Finite Group Databasebeta

← #29748030

Subgroups — #29748030

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449613C211yesmaximal, characteristic
2#21448920C211yesmaximal, characteristic, complemented
3#21447310C211yesmaximal, characteristic, complemented
4#21446968C211yesmaximal, characteristic, complemented
5#21446963C211yesmaximal, characteristic
6#21446752C211yesmaximal, characteristic
7#21446513C211yesmaximal, characteristic
8#21446462C211yesmaximal, characteristic, complemented
9#21446442C211yesmaximal, characteristic
10#21446441C211yesmaximal, characteristic
11#21446412C211yesmaximal, characteristic, complemented
12#21446386C211yesmaximal, characteristic, complemented
13#21446071C211yesmaximal, characteristic, complemented
14#21445651C211yesmaximal, characteristic, complemented
15C61×#2042C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26#16839698—21no—
27#16839687—21no—
28#16839669C2211yescharacteristic
29#16839665C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839658—21no—
33#16839658C2211yescharacteristic
34#16839650—21no—
35#16839650—21no—
36#16839647—21no—
37#16839643—21no—
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×#11758196C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42#16839613—21no—
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#124×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61#14519768—22no—
62#14519768—22no—
63Q8×Dic61—22no—
64Q8×Dic61—41no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196D411yescharacteristic
75C2×#11758196D411yescharacteristic
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519710—41no—
88C4⋊#92418—41no—
89#14519701—22no—
90#14519701—22no—
91#14519698—22no—
92#14519698—22no—
93#14519687—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107#124×C61—22no—
108#124×C61—41no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#103×C61—22no—
116#103×C61—22no—
117#102×C61—22no—
118#102×C61—41no—
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
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—81no—
157C4⋊C4×C61#4112yes—
158C22⋊C4×C61—42no—
159C22⋊C4×C61—42no—
160C42×C61—21no—
161C42×C61—21no—
162C22⋊Dic61—42no—
163C22⋊Dic61—42no—
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—42no—
174C4⋊Dic61#4112yes—
175C4⋊Dic61#4112yes—
176#11758196—24no—
177#11758196—41no—
178#11758196—41no—
179#11758196—42no—
180#11758196—81no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—42no—
186C4×Dic61—42no—
187C4×Dic61—81no—
188C4×Dic61—81no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61C22×D411yescharacteristic
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—41no—
206Q8×C61—41no—
207D4×C61—41no—
208D4×C61—41no—
209D4×C61—41no—
210D4×C61—41no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
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—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C612−1+411yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C8×C61—41no—
257C8×C61—41no—
258#92418—41no—
259#92418—41no—
260#26719—611nomaximal, Sylow
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61#52511yescharacteristic
269C22×C61#54611yescharacteristic
270C22×C61#54811yescharacteristic
271C22×C61#57511yescharacteristic
272C22×C61#57511yescharacteristic
273C22×C61#57611yescharacteristic
274C22×C61#57811yescharacteristic
275C4×C61—42no—
276C4×C61—42no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61#52911yescharacteristic
281C4×C61#54211yescharacteristic
282C4×C61#55211yescharacteristic
283C4×C61#55911yescharacteristic
284Dic61—41no—
285Dic61—42no—
286Dic61—42no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292#2448—611no—
293#2370—611no—
294#2086—611no—
295#2042D6111yescharacteristic
296#2026—611no—
297#2025—611no—
298#1983—611no—
299#1936—611no—
300#1928—611no—
301#1925—611no—
302#1925—611no—
303#1921—611no—
304#1917—611no—
305#1863—611no—
306#1763—611no—
307C2×C61—161no—
308C2×C61#245811yescharacteristic
309C2×C61#311111yescharacteristic
310C2×C61#312411yescharacteristic
311C2×C61#330911yescharacteristic
312C2×C61#332711yescharacteristic
313C2×C61#337011yescharacteristic
314C2×C61#337711yescharacteristic
315C2×#125—611no—
316C2×#124C2×D6111yescharacteristic
317C2×C4×Q8—611no—
