Finite Group Databasebeta

← #29655564

Subgroups — #29655564

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457021C211yesmaximal, characteristic, complemented
2#21456983C211yesmaximal, characteristic
3#21453886C211yesmaximal, characteristic, complemented
4#21444861C212yesmaximal, complemented
5#21444850C212yesmaximal
6#21444827C212yesmaximal, complemented
7#21444816C212yesmaximal
8#21444383C211yesmaximal, characteristic, complemented
9#21444357C211yesmaximal, characteristic
10#21444175C211yesmaximal, characteristic, complemented
11C61×#3563C211yesmaximal, characteristic
12C2×#14519769C2211yescharacteristic
13#16840168—21no—
14#16840161—21no—
15#16840130—21no—
16#16840116—21no—
17#16840095C2211yescharacteristic
18#16840095C2212yes—
19#16840090C2211yescharacteristic
20#16840090C2212yes—
21#16840078C2211yescharacteristic
22#16840073C2211yescharacteristic
23#16840069—21no—
24#16840069—22no—
25#16840062—21no—
26#16839819C2211yescharacteristic
27#16839815C2211yescharacteristic
28#16839701C2212yes—
29#16839700—21no—
30#16839695—22no—
31#16839693—21no—
32#16839692C2212yes—
33#16839628C2211yescharacteristic
34#16839499C2212yes—
35#16839499C2212yes—
36#16839499C2212yes—
37#16839496C2212yes—
38#16839488C2212yes—
39#16839488C2212yes—
40C61×#564—21no—
41C61×#563—21no—
42C61×#557—21no—
43C61×#554—21no—
44C61×#552—21no—
45C61×#552—21no—
46C61×#552—22no—
47C61×#544—21no—
48C61×#544C2212yes—
49C61×#541—21no—
50C61×#541—22no—
51C61×#539C2212yes—
52C61×#533C2211yescharacteristic
53C61×#529C2211yescharacteristic
54C2×#125×C61C2211yescharacteristic
55#14519769—21no—
56#14519769—21no—
57#14519769—21no—
58#14519769—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758196—22no—
65#14519710—21no—
66#14519710—21no—
67#14519710—22no—
68#14519708—21no—
69#14519708—21no—
70#14519708—22no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73#14519701—21no—
74#14519701—22no—
75#14519701—21no—
76#14519701—22no—
77#14519698—21no—
78#14519698—21no—
79C22⋊#92418C2312yes—
80C22⋊#92418C2311yescharacteristic
81C22⋊#92418C2311yescharacteristic
82#14519688—21no—
83C4×#92418—21no—
84#14519684—22no—
85#14519684—22no—
86#14519684—21no—
87#14519684—22no—
88#14519684C2312yes—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61D411yescharacteristic
94#126×C61—22no—
95#126×C61—22no—
96#126×C61—22no—
97#126×C61—22no—
98#126×C61—21no—
99#125×C61—22no—
100#125×C61—22no—
101#125×C61—22no—
102#125×C61—22no—
103#125×C61—21no—
104#125×C61—22no—
105#125×C61—22no—
106#125×C61—22no—
107#125×C61—22no—
108#125×C61—21no—
109#125×C61—21no—
110#125×C61—21no—
111#125×C61—21no—
112#125×C61—21no—
113#125×C61D411yescharacteristic
114#124×C61D411yescharacteristic
115#123×C61—22no—
116#123×C61—22no—
117#123×C61—21no—
118C22⋊Q8×C61—21no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61—22no—
121C22⋊Q8×C61—22no—
122C22⋊Q8×C61—22no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—22no—
125C22⋊Q8×C61C2312yes—
126C22⋊Q8×C61C2311yescharacteristic
127#121×C61C2311yescharacteristic
128C4×Q8×C61—21no—
129C4×Q8×C61—22no—
130C4×Q8×C61—21no—
131C4×Q8×C61—21no—
132C4×Q8×C61D412yes—
133C4×Q8×C61D411yescharacteristic
134C4×D4×C61—22no—
135C4×D4×C61—21no—
136C4×D4×C61D411yescharacteristic
137#117×C61—22no—
138#117×C61—21no—
139#117×C61—21no—
140#117×C61—22no—
141#117×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61D411yescharacteristic
144C2×C4⋊C4×C61D412yes—
145C2×C4⋊C4×C61D411yescharacteristic
146C2×C4⋊C4×C61C2312yes—
147C2×C42×C61C2311yescharacteristic
148C2×Q8×C61—22no—
149C2×Q8×C61—21no—
150C2×Q8×C61C2×D412yes—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—22no—
155C22×C4×C61—21no—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—22no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D412yes—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—21no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C42×C61—22no—
202C42×C61—22no—
203C42×C61—22no—
204C42×C61—22no—
205C42×C61—21no—
206C42×C61C2×D411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61C2×D411yescharacteristic
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—81no—
215#11758196—41no—
216#11758196—41no—
217#11758196—41no—
218#11758196—41no—
219#11758196—82no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—22no—
223C23×C61—21no—
224C23×C61C22×D411yescharacteristic
225Q8×C61—42no—
226Q8×C61—41no—
227Q8×C61—42no—
228Q8×C61—82no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—82no—
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—41no—
262#92418—41no—
263#92418—41no—
264#92418—42no—
265#18469—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61C2×#12011yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57811yescharacteristic
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—42no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—41no—
287Dic61—82no—
288Dic61—81no—
289Dic61—81no—
290#3563D6111yescharacteristic
