Finite Group Databasebeta

← #29783220

Subgroups — #29783220

Order 15616.

674 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#21451785C211yesmaximal, characteristic, complemented
2#21448456C211yesmaximal, characteristic
3#21447317C211yesmaximal, characteristic, complemented
4#21447214C211yesmaximal, characteristic, complemented
5#21447207C211yesmaximal, characteristic
6#21446675C211yesmaximal, characteristic
7#21446551C211yesmaximal, characteristic, complemented
8#21446518C211yesmaximal, characteristic
9#21446506C211yesmaximal, characteristic, complemented
10#21446463C211yesmaximal, characteristic, complemented
11#21446403C211yesmaximal, characteristic, complemented
12#21446155C211yesmaximal, characteristic
13#21446064C211yesmaximal, characteristic
14#21445678C211yesmaximal, characteristic, complemented
15C61×#2055C211yesmaximal, characteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27C61⋊#401—21no—
28#16839704—21no—
29#16839703—21no—
30#16839702—21no—
31#16839669C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839656—21no—
40#16839650—21no—
41#16839623—21no—
42#16839620—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44#16839608C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62D4×Dic61—22no—
63D4×Dic61—22no—
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×C4×Dic61—21no—
77C2×C4×Dic61C2311yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79#14519710—22no—
80#14519710—41no—
81#14519709—22no—
82C4⋊#92418—22no—
83#14519701—22no—
84#14519701—22no—
85#14519698—22no—
86#14519698—41no—
87#14519687—41no—
88C8×Dic61—41no—
89#14519684—21no—
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#14519684—21no—
99#14519684D411yescharacteristic
100#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C2×SD16×C61—22no—
106C2×SD16×C61—22no—
107C2×SD16×C61—41no—
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61C2311yescharacteristic
115C4⋊C8×C61—22no—
116#103×C61—41no—
117#102×C61—22no—
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—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61#4112yes—
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—24no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—81no—
145C22×Dic61—81no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150SD16×C61—44no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61#4112yes—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C22⋊C4×C61—81no—
162C22⋊C4×C61—81no—
163C42×C61—21no—
164C42×C61—21no—
165C22⋊Dic61—24no—
166C22⋊Dic61—81no—
167C22⋊Dic61—81no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—81no—
176C4⋊Dic61#4112yes—
177#11758196—81no—
178#11758196—81no—
179#11758196—81no—
180#11758196—81no—
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#4112yes—
192C23×C61—22no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61C22×D411yescharacteristic
200Q8×C61—41no—
201Q8×C61—41no—
202Q8×C61—41no—
203Q8×C61—41no—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—41no—
207D4×C61—41no—
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—41no—
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—
253C2×Dic61—84no—
254C8×C61—22no—
255C8×C61—41no—
256#92418—22no—
257#92418—41no—
258#29441—611nomaximal, Sylow
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61#52511yescharacteristic
267C22×C61#53211yescharacteristic
268C22×C61#53711yescharacteristic
269C22×C61#57611yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57711yescharacteristic
272C22×C61#57811yescharacteristic
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61#55111yescharacteristic
279C4×C61#55211yescharacteristic
280C4×C61#55311yescharacteristic
281C4×C61#55611yescharacteristic
282Dic61—41no—
283Dic61—42no—
284Dic61—41no—
285Dic61—41no—
286Dic61—41no—
287Dic61—81no—
288Dic61—81no—
289Dic61—81no—
290#2645—611no—
291#2319—611no—
292#2088—611no—
293#2068—611no—
294#2067—611no—
295#2055D6111yescharacteristic
296#1968—611no—
297#1942—611no—
298#1936—611no—
299#1935—611no—
300#1928—611no—
301#1920—611no—
302#1877—611no—
303#1862—611no—
304#1768—611no—
305C2×C61—82no—
306C2×C61#260111yescharacteristic
307C2×C61#311811yescharacteristic
308C2×C61#312011yescharacteristic
309C2×C61#317411yescharacteristic
310C2×C61#317511yescharacteristic
311C2×C61#325711yescharacteristic
312C2×C61#325811yescharacteristic
313C22×SD16C2×D6111yescharacteristic
314C2×C4×D4—611no—
315C2×#107—611no—
316C2×#107—611no—
317C2×#106—611no—
318C2×C4⋊C8—611no—
