Finite Group Databasebeta

← #29748219

Subgroups — #29748219

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450162C211yesmaximal, characteristic
2#21448850C211yesmaximal, characteristic
3#21447139C211yesmaximal, characteristic, complemented
4#21447019C211yesmaximal, characteristic
5#21446968C211yesmaximal, characteristic, complemented
6#21446775C211yesmaximal, characteristic, complemented
7#21446497C211yesmaximal, characteristic, complemented
8#21446463C211yesmaximal, characteristic, complemented
9#21446422C211yesmaximal, characteristic
10#21446393C211yesmaximal, characteristic, complemented
11#21446162C211yesmaximal, characteristic, complemented
12#21446132C211yesmaximal, characteristic
13#21445915C211yesmaximal, characteristic
14#103×Dic61C211yesmaximal, characteristic, complemented
15C61×#2001C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839700—21no—
28#16839687—21no—
29#16839669C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839656—21no—
34#16839655—21no—
35#16839655—21no—
36#16839648—21no—
37#16839627—21no—
38#16839622C2211yescharacteristic
39C4⋊C4×Dic61—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C22⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×Q16×C61C2211yescharacteristic
53C2×C22⋊Q8×C61C2211yescharacteristic
54C2×C22⋊Q8×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61#14519768—22no—
62#14519768—41no—
63Q8×Dic61—22no—
64Q8×Dic61—41no—
65#14519736—22no—
66#14519736—41no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61D411yescharacteristic
78C2×C22⋊Dic61D411yescharacteristic
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519710—22no—
86#14519706—41no—
87#14519701—22no—
88#14519701—22no—
89#14519701—41no—
90#14519701—41no—
91C22⋊#92418—41no—
92C8×Dic61—41no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×Q8×C61C2311yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107C23×C4×C61C2311yescharacteristic
108C2×Q16×C61—22no—
109C2×Q16×C61—41no—
110C2×Q16×C61—41no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61—22no—
116C22⋊Q8×C61—41no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123#103×C61—22no—
124#103×C61—22no—
125C22⋊C8×C61—41no—
126C24×C61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159Q16×C61—44no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61#4112yes—
171C4⋊C4×C61#4112yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—42no—
174C22⋊Dic61—22no—
175C22⋊Dic61—41no—
176C22⋊Dic61—42no—
177C22⋊Dic61—42no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188C4⋊Dic61#4112yes—
189#11758196—42no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—42no—
195C4×Dic61—42no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61C22×D411yescharacteristic
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61#13512yes—
269C8×C61—41no—
270C8×C61—41no—
271#92418—41no—
272#92418—41no—
273#26732—611nomaximal, Sylow
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61#52511yescharacteristic
282C22×C61#54611yescharacteristic
283C22×C61#54611yescharacteristic
284C22×C61C2×#13511yescharacteristic
285C22×C61#57511yescharacteristic
286C22×C61#57811yescharacteristic
287C22×C61#57811yescharacteristic
288C4×C61—42no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—81no—
293C4×C61#52911yescharacteristic
294C4×C61#53811yescharacteristic
295C4×C61#55511yescharacteristic
296C4×C61#55911yescharacteristic
297Dic61—22no—
298Dic61—41no—
299Dic61—41no—
300Dic61—42no—
301Dic61—41no—
302Dic61—81no—
303Dic61—81no—
304Dic61—81no—
305#2489—611no—
306#2362—611no—
307#2056—611no—
308#2036—611no—
309#2026—611no—
310#2001D6111yescharacteristic
311#1989—611no—
312#1933—611no—
313#1928—611no—
314#1923—611no—
315#1918—611no—
316#1878—611no—
317#1873—611no—
318#1825—611no—
319C4×#103—611no—
320C2×C61—81no—
321C2×C61#245911yescharacteristic
322C2×C61#310211yescharacteristic
323C2×C61#312711yescharacteristic
324C2×C61#330011yescharacteristic
325C2×C61#330211yescharacteristic
326C2×C61#331511yescharacteristic
327C2×C61#332711yescharacteristic
328C22×Q16C2×D6111yescharacteristic
329C2×#125—611no—
