
68#14519865—22no—
69C2×#11758213—21no—
70C2×#11758213—22no—
71C2×#11758213—22no—
72C2×#11758213—21no—
73C2×#11758213D411yescharacteristic
74C2×#11758213D411yescharacteristic
75C2×#11758213C2311yescharacteristic
76C2×#11758213C2311yescharacteristic
77C2×#11758213C2311yescharacteristic
78C2×#11758213C2311yescharacteristic
79D61⋊M16—22no—
80D61⋊M16—22no—
81C2×#11758217D411yescharacteristic
82C2×#11758217D411yescharacteristic
83C2×D61⋊C8—21no—
84C2×D61⋊C8C2311yescharacteristic
85C2×D61⋊C8C2311yescharacteristic
86C22×#92420—21no—
87C2×C22⋊#6923—21no—
88C2×C4×#6923C2311yescharacteristic
89C2×C4×#6923C2311yescharacteristic
90#14519848—22no—
91#14519848—22no—
92#14519844—21no—
93C4⋊#92420—21no—
94#14519842—22no—
95#14519842—22no—
96#14519834—21no—
97#14519834—21no—
98#14519834—21no—
99C22⋊#92420—21no—
100#14519832—21no—
101#14519832—21no—
102#14519832—21no—
103#14519832—21no—
104#14519832—21no—
105#14519831—22no—
106C4×#92420—21no—
107C4×#92420—21no—
108C4×#92420—21no—
109C4×#92420—21no—
110C4×#92420—21no—
111C8×#6923—22no—
112(C2×#6923)⋊C4—21no—
113C2×#11758226C2311yescharacteristic
114C22×C4×D61C2311yescharacteristic
115C2×C8×D61C2311yescharacteristic
116C2×C8×D61C2311yescharacteristic
117#14519762—21no—
118#14519753C2×C411yescharacteristic
119#14519752C2×C411yescharacteristic
120#14519751C2×C411yescharacteristic
121#14519747—21no—
122#14519739—21no—
123#14519737—21no—
124#14519736C2×C411yescharacteristic
125#14519735C2311yescharacteristic
126#14519735C2311yescharacteristic
127#14519735C2311yescharacteristic
128#14519734—21no—
129#14519733—21no—
130C4⋊C4×D61C2×C411yescharacteristic
131C4⋊C4×D61C2×C411yescharacteristic
132C22⋊C4×D61—21no—
133C42×D61—21no—
134#14519711C2×C411yescharacteristic
135#14519710C2×C411yescharacteristic
136#14519710C2×C411yescharacteristic
137#14519709C2×C411yescharacteristic
138#14519708C2×C411yescharacteristic
139#14519708C2×C411yescharacteristic
140#14519701C2×C411yescharacteristic
141#14519701C2×C411yescharacteristic
142#14519699C2×C411yescharacteristic
143#14519699C2×C411yescharacteristic
144#14519697C2×C411yescharacteristic
145#14519697C2×C411yescharacteristic
146C22⋊#92418C2×C411yescharacteristic
147#14519689C2×C411yescharacteristic
148#14519689C2×C411yescharacteristic
149C22⋊Q8×C61C2×C411yescharacteristic
150#117×C61C2×C411yescharacteristic
151#107×C61C2×C411yescharacteristic
152#106×C61C2×C411yescharacteristic
153#103×C61C2×C411yescharacteristic
154#103×C61C2×C411yescharacteristic
155C22⋊C8×C61C2×C411yescharacteristic
156C23×D61#4111yescharacteristic
157C22×#6923—21no—
158C22×#6923—21no—
159C2×Q8×C61C22×C411yescharacteristic
160C22×C4×C61C22×C411yescharacteristic
161C2×C22⋊D61—21no—
162C22×Dic61C2×D411yescharacteristic
163#11758226—21no—
164#11758226—21no—
165#11758226—21no—
166#11758226—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244C22×C411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61C22×C411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×C4×D61#4111yescharacteristic
180C2×C4×D61C2411yescharacteristic
181C22⋊#6923—42no—
182#11758217—22no—
183#11758217—22no—
184C2×#92420—24no—
185C2×#92420—21no—
186C2×#92420—21no—
187C2×#92420—21no—
188C2×#92420—21no—
189C2×#92420C2×D411yescharacteristic
190C2×#92420C2×D411yescharacteristic
