Finite Group Databasebeta

← #29779477

Subgroups — #29779477

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450414C211yesmaximal, characteristic
2#21449678C211yesmaximal, characteristic, complemented
3#21447321C211yesmaximal, characteristic, complemented
4#21447241C211yesmaximal, characteristic, complemented
5#21447136C211yesmaximal, characteristic
6#21446963C211yesmaximal, characteristic
7#21446782C211yesmaximal, characteristic, complemented
8#21446674C211yesmaximal, characteristic
9#21446518C211yesmaximal, characteristic
10#21446509C211yesmaximal, characteristic, complemented
11#21446407C211yesmaximal, characteristic, complemented
12#21446171C211yesmaximal, characteristic
13#21445691C211yesmaximal, characteristic, complemented
14#21445648C211yesmaximal, characteristic, complemented
15C61×#1936C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839702—21no—
28#16839692—21no—
29#16839690—21no—
30#16839687—21no—
31#16839668C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839661—21no—
36#16839661C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839655—21no—
39#16839650—21no—
40#16839647—21no—
41#16839627C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#102×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×#11758200—22no—
61C2×#11758200—22no—
62C2×#11758200—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519768—22no—
66#14519768—41no—
67#14519742—22no—
68#14519742—41no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519708—22no—
88C4⋊#92418—22no—
89#14519702—22no—
90#14519702—22no—
91#14519700—22no—
92#14519698—41no—
93#14519687—41no—
94C8×Dic61—41no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#102×C61—22no—
117#102×C61—22no—
118#97×C61—22no—
119#95×C61—21no—
120C24×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122#4112yes—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61#4112yes—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61#4112yes—
160C22⋊C4×C61—42no—
161C22⋊C4×C61—81no—
162C42×C61—21no—
163C42×C61—21no—
164C22⋊Dic61—42no—
165C22⋊Dic61—81no—
166C22⋊Dic61—81no—
167C22⋊Dic61—81no—
168#11758200—44no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—41no—
176C4⋊Dic61—41no—
177C4⋊Dic61—42no—
178C4⋊Dic61—42no—
179#11758196—42no—
180#11758196—42no—
181#11758196—81no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—42no—
187C4×Dic61—81no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418#4112yes—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61C22×D411yescharacteristic
202D4×C61—41no—
203D4×C61—41no—
204D4×C61—41no—
205D4×C61—41no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
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×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—21no—
228C2×Dic61—21no—
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—41no—
241C2×Dic61—41no—
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—
251Dic122—22no—
252Dic122—22no—
253Dic122—41no—
254Dic122—41no—
255C8×C61—22no—
256C8×C61—41no—
257#92418—22no—
258#92418—41no—
259#29102—611nomaximal, Sylow
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61#53211yescharacteristic
268C22×C61#54611yescharacteristic
269C22×C61#54811yescharacteristic
270C22×C61#57511yescharacteristic
271C22×C61#57511yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57711yescharacteristic
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—42no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61#52911yescharacteristic
280C4×C61#53911yescharacteristic
281C4×C61#55111yescharacteristic
282C4×C61#55611yescharacteristic
283Dic61—42no—
284Dic61—41no—
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288Dic61—81no—
289Dic61—81no—
290Dic61—81no—
291#2513—611no—
292#2453—611no—
293#2088—611no—
294#2074—611no—
295#2055—611no—
296#2025—611no—
297#1991—611no—
298#1968—611no—
299#1936—611no—
300#1936D6111yescharacteristic
301#1935—611no—
302#1920—611no—
303#1880—611no—
304#1771—611no—
305#1763—611no—
306C2×C61—161no—
307C2×C61#256011yescharacteristic
308C2×C61#317411yescharacteristic
309C2×C61#318311yescharacteristic
310C2×C61#330711yescharacteristic
311C2×C61#332111yescharacteristic
312C2×C61#337111yescharacteristic
313C2×C61#337111yescharacteristic
314C22×SD16—611no—
315C2×#125—611no—
316C2×C4×Q8—611no—
