Finite Group Databasebeta

← #29748642

Subgroups — #29748642

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450357C211yesmaximal, characteristic, complemented
2#21448888C211yesmaximal, characteristic
3#21447234C211yesmaximal, characteristic
4#21447192C211yesmaximal, characteristic
5#21447061C211yesmaximal, characteristic
6#21447061C211yesmaximal, characteristic
7#21446495C211yesmaximal, characteristic, complemented
8#21446454C211yesmaximal, characteristic, complemented
9#21446405C211yesmaximal, characteristic, complemented
10#21446398C211yesmaximal, characteristic, complemented
11#21446090C211yesmaximal, characteristic, complemented
12#21446083C211yesmaximal, characteristic
13#21445835C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic
15C61×#1916C211yesmaximal, characteristic, complemented
16C2×#14519765C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29#16839703—21no—
30#16839693—21no—
31#16839668C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839652—21no—
38#16839651—21no—
39#16839643—21no—
40#16839633—21no—
41#16839628C2211yescharacteristic
42#16839626—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839609—21no—
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C2×C4×C8×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#336C2211yescharacteristic
56C61×#336C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62#14519765—22no—
63#14519765—22no—
64C4×Dic122—22no—
65C4×Dic122—41no—
66C4×D244—22no—
67C4×D244—41no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87#14519709—22no—
88#14519709—22no—
89#14519702—22no—
90#14519702—22no—
91#14519702—22no—
92#14519702—41no—
93#14519699—22no—
94#14519699—22no—
95#14519699—22no—
96#14519699—41no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C4⋊C8×C61—41no—
117C4×C8×C61—41no—
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C23×D61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244#4112yes—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—81no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122#4112yes—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—41no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—21no—
170#11758198—24no—
171#11758198—24no—
172#11758198—42no—
173#11758198—42no—
174#11758198—81no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61#4112yes—
186C4⋊Dic61#4112yes—
187#11758196—42no—
188#11758196—42no—
189#11758196—81no—
190#11758196—81no—
191#11758196—81no—
192C4×Dic61—42no—
193C4×Dic61—42no—
194C4×Dic61—81no—
195C23×C61C22×D411yescharacteristic
196C22×D61—22no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61#13512yes—
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248D244—22no—
249D244—22no—
250D244—41no—
251D244—41no—
252C4×D61—84no—
253Dic122—22no—
254Dic122—22no—
255Dic122—41no—
256Dic122—41no—
257C8×C61—41no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261#26760—611nomaximal, Sylow
262C22×C61#52511yescharacteristic
263C22×C61#54611yescharacteristic
264C22×C61#54811yescharacteristic
265C22×C61C2×#13511yescharacteristic
266C22×C61#57511yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57711yescharacteristic
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—22no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61#52911yescharacteristic
283C4×C61#53911yescharacteristic
284C4×C61#55211yescharacteristic
285C4×C61#55311yescharacteristic
286Dic61—42no—
287Dic61—42no—
288Dic61—42no—
289Dic61—81no—
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293#2508—611no—
294#2367—611no—
295#2073—611no—
296#2065—611no—
297#2042—611no—
298#2042—611no—
299#1933—611no—
300#1927—611no—
301#1920—611no—
302#1919—611no—
303#1916D6111yescharacteristic, complemented
304#1865—611no—
305#1864—611no—
306#1805—611no—
307C4×#102—611no—
308C2×C61#246011yescharacteristic
309C2×C61#310211yescharacteristic
310C2×C61#312511yescharacteristic
311C2×C61#329611yescharacteristic
312C2×C61#332111yescharacteristic
313C2×C61#336011yescharacteristic
314C2×C61#337111yescharacteristic
315D61—82no—
316C2×#124—611no—
317C2×C4×Q8—611no—
318C2×C4×D4—611no—
319C2×#106—611no—
320C2×#106—611no—
321C2×C4⋊C8C2×D6111yescharacteristic
322C2×#103—611no—
323C2×#103—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#102—611no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#102—611no—