191C2×#92420C2×D411yescharacteristic
192C2×#92420C2×D411yescharacteristic
193C4×#6923—21no—
194C4×#6923—21no—
195C4×#6923—21no—
196C4×#6923—21no—
197#11758213—24no—
198#11758213—21no—
199#11758213—21no—
200#11758213—22no—
201#11758213—22no—
202#11758213—21no—
203#11758213—21no—
204#11758213—21no—
205#11758213—21no—
206#11758213—42no—
207#11758213#4112yes—
208#11758213#4112yes—
209D61⋊C8—21no—
210D61⋊C8—21no—
211D61⋊C8—42no—
212D61⋊C8#4112yes—
213D61⋊C8#4112yes—
214C2×C8×C61C22×C411yescharacteristic
215C2×C8×C61C22×C411yescharacteristic
216C4⋊C4×C61—21no—
217C4⋊C4×C61C22×C411yescharacteristic
218C4⋊C4×C61C22×C411yescharacteristic
219C4⋊C4×C61C22×C411yescharacteristic
220C22⋊C4×C61—21no—
221C22⋊C4×C61—21no—
222C42×C61—21no—
223C22⋊Dic61—21no—
224#11758198—21no—
225#11758198—21no—
226#11758198—21no—
227#11758198—21no—
228#11758198—21no—
229#11758198—21no—
230C4⋊Dic61C22×C411yescharacteristic
231C4⋊Dic61C22×C411yescharacteristic
232C4⋊Dic61C22×C411yescharacteristic
233#11758196—21no—
234#11758196—21no—
235#11758196—21no—
236C4×Dic61—21no—
237C4×Dic61—21no—
238C4×Dic61—21no—
239C4×Dic61—21no—
240C2×#92418C22×C411yescharacteristic
241C2×#92418C22×C411yescharacteristic
242C8×D61—21no—
243C8×D61—21no—
244C8×D61—21no—
245C8×D61—21no—
246C23×C61C4×D411yescharacteristic
247C22×D61—21no—
248C22×D61—21no—
249C22×D61—21no—
250C22×D61—41no—
251C22×D61—41no—
252C22×D61C4×D411yescharacteristic
253C22×D61C2×#4111yescharacteristic
254C2×#6923—22no—
255C2×#6923—41no—
256C2×#6923—41no—
257C2×#6923—41no—
258C2×#6923—41no—
259C2×#6923—41no—
260Q8×C61—21no—
261Q8×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61C4×D411yescharacteristic
269C2×C4×C61C23×C411yescharacteristic
270C22⋊D61—81no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61C4×D411yescharacteristic
277C2×Dic61C22×D411yescharacteristic
278D244—21no—
279D244—21no—
280D244—41no—
281D244—41no—
282D244—81no—
283C4×D61—21no—
284C4×D61—21no—
285C4×D61—21no—
286C4×D61—21no—
287C4×D61—21no—
288C4×D61—21no—
289C4×D61—41no—
290C4×D61—41no—
291C4×D61—81no—
292C4×D61—81no—
293C4×D61C2×#4111yescharacteristic
294C4×D61C2×#4111yescharacteristic
295C4×D61C2×#4111yescharacteristic
296C4×D61C2×#4111yescharacteristic
297#92420—21no—
298#92420—21no—
299#92420—22no—
300#92420—22no—
301#92420—41no—
302#92420—41no—
303#92420#12312yes—
304#92420#12312yes—
305C8×C61—21no—
306C8×C61—21no—
307#92418—21no—
308#92418—21no—
309#21594—611nomaximal, Sylow
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61C2×C4×D411yescharacteristic
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—21no—
317C2×D61—21no—
318C2×D61—21no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—81no—
323C2×D61—81no—
324C2×D61#52911yescharacteristic
325C2×D61#52911yescharacteristic
326#6923—42no—
327#6923—81no—
328C4×C61—21no—
329C4×C61—41no—
330C4×C61—41no—
331C4×C61—41no—
332C4×C61—41no—
333C4×C61C4×#4111yescharacteristic
334C4×C61C4×#4111yescharacteristic
335Dic61—21no—
336Dic61—41no—
337Dic61—41no—
338Dic61—41no—
339Dic61C2×#12311yescharacteristic
340Dic61#52511yescharacteristic
341#3091—611no—
342#2930—611no—
343#2873—611no—
344#1929—611no—
345#1929—611no—
346#1922—611no—
347#1922—611no—
348#1871—611no—
349#1855—611no—
350#1826—611no—
351#1773—611no—
352#1772—611no—
353#1765—611no—
354#1764—611no—