330C2×C22⋊Q8C2×D6111yescharacteristic
331C2×C22⋊Q8C2×D6111yescharacteristic
332C2×C4×Q8—611no—
333C2×#117—611no—
334C2×#107—611no—
335C2×C4⋊C8—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103C2×D6111yescharacteristic
341C2×#103C2×D6111yescharacteristic
342C2×C22⋊C8—611no—
343C2×C22⋊C8C2×D6111yescharacteristic
344C2×C4×C8—611no—
345#400—1221no—
346#397—1221no—
347#391—611no—
348#391—611no—
349#391—611no—
350#389—611no—
351#388—1221no—
352#388—1221no—
353#388—1221no—
354#387—1221no—
355#382—1221no—
356#380—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360C4×C22⋊C4—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340C2×D6111yescharacteristic
369#336—611no—
370#336—611no—
371C61#2673211yescharacteristic, complemented, Sylow
372C22×Q8C22×D6111yescharacteristic
373C22×Q8C22×D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375C2×Q16—22no—
376C2×Q16—41no—
377C2×Q16—41no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6111yescharacteristic
382#125—1222no—
383#125—2441no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—41no—
388C4×Q8—1222no—
389C4×Q8—2441no—
390#117—1222no—
391#117—2441no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—611no—
407C2×C22⋊C4—611no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416#107—1222no—
417C4⋊C8—2441no—
418#103—22no—
419#103—22no—
420#103—1222no—
421#103—1222no—
422#103—2441no—
423#103—2441no—
424C22⋊C8—41no—
425C22⋊C8—2441no—
426C4×C8—2441no—
427#95—611no—
428#95—611no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439C24D4×D6111yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8#1175822412yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4D4×D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472Q16—44no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—612no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4#1175822412yes—
502C4⋊C4#1175822412yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—1222no—
506C22⋊C4—2441no—
507C22⋊C4—2442no—
508C22⋊C4—2442no—
509C42—1221no—
510C42—1221no—
511C42—1221no—
512C42—1221no—
513C42—2442no—
514C42—2442no—
515C42—2442no—
516C42—2442no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C2×D4×D6111yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—612no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4#41×D6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C2×C4C2×#1175822411yescharacteristic
579C2×C4C2×#1175822411yescharacteristic
580C2×C4#1451982211yescharacteristic
581C2×C4#1451982211yescharacteristic
582C8—41no—
583C8—41no—
584C8—2441no—
585C8—2441no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1684026811yescharacteristic
594C22#1684039911yescharacteristic
595C22#1684040111yescharacteristic
596C22#135×D6111yescharacteristic
597C22#1684056111yescharacteristic
598C22#1684057511yescharacteristic
599C22#1684057811yescharacteristic
600C4—41no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—81no—
605C4—1222no—
606C4—4881no—
607C4—4881no—
608C4—4881no—
609C4—2441no—
610C4—2441no—
611C4—2441no—
612C4—2442no—
613C4#1684029311yescharacteristic
614C4#1684034711yescharacteristic
615C4#1684047611yescharacteristic
616C4#1684049811yescharacteristic
617C2—81no—
618C2#2144975111yescharacteristic
619C2#2145583811yescharacteristic
620C2#2145611311yescharacteristic
621C2#2145770111yescharacteristic
622C2#2145772911yescharacteristic
623C2#2145791011yescharacteristic
624C2#2145809111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2001C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2001D61
hypocenterC61#26732
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
3. #21447139C2semidirect
5. #21446968C2semidirect
6. #21446775C2semidirect
7. #21446497C2semidirect
8. #21446463C2semidirect
10. #21446393C2semidirect
11. #21446162C2semidirect
14. #103×Dic61C2semidirect
371. C61#26732semidirect

Maximal subgroups

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

subgroupindex
#214501622
#214488502
#214471392
#214470192
#214469682
#214467752
#214464972
#214464632
#214464222
#214463932
#214461622
#214461322
#214459152
#103×Dic612
C61×#20012
#2673261