Finite Group Databasebeta

← #29725155

Subgroups — #29725155

Order 15616.

750 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#21450740C211yesmaximal, characteristic
2#21448731C211yesmaximal, characteristic, complemented
3#21447379C211yesmaximal, characteristic, complemented
4#21447226C211yesmaximal, characteristic
5#21447223C211yesmaximal, characteristic
6#21446753C211yesmaximal, characteristic, complemented
7#21446576C211yesmaximal, characteristic, complemented
8#21446440C211yesmaximal, characteristic
9#21446437C211yesmaximal, characteristic
10#21446389C211yesmaximal, characteristic, complemented
11#21446387C211yesmaximal, characteristic, complemented
12#21445945C211yesmaximal, characteristic, complemented
13#21445818C211yesmaximal, characteristic
14#21445633C211yesmaximal, characteristic
15C61×#2099C211yesmaximal, characteristic, complemented
16C2×#14519755—21no—
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742—21no—
19C2×#14519739C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839686—21no—
29#16839684—21no—
30#16839664C2211yescharacteristic
31#16839663C2211yescharacteristic
32#16839658—21no—
33#16839658C2211yescharacteristic
34#16839657—21no—
35#16839657—21no—
36#16839631C2211yescharacteristic
37#16839631C2211yescharacteristic
38#16839631C2211yescharacteristic
39#16839621—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#384C2211yescharacteristic
54C61×#340C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122D411yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519755—44no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67#14519742—44no—
68#14519739—22no—
69#14519739—22no—
70C2×C4⋊Dic61D411yescharacteristic
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×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×#11758196D411yescharacteristic
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519710—22no—
95#14519706—22no—
96#14519699—22no—
97#14519699—22no—
98#14519697—22no—
99#14519697—22no—
100C8×Dic61—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#95×C61—21no—
116C23×D61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244#4112yes—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122D814yes—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61—81no—
160C42×C61—21no—
161C42×C61—21no—
162#11758198—42no—
163#11758198—42no—
164#11758198—42no—
165#11758198—81no—
166#11758198—81no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61#4112yes—
174#11758196—22no—
175#11758196—22no—
176#11758196—42no—
177#11758196—42no—
178#11758196—42no—
179#11758196—42no—
180#11758196—81no—
181#11758196—81no—
182#11758196—81no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—42no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C4×Dic61#4112yes—
189C4×Dic61#4112yes—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
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×D812yes—
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61—21no—
235C2×Dic61—21no—
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#12312yes—
245C2×Dic61#12312yes—
246C2×Dic61C2×D812yes—
247C2×Dic61C2×D812yes—
248C2×Dic61#13712yes—
249D244—22no—
250D244—22no—
251D244—41no—
252D244—41no—
253C4×D61—84no—
254Dic122—28no—
255C8×C61—41no—
256C8×C61—41no—
257#92418—41no—
258#92418—41no—
259#24990—611nomaximal, Sylow
260C22×C61C2×#12311yescharacteristic
261C22×C61#54811yescharacteristic
262C22×C61#54811yescharacteristic
263C22×C61C22×D811yescharacteristic
264C22×C61C2×#13711yescharacteristic
265C22×C61#57711yescharacteristic
266C22×C61#57711yescharacteristic
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C4×C61—22no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61#52911yescharacteristic
281C4×C61#54111yescharacteristic
282C4×C61#55211yescharacteristic
283C4×C61#55611yescharacteristic
284Dic61—22no—
285Dic61—22no—
286Dic61—41no—
287Dic61—42no—
288Dic61—81no—
289Dic61—81no—
290Dic61#48014yes—
291#2538—611no—
292Q8×C4⋊C4—611no—
293#2099—611no—
294#2099D6111yescharacteristic, complemented
295#2071—611no—
296#2071—611no—
297#1984—611no—
298#1947—611no—
299#1925—611no—
300#1925—611no—
301#1918—611no—
302#1918—611no—
303#1834—611no—
304#1802—611no—
305C4×#102—611no—
306C2×C61#242111yescharacteristic
307C2×C61C2×#48011yescharacteristic
308C2×C61#309911yescharacteristic
