Finite Group Databasebeta

← #29652416

Subgroups — #29652416

Order 15616.

1196 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455052C212yesmaximal, complemented
2#21449184C211yesmaximal, characteristic, complemented
3#21444782C214yesmaximal, complemented
4#21444776C212yesmaximal, complemented
5#21444771C212yesmaximal, complemented
6#21444362C212yesmaximal, complemented
7#21444157C211yesmaximal, characteristic, complemented
8C61×#3459C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic, complemented
10C2×#14519807—22no—
11C2×#14519756—21no—
12#16839917—22no—
13#16839917C2214yescomplemented
14#16839900—22no—
15#16839900C2214yes—
16#16839770—22no—
17#16839726C2212yescomplemented
18#16839682C2212yescomplemented
19#16839674C2212yes—
20#16839668—21no—
21#16839642—21no—
22#16839634—21no—
23#16839622C2211yescharacteristic, complemented
24#16839499C2214yescomplemented
25#16839496C2214yes—
26#16839488C2212yescomplemented
27#16839486C2212yes—
28C61×#543C2211yescharacteristic
29C61×#539—22no—
30C61×#539C2212yes—
31C61×#538—22no—
32C61×#538—21no—
33C61×#534—21no—
34C61×#534C2211yescharacteristic
35C61×#529—22no—
36C61×#529—21no—
37C61×#525—22no—
38C61×#525C2212yes—
39C2×C4×D4×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C22×Dic122—22no—
42C22×Dic122—21no—
43C22×Dic122—22no—
44C22×Dic122—44no—
45C22×Dic122C2312yes—
46C23×Dic61—21no—
47#14519807—48no—
48C2×#11758202—24no—
49C2×#11758200—24no—
50C2×#11758194—22no—
51#14519756—44no—
52C2×C4⋊Dic61—21no—
53C2×#11758196—22no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—21no—
56#14519700—24no—
57#14519700—24no—
58C22⋊#92418—22no—
59C22⋊#92418C2314yes—
60#14519684—22no—
61#14519684—22no—
62#14519684C2312yes—
63C2×#41×C61D412yes—
64C22×D4×C61—21no—
65C22×D4×C61D411yescharacteristic
66C23×C4×C61—21no—
67C23×C4×C61D411yescharacteristic
68C23×C4×C61C2311yescharacteristic
69#126×C61—22no—
70#126×C61—22no—
71#126×C61—22no—
72#125×C61—21no—
73#125×C61—22no—
74#124×C61—22no—
75#124×C61—21no—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79#123×C61—22no—
80#123×C61—22no—
81#123×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61D412yes—
87C22⋊Q8×C61C2312yes—
88#121×C61—22no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61—21no—
92#121×C61D412yes—
93#121×C61C2312yes—
94#120×C61—22no—
95#120×C61—22no—
96#120×C61—22no—
97C4×Q8×C61—21no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—21no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106#117×C61—22no—
107#117×C61D412yes—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61C2311yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61D411yescharacteristic
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119#41×C61—44no—
120C2×Q8×C61—21no—
121C2×Q8×C61C2×D412yes—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—41no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61C2×D412yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61—82no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—44no—
153C22×Dic61—42no—
154C2×Dic122—28no—
155C2×Dic122—24no—
156C2×Dic122—24no—
157C2×Dic122—44no—
158C2×Dic122—48no—
159C2×Dic122—44no—
160C2×Dic122—42no—
161C2×Dic122—41no—
162C2×Dic122—41no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×D412yes—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
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—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—44no—
191C22⋊Dic61—42no—
192#11758202—48no—
193#11758200—48no—
194C4⋊Dic61—42no—
195#11758196—44no—
196#11758194—44no—
197C2×#92418—24no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—42no—
208C23×C61—81no—
209C23×C61—81no—
210C23×C61#12012yes—
211C23×C61#13712yescomplemented
212C23×C61C22×D411yescharacteristic
213Q8×C61—42no—
214Q8×C61—81no—
215D4×C61—42no—
216D4×C61—44no—
217D4×C61—44no—
218D4×C61—42no—
219D4×C61—42no—
220D4×C61—82no—
221D4×C61—82no—
222D4×C61—81no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—44no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—81no—
241C2×C4×C61—81no—
242C2×C4×C61—82no—
243C2×C4×C61—81no—
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61C22×D412yes—
247C2×Dic61—24no—
248C2×Dic61—48no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254Dic122—28no—
255Dic122—44no—
256Dic122—44no—
257Dic122—44no—
258#92418—44no—
259#18136—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—81no—
275C22×C61—84no—
276C22×C61C2×#12011yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57511yescharacteristic
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—42no—
285C4×C61—41no—
286C4×C61—42no—
287C4×C61B(4,𝔽2)12yes—
288Dic61—44no—
289Dic61—82no—
290#3459D6111yescharacteristic
291#3004—612no—
292#2409—611no—
293#1603—614no—
294#1602—612no—
295#1601—612no—
296#1513—612no—
