Finite Group Databasebeta

← #29783239

Subgroups — #29783239

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451780C211yesmaximal, characteristic
2#21448464C211yesmaximal, characteristic, complemented
3#21447318C211yesmaximal, characteristic, complemented
4#21447198C211yesmaximal, characteristic, complemented
5#21447191C211yesmaximal, characteristic
6#21447132C211yesmaximal, characteristic
7#21446547C211yesmaximal, characteristic, complemented
8#21446519C211yesmaximal, characteristic
9#21446512C211yesmaximal, characteristic, complemented
10#21446448C211yesmaximal, characteristic
11#21446406C211yesmaximal, characteristic, complemented
12#21446176C211yesmaximal, characteristic
13#21446089C211yesmaximal, characteristic, complemented
14#21445677C211yesmaximal, characteristic, complemented
15C61×#1968C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×#14519709C2211yescharacteristic
18C2×#14519709C2211yescharacteristic
19C2×#14519706C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519688C2211yescharacteristic
26#16839690—21no—
27#16839690—21no—
28#16839687—21no—
29#16839685—21no—
30#16839669C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839647—21no—
39#16839645—21no—
40#16839629—21no—
41#16839626—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43#16839608C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61Q8×Dic61—22no—
62Q8×Dic61—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2311yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81#14519709—22no—
82#14519709—41no—
83#14519706—22no—
84#14519706—22no—
85#14519701—22no—
86#14519701—41no—
87#14519698—22no—
88#14519698—41no—
89#14519688—41no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684D411yescharacteristic
99#14519684D411yescharacteristic
100#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102C22×Q8×C61C2311yescharacteristic
103C22×D4×C61C2311yescharacteristic
104C2×SD16×C61—22no—
105C2×SD16×C61—22no—
106C2×SD16×C61—41no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#106×C61—22no—
115#103×C61—22no—
116#102×C61—22no—
117C4×C8×C61—41no—
118#95×C61—21no—
119#95×C61—21no—
120C24×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61#4112yes—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147SD16×C61—44no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61#4112yes—
153C2×C8×C61#4112yes—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C22⋊C4×C61—81no—
159C22⋊C4×C61—81no—
160C42×C61—21no—
161C42×C61—21no—
162C22⋊Dic61—81no—
163C22⋊Dic61—81no—
164C22⋊Dic61—81no—
165C22⋊Dic61—81no—
166C4⋊Dic61—24no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—81no—
175C4⋊Dic61#4112yes—
176#11758196—81no—
177#11758196—81no—
178#11758196—81no—
179#11758196—81no—
180C4×Dic61—24no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—81no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61C22×D411yescharacteristic
200Q8×C61—22no—
201Q8×C61—22no—
202Q8×C61—41no—
203Q8×C61—41no—
204D4×C61—41no—
205D4×C61—41no—
206D4×C61—41no—
207D4×C61—41no—
208C2×C4×C61—22no—
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—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—22no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C8×C61—22no—
254C8×C61—22no—
255#92418—41no—
256#92418—41no—
257#29442—611nomaximal, Sylow
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61#52511yescharacteristic
266C22×C61#53211yescharacteristic
267C22×C61#53711yescharacteristic
268C22×C61#57711yescharacteristic
269C22×C61#57711yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57711yescharacteristic
272C4×C61—42no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61#55111yescharacteristic
278C4×C61#55211yescharacteristic
279C4×C61#55311yescharacteristic
280C4×C61#55611yescharacteristic
281Dic61—41no—
282Dic61—42no—
283Dic61—41no—
284Dic61—41no—
285Dic61—41no—
286Dic61—81no—
287Dic61—81no—
288Dic61—81no—
289#2644—611no—
290#2320—611no—
291#2088—611no—
292#2066—611no—
293#2065—611no—
294#2055—611no—
295#1968D6111yescharacteristic
296#1941—611no—
297#1936—611no—
298#1935—611no—
299#1926—611no—
300#1920—611no—
301#1880—611no—
302#1865—611no—
303#1768—611no—
304C2×C61—161no—
305C2×C61#260111yescharacteristic
306C2×C61#312511yescharacteristic
307C2×C61#312611yescharacteristic
308C2×C61#317311yescharacteristic
309C2×C61#317411yescharacteristic
310C2×C61#326011yescharacteristic