291#3236—611no—
292#3232—611no—
293#2822—611no—
294#1617—612no—
295#1615—612no—
296#1611—612no—
297#1609—612no—
298#1518—611no—
299#1512—611no—
300#1470—611no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—81no—
304C2×C61#301311yescharacteristic
305C2×C61#302011yescharacteristic
306C2×C61#302811yescharacteristic
307#564—21no—
308#563—21no—
309#557—21no—
310#554—21no—
311#552—21no—
312#552—22no—
313#552—21no—
314#544—21no—
315#544C2×D6112yes—
316#541—21no—
317#541—22no—
318#539C2×D6112yes—
319#533C2×D6111yescharacteristic
320#529C2×D6111yescharacteristic
321C2×#125—611no—
322C2×#125C2×D6111yescharacteristic
323#501—1221no—
324#499—1221no—
325#491—1221no—
326#488—1221no—
327#484—612no—
328#484—611no—
329#483—612no—
330#483—611no—
331#481—611no—
332#480—611no—
333#479—1221no—
334#479—1222no—
335#478—1221no—
336#432—611no—
337#431—611no—
338#400—612no—
339#400—1221no—
340#398—612no—
341#398—1221no—
342#398—1222no—
343#382—611no—
344#328—612no—
345#328—612no—
346#328—612no—
347#327—612no—
348#324—612no—
349#324—612no—
350C61#1846911yescharacteristic, complemented, Sylow
351C4⋊Q8—21no—
352C4⋊Q8—22no—
353C4⋊Q8—22no—
354C4⋊Q8C22⋊D6111yescharacteristic
355C4⋊Q8C22⋊D6111yescharacteristic
356#126—22no—
357#126—22no—
358#126—22no—
359#126—22no—
360#126—21no—
361#125—22no—
362#125—22no—
363#125—22no—
364#125—22no—
365#125—21no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—22no—
370#125—21no—
371#125—21no—
372#125—21no—
373#125—21no—
374#125—21no—
375#125—1221no—
376#125—1221no—
377#125—1221no—
378#125—1221no—
379#125C22⋊D6111yescharacteristic
380#124C22⋊D6111yescharacteristic
381#123—22no—
382#123—21no—
383#123—22no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—21no—
390C22⋊Q8—22no—
391C22⋊Q8C22×D6111yescharacteristic
392C22⋊Q8C22×D6112yes—
393#121C22×D6111yescharacteristic
394C4×Q8—21no—
395C4×Q8—22no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8C22⋊D6111yescharacteristic
399C4×Q8C22⋊D6112yes—
400C4×D4—21no—
401C4×D4—22no—
402C4×D4C22⋊D6111yescharacteristic
403#117—22no—
404#117—21no—
405#117—21no—
406#117—22no—
407#117—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22⋊D6112yes—
417C2×C4⋊C4C22⋊D6111yescharacteristic
418C2×C4⋊C4C22×D6112yes—
419C2×C42C22×D6111yescharacteristic
420#107—1221no—
421#107—1221no—
422#107—1222no—
423#106—1221no—
424#106—1221no—
425#106—1222no—
426C4⋊C8—1221no—
427C4⋊C8—1221no—
428#103—1221no—
429#103—1222no—
430#103—1221no—
431#103—1222no—
432#102—1221no—
433#102—1221no—
434C22⋊C8—612no—
435C22⋊C8—611no—
436C22⋊C8—611no—
437#97—1221no—
438C4×C8—1221no—
439#95—612no—
440#95—1221no—
441#95—1222no—
442#95—1222no—
443#95—1222no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8C2×C22⋊D6112yes—
447C2×Q8C2×C22⋊D6111yescharacteristic
448C2×D4—21no—
449C2×D4C2×C22⋊D6111yescharacteristic
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6112yes—
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1222no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—4882no—
493C4⋊C4—4881no—
494C4⋊C4—2441no—
495C4⋊C4—2441no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C4⋊C4C2×C22⋊D6111yescharacteristic
500C4⋊C4C2×C22⋊D6112yes—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42C2×C22⋊D6111yescharacteristic
516C42C2×C22⋊D6111yescharacteristic
517C42C2×C22⋊D6111yescharacteristic
518C42C2×C22⋊D6111yescharacteristic
519C23—21no—
520C23C22×C22⋊D6111yescharacteristic
521Q8—41no—
522Q8—42no—
523Q8—42no—
524Q8—82no—
525D4—41no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—82no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4#1451974611yescharacteristic
553C2×C4#1451974611yescharacteristic
554C2×C4#1451974611yescharacteristic
555C2×C4#1451974611yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4C22×C22⋊D6111yescharacteristic
558C8—2441no—
559C8—2441no—
560C8—2442no—
561C22—21no—
562C22—21no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×#1451974611yescharacteristic
567C22#1684057011yescharacteristic
568C22#1684057711yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—41no—
582C4—4882no—
583C4—4881no—
584C4—4881no—
585C2—21no—
586C2—21no—
587C2—81no—
588C2#2145513411yescharacteristic
589C2#2145518311yescharacteristic
590C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3563C2
socleC22×C61C2×#120
prime radical#121×C61C23
hypercenter#3563D61
hypocenterC61#18469
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3563

Complements

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

NH ≅ G/Ntype
1. #21457021C2semidirect
3. #21453886C2semidirect
4. #21444861C2semidirect
6. #21444827C2semidirect
8. #21444383C2semidirect
10. #21444175C2semidirect
350. C61#18469semidirect

Maximal subgroups

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

subgroupindex
#214570212
#214569832
#214538862
#214448612
#214448502
#214448272
#214448162
#214443832
#214443572
#214441752
C61×#35632
#1846961