297#1465—611no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—42no—
301C2×C61—42no—
302C2×C61—82no—
303C2×C61#300711yescharacteristic
304C2×C61#302211yescharacteristic
305C2×C61C2×B(4,𝔽2)11yescharacteristic
306C23×Q8—611no—
307C2×#137—1222no—
308#543C2×D6111yescharacteristic
309#539—22no—
310#539C2×D6112yes—
311#538—22no—
312#538—21no—
313#534—21no—
314#534C2×D6111yescharacteristic
315#529—22no—
316#529—21no—
317#525—22no—
318#525C2×D6112yes—
319C2×C22⋊Q8—1221no—
320C2×C4×D4—21no—
321C2×C4×D4C2×D6111yescharacteristic
322#451—614no—
323#451—1222no—
324#448—614no—
325#448—1222no—
326#419—1222no—
327#407—612no—
328#395—612no—
329#393—612no—
330#391—1221no—
331#386—1221no—
332#385—1221no—
333#380—611no—
334#328—614no—
335#327—614no—
336#324—612no—
337#323—612no—
338C61#1813611yescharacteristic, complemented, Sylow
339C2×#41C22⋊D6112yes—
340C22×Q8—612no—
341C22×Q8—1221no—
342C22×Q8—1222no—
343C22×Q8—1222no—
344C22×Q8—2444no—
345C22×D4—21no—
346C22×D4C22⋊D6111yescharacteristic
347C23×C4—21no—
348C23×C4—1221no—
349C23×C4C22⋊D6111yescharacteristic
350C23×C4C22×D6111yescharacteristic
351#137—2448no—
352C2×Q16—1224no—
353C2×SD16—1224no—
354C2×M16—1222no—
355#126—22no—
356#126—22no—
357#126—22no—
358#125—21no—
359#125—22no—
360#124—22no—
361#124—21no—
362#123—22no—
363#123—22no—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—2444no—
373C22⋊Q8C22⋊D6112yes—
374C22⋊Q8C22×D6112yes—
375#121—22no—
376#121—22no—
377#121—22no—
378#121—21no—
379#121C22⋊D6112yes—
380#121C22×D6112yes—
381#120—22no—
382#120—22no—
383#120—22no—
384C4×Q8—21no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—21no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393#117—22no—
394#117C22⋊D6112yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4C22⋊D6111yescharacteristic
404C2×C22⋊C4C22×D6111yescharacteristic
405C2×C42—21no—
406C2×C42C22⋊D6111yescharacteristic
407#103—1224no—
408#103—1224no—
409C22⋊C8—614no—
410C22⋊C8—1222no—
411#95—612no—
412#95—1222no—
413#95—1222no—
414C24—21no—
415C24C2×C22⋊D6111yescharacteristic
416C24C2×C22⋊D6111yescharacteristic
417#41—44no—
418C2×Q8—21no—
419C2×Q8—1224no—
420C2×Q8—1228no—
421C2×Q8—1224no—
422C2×Q8—2442no—
423C2×Q8—2444no—
424C2×Q8—2444no—
425C2×Q8—2448no—
426C2×Q8—2441no—
427C2×Q8—2441no—
428C2×Q8C2×C22⋊D6112yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—41no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×C22⋊D6112yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—2444no—
453C22×C4—2442no—
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C2×C22⋊D6112yes—
460C22×C4C23×D6111yescharacteristic
461Q16—2448no—
462SD16—2448no—
463M16—2444no—
464C2×C8—1224no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—2442no—
475C4⋊C4—2444no—
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—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—2444no—
491C22⋊C4—2442no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—81no—
507C23—81no—
508C23#1451974612yes—
509C23#1451980712yescomplemented
510C23C22×C22⋊D6111yescharacteristic
511Q8—42no—
512Q8—81no—
513Q8—1228no—
514Q8—2444no—
515Q8—2444no—
516Q8—2444no—
517D4—42no—
518D4—44no—
519D4—44no—
520D4—42no—
521D4—42no—
522D4—82no—
523D4—82no—
524D4—81no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—1224no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2448no—
552C2×C4—2442no—
553C2×C4#1451974611yescharacteristic
554C2×C4#1451974611yescharacteristic
555C2×C4C22×C22⋊D6112yes—
556C8—2444no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—84no—
573C22C2×#1451974611yescharacteristic
574C22C2×#1451980711yescharacteristic
575C22#1684056011yescharacteristic
576C4—22no—
577C4—21no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—4882no—
585C4—2444no—
586C4#1683994912yes—
587C2—21no—
588C2—21no—
589C2—42no—
590C2—42no—
591C2—82no—
592C2#2145507911yescharacteristic
593C2#2145519811yescharacteristic
594C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3459C2
socleC22×C61C2×#120
hypercenter#3459D61
hypocenterC61#18136
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3459

Complements

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

NH ≅ G/Ntype
1. #21455052C2semidirect
2. #21449184C2semidirect
3. #21444782C2semidirect
4. #21444776C2semidirect
5. #21444771C2semidirect
6. #21444362C2semidirect
7. #21444157C2semidirect
9. C23×Dic122C22semidirect
13. #16839917C22semidirect
17. #16839726C22semidirect
18. #16839682C22semidirect
23. #16839622C22semidirect
24. #16839499C22semidirect
26. #16839488C22semidirect
211. C23×C61#137semidirect
338. C61#18136semidirect
509. C23#14519807semidirect

Maximal subgroups

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

subgroupindex
#214550522
#214491842
#214447822
#214447762
#214447712
#214443622
#214441572
C61×#34592
#1813661