355#1753—611no—
356C2×C61—41no—
357C2×C61C4×#12311yescharacteristic
358C2×C61#295611yescharacteristic
359C2×C61#295711yescharacteristic
360D61—21no—
361D61—21no—
362D61—41no—
363D61—81no—
364C2×#131—611no—
365C22×M16—611no—
366C22×M16—611no—
367#525—611no—
368C2×#117—611no—
369#485—611no—
370#485#692311yescharacteristic, complemented
371#484—611no—
372#483—611no—
373#482—611no—
374#482—611no—
375#481—611no—
376#480—611no—
377#436—1221no—
378#436—1221no—
379#435—1221no—
380#435—1221no—
381C2×#108—611no—
382C2×#108—611no—
383#427—611no—
384#427—611no—
385#424—611no—
386C2×#104—611no—
387C2×#104—611no—
388#416—611no—
389#416—611no—
390#408—611no—
391#407—611no—
392C2×C22⋊C8—611no—
393#405—611no—
394#405—611no—
395#405—611no—
396#405—611no—
397#405—611no—
398#405—1221no—
399#405—1221no—
400C4×M16—1221no—
401C4×M16—1221no—
402C2×#97—611no—
403C2×C4×C8—611no—
404C4×C22⋊C4—611no—
405#341—611no—
406#341—611no—
407#339—611no—
408#339—611no—
409#339—611no—
410#339—611no—
411#339—611no—
412#339—611no—
413#336—611no—
414#336—611no—
415C61#2159411yescharacteristic, complemented, Sylow
416C2×#41—611no—
417C23×C4—611no—
418#131—1222no—
419#131—1222no—
420C2×M16—611no—
421C2×M16—611no—
422C2×M16—611no—
423C2×M16—611no—
424C2×M16—611no—
425C2×M16—611no—
426C2×M16—611no—
427C2×M16—611no—
428C2×M16—1222no—
429C2×M16—1222no—
430C2×M16—1221no—
431C2×M16—1222no—
432C2×M16—1222no—
433C2×M16—1221no—
434C22×C8—611no—
435C22×C8—611no—
436C22×C8—611no—
437C22×C8—611no—
438C22×C8—1221no—
439C22×C8—1221no—
440#123—1221no—
441C22⋊Q8—611no—
442C22⋊Q8C2×#692311yescharacteristic
443#121—611no—
444#121—611no—
445#120—1221no—
446C4×D4—1221no—
447C4×D4—1221no—
448#117—611no—
449#117—611no—
450#117—611no—
451#117—611no—
452#117—1221no—
453#117—1221no—
454#117C2×#692311yescharacteristic
455C2×C4⋊C4—611no—
456C2×C4⋊C4—611no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—1221no—
462#108—1222no—
463#108—1222no—
464#107—611no—
465#107—611no—
466#107—611no—
467#107C2×#692311yescharacteristic
468#106—611no—
469#106—611no—
470#106—611no—
471#106C2×#692311yescharacteristic
472C4⋊C8—1221no—
473C4⋊C8—1221no—
474#104—1222no—
475#104—1222no—
476#103—611no—
477#103—611no—
478#103C2×#692311yescharacteristic
479#103C2×#692311yescharacteristic
480#102—611no—
481#102—611no—
482#102—611no—
483#102—611no—
484C22⋊C8—611no—
485C22⋊C8—611no—
486C22⋊C8—611no—
487C22⋊C8—1221no—
488C22⋊C8—1221no—
489C22⋊C8—1221no—
490C22⋊C8—1221no—
491C22⋊C8C2×#692311yescharacteristic
492#97—1222no—
493#97—1221no—
494#97—1221no—
495#97—1221no—
496#97—1221no—
497#97—1221no—
498C4×C8—1222no—
499C4×C8—1221no—
500C4×C8—1221no—
501C4×C8—1221no—
502C4×C8—1221no—
503C4×C8—1221no—
504#95—1221no—
505C24—611no—
506#41—1221no—
507#41—1221no—
508#41—1221no—
509#41—1221no—
510C2×Q8C22×#692311yescharacteristic
511C2×D4—611no—
512C2×D4—1221no—
513C2×D4—1221no—
514C2×D4—1221no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4C22×#692311yescharacteristic
530M16—612no—
531M16—612no—
532M16—1222no—
533M16—1222no—
534M16—1222no—
535M16—1222no—
536M16—1221no—
537M16—1221no—
538M16—1224no—
539M16—1221no—
540M16—1221no—
541M16—1221no—
542M16—1221no—
543M16—2442no—
544C2×C8—611no—
545C2×C8—611no—
546C2×C8—612no—
547C2×C8—612no—
548C2×C8—611no—
549C2×C8—611no—
550C2×C8—611no—