330C2×C4×C8C2×D6111yescharacteristic
331#400—1221no—
332#398—1221no—
333#391—611no—
334#390—611no—
335#389—611no—
336#389—611no—
337#389—611no—
338#389—611no—
339#388—1221no—
340#388—1221no—
341#387—1221no—
342#385—1221no—
343#382—611no—
344#382—1221no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4C2×D6111yescharacteristic
348C4×C22⋊C4—1221no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#336C2×D6111yescharacteristic
356#336C2×D6111yescharacteristic
357#336C2×D6111yescharacteristic
358#336C2×D6111yescharacteristic
359C61#2676011yescharacteristic, complemented, Sylow
360C22×Q8—611no—
361C22×D4—611no—
362C23×C4—1221no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365C22×C8C22×D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367#124—1222no—
368#124—1222no—
369C4×Q8—1222no—
370C4×Q8—2441no—
371C4×D4—1222no—
372C4×D4—2441no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4D24411yescharacteristic
385C2×C4⋊C4D24411yescharacteristic
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395C2×C42C22×D6111yescharacteristic
396C2×C42C22×D6111yescharacteristic
397#106—1222no—
398#106—1222no—
399C4⋊C8—41no—
400#103—1222no—
401#103—1222no—
402#103—1222no—
403#103—2441no—
404#102—1222no—
405#102—1222no—
406#102—1222no—
407#102—2441no—
408C4×C8—41no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95D24411yescharacteristic
420#95D24411yescharacteristic
421C24—1221no—
422C2×Q8—612no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×D4—612no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—4881no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4C2×D24411yescharacteristic
445C22×C4C2×D24411yescharacteristic
446C22×C4C2×D24411yescharacteristic
447C22×C4C2×D24411yescharacteristic
448C22×C4C2×D24411yescharacteristic
449C22×C4C2×D24411yescharacteristic
450C22×C4C23×D6111yescharacteristic
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C4⋊C4—22no—
464C4⋊C4—41no—
465C4⋊C4—612no—
466C4⋊C4—612no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—4881no—
478C4⋊C4—4881no—
479C4⋊C4—4881no—
480C4⋊C4—2442no—
481C4⋊C4—2442no—
482C22⋊C4—1224no—
483C22⋊C4—1224no—
484C22⋊C4—4881no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—4881no—
494C42—2442no—
495C42—2442no—
496C23—1222no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23C22×D24411yescharacteristic
504Q8—1222no—
505Q8—1222no—
506Q8—2441no—
507Q8—2441no—
508D4—1222no—
509D4—1222no—
510D4—2441no—
511D4—2441no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—4884no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4#1451979912yes—
552C2×C4#41×D6111yescharacteristic
553C2×C4#41×D6111yescharacteristic
554C2×C4#41×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22#1684027411yescharacteristic
570C22#1684040711yescharacteristic
571C22#1684042411yescharacteristic
572C22C2×#1451979911yescharacteristic
573C22#1684056211yescharacteristic
574C22#1684057411yescharacteristic
575C22#1684057411yescharacteristic
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—4881no—
583C4—4881no—
584C4—4881no—
585C4—4881no—
586C4—2442no—
587C4—2442no—
588C4—2442no—
589C4#1684029511yescharacteristic
590C4#1684035811yescharacteristic
591C4#1684044411yescharacteristic
592C4#1684045711yescharacteristic
593C2—4882no—
594C2#2144976911yescharacteristic
595C2#2145583911yescharacteristic
596C2#2145609811yescharacteristic
597C2#2145765611yescharacteristic
598C2#2145800711yescharacteristic
599C2#2145847011yescharacteristic
600C2#2145858911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1916C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1916D61
hypocenterC61#26760
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
1. #21450357C2semidirect
7. #21446495C2semidirect
8. #21446454C2semidirect
9. #21446405C2semidirect
10. #21446398C2semidirect
11. #21446090C2semidirect
13. #21445835C2semidirect
15. C61×#1916C2semidirect
303. #1916D61semidirect
359. C61#26760semidirect

Maximal subgroups

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

subgroupindex
#214503572
#214488882
#214472342
#214471922
#214470612
#214464952
#214464542
#214464052
#214463982
#214460902
#214460832
#214458352
C4×#145196992
C61×#19162
#2676061