317C2×#106—611no—
318C2×C4⋊C8—611no—
319C2×#103—611no—
320C2×#103—611no—
321C2×#103—611no—
322C2×#102—611no—
323C2×#102C2×D6111yescharacteristic
324C2×#102C2×D6111yescharacteristic
325C2×#97—611no—
326C2×#97C2×D6111yescharacteristic
327C2×C4×C8—611no—
328#400—1221no—
329#398—1221no—
330#397—1221no—
331#397—1221no—
332#391—611no—
333#391—611no—
334#390—611no—
335#390—611no—
336#390C2×D6111yescharacteristic
337#389—611no—
338#389—611no—
339#389—1221no—
340#389C2×D6111yescharacteristic
341#388—1221no—
342#388—1221no—
343#387—1221no—
344#382—611no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340C2×D6111yescharacteristic
354#340C2×D6111yescharacteristic
355#336—611no—
356#336—611no—
357C61#2910211yescharacteristic, complemented, Sylow
358C22×Q8—611no—
359C22×D4C22×D6111yescharacteristic
360C2×SD16—1222no—
361C2×SD16—1222no—
362C2×SD16—2441no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8C22×D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367#125—1222no—
368#125—2441no—
369C4×Q8—1222no—
370C4×Q8—2441no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42C22×D6111yescharacteristic
396#106—1222no—
397C4⋊C8—1222no—
398#103—1222no—
399#103—1222no—
400#103—1222no—
401#102—22no—
402#102—22no—
403#102—2441no—
404#97—22no—
405#97—2441no—
406C4×C8—2441no—
407#95—21no—
408#95—611no—
409#95—611no—
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—
419C24—21no—
420C2×Q8—612no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4D4×D6111yescharacteristic
444C22×C4D4×D6111yescharacteristic
445C22×C4C23×D6111yescharacteristic
446SD16—2444no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—612no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8#1175822412yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—4881no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4—2442no—
475C4⋊C4—2442no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4#1175822212yes—
479C22⋊C4—42no—
480C22⋊C4—81no—
481C22⋊C4—4881no—
482C22⋊C4—4881no—
483C22⋊C4—4881no—
484C22⋊C4—2442no—
485C42—21no—
486C42—21no—
487C42—1221no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—4881no—
492C42—2442no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C2×D4×D6111yescharacteristic
501Q8—1222no—
502Q8—1222no—
503Q8—2441no—
504Q8—2441no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
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—1221no—
535C2×C4—1221no—
536C2×C4—1222no—
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#41×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C2×C4C2×#1175822411yescharacteristic
551C2×C4C2×#1175822211yescharacteristic
552C2×C4#1451982311yescharacteristic
553C2×C4#1451982411yescharacteristic
554C8—22no—
555C8—41no—
556C8—1222no—
557C8—2441no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22#1684030711yescharacteristic
566C22#1684040811yescharacteristic
567C22#1684042011yescharacteristic
568C22#1684055811yescharacteristic
569C22#1684055811yescharacteristic
570C22#1684057111yescharacteristic
571C22#1684057211yescharacteristic
572C4—41no—
573C4—42no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—4881no—
578C4—4881no—
579C4—4881no—
580C4—2441no—
581C4—2441no—
582C4—2441no—
583C4—2441no—
584C4—2442no—
585C4#1684029411yescharacteristic
586C4#1684035611yescharacteristic
587C4#1684043711yescharacteristic
588C4#1684048011yescharacteristic
589C2—161no—
590C2#2145096211yescharacteristic
591C2#2145644211yescharacteristic
592C2#2145654311yescharacteristic
593C2#2145780011yescharacteristic
594C2#2145801011yescharacteristic
595C2#2145858211yescharacteristic
596C2#2145858611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1936C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1936D61
hypocenterC61#29102
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #21449678C2semidirect
3. #21447321C2semidirect
4. #21447241C2semidirect
7. #21446782C2semidirect
10. #21446509C2semidirect
11. #21446407C2semidirect
13. #21445691C2semidirect
14. #21445648C2semidirect
357. C61#29102semidirect

Maximal subgroups

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

subgroupindex
#214504142
#214496782
#214473212
#214472412
#214471362
#214469632
#214467822
#214466742
#214465182
#214465092
#214464072
#214461712
#214456912
#214456482
C61×#19362
#2910261