309C2×C61#335811yescharacteristic
310C2×C61#335811yescharacteristic
311C2×C61#336911yescharacteristic
312C2×C61#337211yescharacteristic
313D61—82no—
314C2×C22⋊Q8—1221no—
315C2×C4×Q8—611no—
316C2×C4×Q8—1221no—
317C2×C4×D4—611no—
318C2×#107—611no—
319C2×#107—611no—
320C2×C4⋊C8—611no—
321C2×#102—611no—
322C2×#102—611no—
323C2×#102—611no—
324C2×#102—611no—
325C2×C4×C8—611no—
326#396—1221no—
327#396—1221no—
328#390—611no—
329#390—611no—
330#389—611no—
331#389—1221no—
332#389—1221no—
333#389—1221no—
334#389C2×D6111yescharacteristic
335#384—611no—
336#384—611no—
337#384—611no—
338#384C2×D6111yescharacteristic
339#380—1221no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—611no—
342C4×C4⋊C4—611no—
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340C2×D6111yescharacteristic
350#340C2×D6111yescharacteristic
351#340C2×D6111yescharacteristic
352#340C2×D6111yescharacteristic
353#336—611no—
354#336—611no—
355#336—611no—
356#336C2×D6111yescharacteristic
357C61#2499011yescharacteristic, complemented, Sylow
358C22×Q8—611no—
359C22×D4—611no—
360C23×C4—1221no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365C22⋊Q8—2444no—
366C4×Q8—1222no—
367C4×Q8—1222no—
368C4×Q8—2444no—
369C4×D4—1222no—
370C4×D4—1222no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42C22×D6111yescharacteristic
402#107—1222no—
403#107—1222no—
404C4⋊C8—1222no—
405#102—1222no—
406#102—1222no—
407#102—1222no—
408#102—1222no—
409C4×C8—1222no—
410#95—21no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415C24—1221no—
416C2×Q8—614no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×D4—612no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—612no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—4881no—
475C4⋊C4—4881no—
476C4⋊C4—4881no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4—2442no—
480C4⋊C4—2442no—
481C22⋊C4—4881no—
482C22⋊C4—4881no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C22⋊C4—2442no—
486C42—21no—
487C42—21no—
488C42—612no—
489C42—612no—
490C42—1221no—
491C42—1221no—
492C42—2442no—
493C42—2442no—
494C42—2442no—
495C23—1222no—
496C23—2441no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23C2×D4×D6111yescharacteristic
503Q8—1228no—
504D4—1222no—
505D4—1222no—
506D4—2441no—
507D4—2441no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
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—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—4884no—
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×C4D8×D6112yes—
548C2×C4#41×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C2×C4#41×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C2×C4#1451982411yescharacteristic
553C2×C4#1451982511yescharacteristic
554C8—41no—
555C8—41no—
556C8—2441no—
557C8—2441no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22#123×D6111yescharacteristic
566C22#1684042011yescharacteristic
567C22#1684042211yescharacteristic
568C22C2×D8×D6111yescharacteristic
569C22#137×D6111yescharacteristic
570C22#1684056811yescharacteristic
571C22#1684057111yescharacteristic
572C4—22no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—614no—
579C4—1222no—
580C4—1222no—
581C4—4881no—
582C4—4881no—
583C4—2442no—
584C4—2441no—
585C4#1684029511yescharacteristic
586C4#1684037111yescharacteristic
587C4#1684044411yescharacteristic
588C4#1684047911yescharacteristic
589C2—4882no—
590C2#2144931711yescharacteristic
591C2#480×D6111yescharacteristic
592C2#2145580311yescharacteristic
593C2#2145845111yescharacteristic
594C2#2145845311yescharacteristic
595C2#2145856211yescharacteristic
596C2#2145859711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2099C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2099D61
hypocenterC61#24990
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
2. #21448731C2semidirect
3. #21447379C2semidirect
6. #21446753C2semidirect
7. #21446576C2semidirect
10. #21446389C2semidirect
11. #21446387C2semidirect
12. #21445945C2semidirect
15. C61×#2099C2semidirect
294. #2099D61semidirect
357. C61#24990semidirect

Maximal subgroups

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

subgroupindex
#214507402
#214487312
#214473792
#214472262
#214472232
#214467532
#214465762
#214464402
#214464372
#214463892
#214463872
#214459452
#214458182
#214456332
C61×#20992
#2499061