551C2×C8—611no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8—1221no—
557C2×C8—1221no—
558C2×C8—1221no—
559C2×C8—1221no—
560C2×C8—1224no—
561C2×C8—1221no—
562C2×C8—1221no—
563C2×C8—2442no—
564C2×C8C22×#692311yescharacteristic
565C2×C8C22×#692311yescharacteristic
566C4⋊C4—21no—
567C4⋊C4—611no—
568C4⋊C4—611no—
569C4⋊C4—611no—
570C4⋊C4—1221no—
571C4⋊C4—1221no—
572C4⋊C4—1221no—
573C4⋊C4C22×#692311yescharacteristic
574C4⋊C4C22×#692311yescharacteristic
575C4⋊C4C22×#692311yescharacteristic
576C22⋊C4—21no—
577C22⋊C4—21no—
578C22⋊C4—1221no—
579C22⋊C4—1221no—
580C22⋊C4—1221no—
581C22⋊C4—1221no—
582C22⋊C4—1221no—
583C22⋊C4—1221no—
584C22⋊C4—1221no—
585C22⋊C4—2442no—
586C42—21no—
587C42—1221no—
588C42—1221no—
589C42—1221no—
590C42—1221no—
591C42—1221no—
592C42—1221no—
593C42—1221no—
594C42—1221no—
595C23—611no—
596C23—611no—
597C23—1221no—
598C23—1221no—
599C23—1221no—
600C23—2441no—
601C23—2441no—
602C23D4×#692311yescharacteristic
603Q8—21no—
604Q8—21no—
605D4—1221no—
606D4—1221no—
607D4—4881no—
608D4—4881no—
609D4—2441no—
610D4—2441no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—611no—
618C2×C4—611no—
619C2×C4—611no—
620C2×C4—611no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1222no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—4881no—
636C2×C4—4881no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4D4×#692311yescharacteristic
645C2×C4C23×#692311yescharacteristic
646C8—21no—
647C8—21no—
648C8—612no—
649C8—612no—
650C8—1221no—
651C8—1221no—
652C8—1222no—
653C8—1222no—
654C8—1221no—
655C8—1221no—
656C8—2441no—
657C8—2441no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—611no—
662C22—611no—
663C22—1221no—
664C22—1221no—
665C22—1221no—
666C22—1221no—
667C22—1221no—
668C22—4881no—
669C22—4881no—
670C22—2441no—
671C22—2441no—
672C22—2441no—
673C22C2×D4×#692311yescharacteristic
674C4—21no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—611no—
680C4—611no—
681C4—1221no—
682C4—4881no—
683C4—2442no—
684C4—2441no—
685C4—2441no—
686C4—2441no—
687C4#41×#692311yescharacteristic
688C4#41×#692311yescharacteristic
689C2—41no—
690C2—1221no—
691C2—1221no—
692C2—4881no—
693C2—2441no—
694C2#123×#692311yescharacteristic
695C2#2146259411yescharacteristic
696C2#2146259611yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#485C4
socleC22×C61C2×C4×D4
prime radical#16840275C22
cosocleC2×C4×D61C24
hypercenter#485#6923
hypocenterC61#21594
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #21462384C2semidirect
3. #21461903C2semidirect
4. #21461903C2semidirect
5. #21461886C2semidirect
6. #21461884C2semidirect
8. #21461771C2semidirect
9. #21461727C2semidirect
10. #21461627C2semidirect
11. #21461623C2semidirect
12. #21461600C2semidirect
13. #21461596C2semidirect
14. #21461562C2semidirect
54. #16840100C4semidirect
55. #16840096C4semidirect
56. #16840091C4semidirect
57. #16840087C4semidirect
58. #16840086C4semidirect
59. #16840082C4semidirect
60. #16840077C4semidirect
66. C61×#485C4semidirect
370. #485#6923semidirect
415. C61#21594semidirect

Maximal subgroups

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

subgroupindex
#214625392
#214623842
#214619032
#214618862
#214618842
#214618002
#214617712
#214617272
#214616272
#214616232
#214616002
#214615962
#214615622
#214557372
#2159461