318C2×#107—611no—
319C2×#107—611no—
320C2×C4⋊C8—611no—
321C2×#103—611no—
322C2×#103—611no—
323C2×#103C2×D6111yescharacteristic
324C2×#103C2×D6111yescharacteristic
325C2×#102—611no—
326C2×#102—611no—
327C2×#102C2×D6111yescharacteristic
328C2×#102C2×D6111yescharacteristic
329C2×#97—611no—
330#399—1221no—
331#397—1221no—
332#391—611no—
333#390—611no—
334#390—611no—
335#389—611no—
336#389—611no—
337#389—1221no—
338#389C2×D6111yescharacteristic
339#388—1221no—
340#388—1221no—
341#387—1221no—
342#387—1221no—
343C4×C4⋊C4—611no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4—1221no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#336—611no—
355#336—611no—
356#336—611no—
357#336C2×D6111yescharacteristic
358C61#2671911yescharacteristic, complemented, Sylow
359C22×Q8C22×D6111yescharacteristic
360C22×D4C22×D6111yescharacteristic
361C22×C8—611no—
362C22×C8—611no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365#125—1222no—
366#125—1222no—
367#124—22no—
368#124—41no—
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—611no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
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×C42—1221no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398#107—1222no—
399#107—2441no—
400C4⋊C8—2441no—
401#103—22no—
402#103—22no—
403#103—1222no—
404#103—1222no—
405#102—22no—
406#102—41no—
407#102—1222no—
408#102—1222no—
409#97—2441no—
410#95—21no—
411#95—611no—
412#95—611no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1175822412yes—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4D4×D6111yescharacteristic
445C22×C4D4×D6111yescharacteristic
446C22×C4C23×D6111yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—81no—
463C4⋊C4—612no—
464C4⋊C4—612no—
465C4⋊C4—1224no—
466C4⋊C4—1224no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—4881no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4#1175822412yes—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—4881no—
482C22⋊C4—4881no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C42—21no—
486C42—21no—
487C42—1221no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—4881no—
492C42—4881no—
493C42—2442no—
494C42—2442no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C2×D4×D6111yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1222no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1222no—
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#41×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C2×C4C2×#1175822411yescharacteristic
553C2×C4C2×#1175822411yescharacteristic
554C2×C4#1451982211yescharacteristic
555C2×C4#1451982411yescharacteristic
556C8—41no—
557C8—41no—
558C8—2441no—
559C8—2441no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22#1684027211yescharacteristic
568C22#1684039911yescharacteristic
569C22#1684042011yescharacteristic
570C22#1684055911yescharacteristic
571C22#1684056111yescharacteristic
572C22#1684056611yescharacteristic
573C22#1684057611yescharacteristic
574C4—42no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—81no—
579C4—4881no—
580C4—4881no—
581C4—2441no—
582C4—2442no—
583C4—2442no—
584C4—2441no—
585C4—2441no—
586C4—2441no—
587C4#1684029311yescharacteristic
588C4#1684037911yescharacteristic
589C4#1684044911yescharacteristic
590C4#1684049611yescharacteristic
591C2—161no—
592C2#2144974211yescharacteristic
593C2#2145592711yescharacteristic
594C2#2145607811yescharacteristic
595C2#2145782911yescharacteristic
596C2#2145808011yescharacteristic
597C2#2145857111yescharacteristic
598C2#2145866711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2042C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2042D61
hypocenterC61#26719
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
2. #21448920C2semidirect
3. #21447310C2semidirect
4. #21446968C2semidirect
8. #21446462C2semidirect
11. #21446412C2semidirect
12. #21446386C2semidirect
13. #21446071C2semidirect
14. #21445651C2semidirect
358. C61#26719semidirect

Maximal subgroups

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

subgroupindex
#214496132
#214489202
#214473102
#214469682
#214469632
#214467522
#214465132
#214464622
#214464422
#214464412
#214464122
#214463862
#214460712
#214456512
C61×#20422
#2671961