Finite Group Databasebeta

← #29654090

Subgroups — #29654090

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456967C211yesmaximal, characteristic, complemented
2#21456944C211yesmaximal, characteristic, complemented
3#21451640C211yesmaximal, characteristic, complemented
4#21444867C214yesmaximal, complemented
5#21444850C214yesmaximal, complemented
6#21444383C211yesmaximal, characteristic, complemented
7#21444379C211yesmaximal, characteristic, complemented
8#21444133C211yesmaximal, characteristic, complemented
9C61×#3455C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840165—21no—
12#16840158—21no—
13#16840112—21no—
14#16840108—21no—
15#16840095C2214yes—
16#16840078C2214yes—
17#16840057—22no—
18#16840052—22no—
19#16839815C2211yescharacteristic
20#16839815C2211yescharacteristic
21#16839701C2212yes—
22#16839698C2212yes—
23#16839682—21no—
24#16839674—21no—
25#16839660—21no—
26#16839628C2211yescharacteristic
27#16839622—21no—
28#16839499C2214yes—
29#16839496C2214yes—
30#16839488C2212yes—
31#16839486C2212yes—
32C61×#542—21no—
33C61×#542—21no—
34C61×#541—21no—
35C61×#540—21no—
36C61×#539—21no—
37C61×#539C2212yes—
38C61×#538—21no—
39C61×#538C2212yes—
40C61×#533C2211yescharacteristic
41C61×#532C2211yescharacteristic
42C61×#529—22no—
43C61×#529—22no—
44C61×#525—21no—
45C4×#41×C61—21no—
46C2×#117×C61C2211yescharacteristic
47C22×Dic122—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—22no—
56C2×C22⋊Dic61—22no—
57#14519710—24no—
58#14519708—24no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519701—24no—
62#14519698—24no—
63C22⋊#92418C2312yes—
64C22⋊#92418C2312yes—
65C4×#92418—21no—
66C4×#92418—21no—
67#14519684—22no—
68#14519684—21no—
69#14519684—22no—
70#14519684C2312yes—
71C2×#41×C61—21no—
72C23×C4×C61D411yescharacteristic
73C4⋊Q8×C61D411yescharacteristic
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76#127×C61D411yescharacteristic
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—22no—
80#126×C61—22no—
81#125×C61—22no—
82#125×C61—22no—
83#125×C61—21no—
84#124×C61—22no—
85#124×C61—22no—
86#124×C61—21no—
87#123×C61—22no—
88#123×C61—22no—
89#123×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92C22⋊Q8×C61—22no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61C2312yes—
98#121×C61—22no—
99#121×C61—21no—
100#121×C61—21no—
101#121×C61—22no—
102#121×C61—21no—
103#121×C61C2312yes—
104#120×C61—22no—
105C4×Q8×C61—21no—
106C4×Q8×C61—21no—
107C4×Q8×C61—21no—
108C4×Q8×C61D412yes—
109C4×D4×C61—21no—
110C4×D4×C61—21no—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C4×D4×C61—21no—
116C4×D4×C61D412yes—
117#117×C61—22no—
118#117×C61—21no—
119#117×C61—21no—
120#117×C61—21no—
121#117×C61—21no—
122#117×C61—21no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61C2311yescharacteristic
127C2×C42×C61—22no—
128C2×C42×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C24×C61C2×D411yescharacteristic
131#41×C61—42no—
132#41×C61—42no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C2×D412yes—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D412yes—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—22no—
156C22×Dic61—22no—
157C2×Dic122—41no—
158C2×Dic122—41no—
159C2×Dic122—82no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D412yes—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C42×C61—22no—
190C42×C61—22no—
191C42×C61—21no—
192C42×C61—22no—
193C42×C61—21no—
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C22⋊Dic61—82no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203C4⋊Dic61—42no—
204#11758196—82no—
205C2×#92418—22no—
206C2×#92418—22no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61C22×D411yescharacteristic
214Q8×C61—42no—
215Q8×C61—42no—
216Q8×C61—81no—
217D4×C61—42no—
218D4×C61—42no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—81no—
222D4×C61—82no—
223D4×C61—82no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—44no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—82no—
240C2×C4×C61—82no—
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#13512yes—
246C2×C4×C61#13712yes—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255Dic122—82no—
256Dic122—82no—
257#92418—42no—
258#92418—42no—
259#18313—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—42no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61—42no—
268C22×C61—41no—
269C22×C61—81no—
270C22×C61C2×#12011yescharacteristic
271C22×C61C2×#13511yescharacteristic
272C22×C61C2×#13711yescharacteristic
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—42no—
279C4×C61—41no—
280C4×C61—42no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61#45212yes—
284C4×C61#45212yes—
285Dic61—82no—
286Dic61—82no—
287#3455D6111yescharacteristic
288#3230—611no—
289#3227—611no—
290#2628—611no—
291#1618—614no—
292#1615—614no—
293#1518—611no—
294#1517—611no—