311C2×C61#326311yescharacteristic
312C22×SD16C2×D6111yescharacteristic
313C2×C4×Q8—611no—
314C2×#106—611no—
315C2×#106—611no—
316C2×#106C2×D6111yescharacteristic
317C2×C4⋊C8—611no—
318C2×C4⋊C8—611no—
319C2×#103—611no—
320C2×#103—611no—
321C2×#103C2×D6111yescharacteristic
322C2×#102—611no—
323C2×#102—611no—
324C2×#102C2×D6111yescharacteristic
325C2×#97—611no—
326C2×C4×C8C2×D6111yescharacteristic
327#397—1221no—
328#397—1221no—
329#397—1221no—
330#396—1221no—
331#391—611no—
332#391C2×D6111yescharacteristic
333#390—611no—
334#390—611no—
335#390C2×D6111yescharacteristic
336#389—611no—
337#389—611no—
338#389—611no—
339#389—611no—
340#389—611no—
341#387—1221no—
342#387—1221no—
343#383—1221no—
344#382—1221no—
345C4×C4⋊C4—611no—
346#376—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#336—611no—
354#336—611no—
355C61#2944211yescharacteristic, complemented, Sylow
356C22×Q8C22×D6111yescharacteristic
357C22×D4C22×D6111yescharacteristic
358C2×SD16—22no—
359C2×SD16—22no—
360C2×SD16—41no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365C4×Q8—1222no—
366C4×Q8—1222no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42C22×D6111yescharacteristic
390#106—22no—
391#106—1222no—
392#106—2441no—
393C4⋊C8—1222no—
394C4⋊C8—1222no—
395#103—22no—
396#103—1222no—
397#103—2441no—
398#102—22no—
399#102—1222no—
400#102—2441no—
401#97—2441no—
402C4×C8—41no—
403#95—21no—
404#95—21no—
405#95—611no—
406#95—611no—
407#95—611no—
408#95—611no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417C24—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8#1175822212yes—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4D4×D6111yescharacteristic
442C22×C4D4×D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444SD16—44no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8#1175822412yes—
456C2×C8#1175822412yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—612no—
462C4⋊C4—1224no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—4881no—
471C4⋊C4—4881no—
472C4⋊C4—4881no—
473C4⋊C4—4881no—
474C4⋊C4—4881no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—4881no—
478C22⋊C4—4881no—
479C22⋊C4—4881no—
480C22⋊C4—4881no—
481C42—21no—
482C42—21no—
483C42—1224no—
484C42—1221no—
485C42—1221no—
486C42—1221no—
487C42—1221no—
488C42—4881no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C2×D4×D6111yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—1221no—
521C2×C4—1222no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
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#41×D6111yescharacteristic
545C2×C4C2×#1175822411yescharacteristic
546C2×C4C2×#1175822411yescharacteristic
547C2×C4C2×#1175822211yescharacteristic
548C2×C4#1451982211yescharacteristic
549C2×C4#1451982411yescharacteristic
550C8—22no—
551C8—22no—
552C8—2441no—
553C8—2441no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22#1684027111yescharacteristic
562C22#1684031011yescharacteristic
563C22#1684033911yescharacteristic
564C22#1684056811yescharacteristic
565C22#1684056811yescharacteristic
566C22#1684056911yescharacteristic
567C22#1684056911yescharacteristic
568C4—41no—
569C4—41no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—4881no—
574C4—4881no—
575C4—4881no—
576C4—2441no—
577C4—2441no—
578C4—2441no—
579C4—2442no—
580C4—2441no—
581C4#1684043811yescharacteristic
582C4#1684045211yescharacteristic
583C4#1684045511yescharacteristic
584C4#1684048111yescharacteristic
585C2—161no—
586C2#2145141311yescharacteristic
587C2#2145610211yescharacteristic
588C2#2145610711yescharacteristic
589C2#2145643311yescharacteristic
590C2#2145644411yescharacteristic
591C2#2145725911yescharacteristic
592C2#2145728311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1968C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1968D61
hypocenterC61#29442
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #21448464C2semidirect
3. #21447318C2semidirect
4. #21447198C2semidirect
7. #21446547C2semidirect
9. #21446512C2semidirect
11. #21446406C2semidirect
13. #21446089C2semidirect
14. #21445677C2semidirect
355. C61#29442semidirect

Maximal subgroups

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

subgroupindex
#214517802
#214484642
#214473182
#214471982
#214471912
#214471322
#214465472
#214465192
#214465122
#214464482
#214464062
#214461762
#214460892
#214456772
C61×#19682
#2944261