319C2×C4⋊C8C2×D6111yescharacteristic
320C2×#103—611no—
321C2×#103—611no—
322C2×#103C2×D6111yescharacteristic
323C2×#102—611no—
324C2×#102—611no—
325C2×#102C2×D6111yescharacteristic
326C2×#97—611no—
327C2×C4×C8—611no—
328#401—1221no—
329#400—1221no—
330#400—1221no—
331#400—1221no—
332#391—611no—
333#391—611no—
334#391C2×D6111yescharacteristic
335#390—611no—
336#390C2×D6111yescharacteristic
337#389—611no—
338#389—611no—
339#389—611no—
340#389—611no—
341#389—611no—
342#388—1221no—
343#388—1221no—
344#381—1221no—
345#380—1221no—
346C4×C4⋊C4—611no—
347#376—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340C2×D6111yescharacteristic
354#336—611no—
355#336—611no—
356C61#2944111yescharacteristic, complemented, Sylow
357C22×Q8C22×D6111yescharacteristic
358C22×D4C22×D6111yescharacteristic
359C23×C4—1221no—
360C2×SD16—22no—
361C2×SD16—22no—
362C2×SD16—41no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8C22×D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367C4×D4—1222no—
368C4×D4—1222no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—1221no—
388C2×C42C22×D6111yescharacteristic
389#107—1222no—
390#107—2441no—
391#106—1222no—
392C4⋊C8—22no—
393C4⋊C8—1222no—
394#103—41no—
395#103—1222no—
396#103—1222no—
397#102—22no—
398#102—1222no—
399#102—2441no—
400#97—2441no—
401C4×C8—2441no—
402#95—21no—
403#95—21no—
404#95—611no—
405#95—611no—
406#95—611no—
407#95—611no—
408#95—1221no—
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—
418C24—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4#1175822412yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—1224no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—4881no—
444C22×C4—4881no—
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448SD16—44no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—612no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8#1175822412yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—612no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—4881no—
474C4⋊C4—4881no—
475C4⋊C4—4881no—
476C4⋊C4—4881no—
477C4⋊C4—4881no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—1224no—
481C22⋊C4—4881no—
482C22⋊C4—4881no—
483C42—21no—
484C42—21no—
485C42—1221no—
486C42—1221no—
487C42—1221no—
488C42—1221no—
489C42—4881no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D6111yescharacteristic
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
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—41no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1222no—
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—1221no—
534C2×C4—4884no—
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#41×D6111yescharacteristic
547C2×C4#41×D6111yescharacteristic
548C2×C4C2×#1175822411yescharacteristic
549C2×C4C2×#1175822411yescharacteristic
550C2×C4#1451982211yescharacteristic
551C2×C4#1451982411yescharacteristic
552C8—22no—
553C8—41no—
554C8—1222no—
555C8—2441no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#1684027211yescharacteristic
564C22#1684030711yescharacteristic
565C22#1684034111yescharacteristic
566C22#1684056611yescharacteristic
567C22#1684056811yescharacteristic
568C22#1684056911yescharacteristic
569C22#1684057811yescharacteristic
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—4881no—
576C4—4881no—
577C4—4881no—
578C4—2441no—
579C4—2441no—
580C4—2441no—
581C4—2442no—
582C4—2441no—
583C4#1684044311yescharacteristic
584C4#1684044911yescharacteristic
585C4#1684045711yescharacteristic
586C4#1684048111yescharacteristic
587C2—82no—
588C2#2145140911yescharacteristic
589C2#2145600211yescharacteristic
590C2#2145603011yescharacteristic
591C2#2145644011yescharacteristic
592C2#2145645311yescharacteristic
593C2#2145722811yescharacteristic
594C2#2145724211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2055C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2055D61
hypocenterC61#29441
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
1. #21451785C2semidirect
3. #21447317C2semidirect
4. #21447214C2semidirect
7. #21446551C2semidirect
9. #21446506C2semidirect
10. #21446463C2semidirect
11. #21446403C2semidirect
14. #21445678C2semidirect
356. C61#29441semidirect

Maximal subgroups

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

subgroupindex
#214517852
#214484562
#214473172
#214472142
#214472072
#214466752
#214465512
#214465182
#214465062
#214464632
#214464032
#214461552
#214460642
#214456782
C61×#20552
#2944161