295#1458—611no—
296C2×C61—21no—
297C2×C61—21no—
298C2×C61—82no—
299C2×C61—81no—
300C2×C61C2×#45211yescharacteristic
301C2×C61#302111yescharacteristic
302C2×C61#302811yescharacteristic
303#542—21no—
304#542—21no—
305#541—21no—
306#540—21no—
307#539—21no—
308#539C2×D6112yes—
309#538—21no—
310#538C2×D6112yes—
311#533C2×D6111yescharacteristic
312#532C2×D6111yescharacteristic
313C2×C4⋊Q8—611no—
314#529—22no—
315#529—22no—
316#525—21no—
317C4×#41—21no—
318C2×#117C2×D6111yescharacteristic
319#500—1221no—
320#498—1221no—
321#487—1221no—
322#486—1221no—
323#484—614no—
324#481—614no—
325#477—1222no—
326#476—1222no—
327#431—611no—
328#431—611no—
329#400—612no—
330#399—612no—
331#395—1221no—
332#393—1221no—
333#389—1221no—
334#382—611no—
335#380—1221no—
336#328—614no—
337#327—614no—
338#324—612no—
339#323—612no—
340C61#1831311yescharacteristic, complemented, Sylow
341C2×#41—21no—
342C22×Q8—1221no—
343C23×C4C22⋊D6111yescharacteristic
344C4⋊Q8—1221no—
345C4⋊Q8—1221no—
346C4⋊Q8—1221no—
347C4⋊Q8—1221no—
348C4⋊Q8C22⋊D6111yescharacteristic
349C4⋊Q8C22⋊D6111yescharacteristic
350C4⋊Q8C22⋊D6111yescharacteristic
351#127C22⋊D6111yescharacteristic
352#126—22no—
353#126—22no—
354#126—22no—
355#126—22no—
356#125—22no—
357#125—22no—
358#125—21no—
359#124—22no—
360#124—22no—
361#124—21no—
362#123—22no—
363#123—22no—
364#123—22no—
365#123—22no—
366#123—22no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—21no—
372C22⋊Q8C22×D6112yes—
373#121—21no—
374#121—21no—
375#121—22no—
376#121—22no—
377#121—21no—
378#121C22×D6112yes—
379#120—22no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8C22⋊D6112yes—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—21no—
391C4×D4C22⋊D6112yes—
392#117—22no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—21no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—1222no—
405C2×C22⋊C4C22⋊D6111yescharacteristic
406C2×C22⋊C4C22×D6111yescharacteristic
407C2×C42—22no—
408C2×C42C22⋊D6111yescharacteristic
409C2×C42C22×D6111yescharacteristic
410#107—1224no—
411#106—1224no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414#103—1224no—
415#102—1224no—
416C22⋊C8—612no—
417C22⋊C8—612no—
418C4×C8—1221no—
419C4×C8—1221no—
420#95—612no—
421#95—1222no—
422#95—1221no—
423#95—1222no—
424C24C2×C22⋊D6111yescharacteristic
425#41—42no—
426#41—42no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—4882no—
430C2×Q8—2441no—
431C2×Q8—2441no—
432C2×Q8C2×C22⋊D6112yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4C2×C22⋊D6112yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454C2×C8—1222no—
455C2×C8—1222no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—4882no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2442no—
476C4⋊C4C2×C22⋊D6112yes—
477C4⋊C4C2×C22⋊D6112yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—22no—
491C22⋊C4—4882no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—22no—
496C42—21no—
497C42C2×C22⋊D6111yescharacteristic
498C42C2×C22⋊D6111yescharacteristic
499C42C2×C22⋊D6111yescharacteristic
500C42C2×C22⋊D6111yescharacteristic
501C23—21no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—41no—
506C23—41no—
507C23C22×C22⋊D6111yescharacteristic
508Q8—42no—
509Q8—42no—
510Q8—81no—
511Q8—4882no—
512Q8—4882no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—81no—
518D4—82no—
519D4—82no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4#1451974611yescharacteristic
544C2×C4#1451974611yescharacteristic
545C2×C4#1451974611yescharacteristic
546C2×C4#1451974611yescharacteristic
547C2×C4#1451979712yes—
548C2×C4#1451980712yes—
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C8—2442no—
552C8—2442no—
553C22—21no—
554C22—21no—
555C22—42no—
556C22—42no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—81no—
563C22C2×#1451974611yescharacteristic
564C22C2×#1451979711yescharacteristic
565C22C2×#1451980711yescharacteristic
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—41no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—4882no—
577C4—4882no—
578C4#1683992412yes—
579C4#1683992412yes—
580C2—21no—
581C2—21no—
582C2—81no—
583C2—82no—
584C2C2×#1683992411yescharacteristic
585C2#2145519111yescharacteristic
586C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3455C2
socleC22×C61C2×#120
hypercenter#3455D61
hypocenterC61#18313
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3455

Complements

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

NH ≅ G/Ntype
1. #21456967C2semidirect
2. #21456944C2semidirect
3. #21451640C2semidirect
4. #21444867C2semidirect
5. #21444850C2semidirect
6. #21444383C2semidirect
7. #21444379C2semidirect
8. #21444133C2semidirect
340. C61#18313semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214569442
#214516402
#214448672
#214448502
#214443832
#214443792
#214441332
C61×#34552
#1831361