Finite Group Databasebeta

← #29752781

Subgroups — #29752781

Order 15616.

700 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#21450220C211yesmaximal, characteristic
2#21448823C211yesmaximal, characteristic
3#21447241C211yesmaximal, characteristic, complemented
4#21447195C211yesmaximal, characteristic, complemented
5#21447025C211yesmaximal, characteristic
6#21447017C211yesmaximal, characteristic
7#21446588C211yesmaximal, characteristic, complemented
8#21446425C211yesmaximal, characteristic
9#21446408C211yesmaximal, characteristic, complemented
10#21446407C211yesmaximal, characteristic, complemented
11#21446086C211yesmaximal, characteristic, complemented
12#21446010C211yesmaximal, characteristic
13#21445904C211yesmaximal, characteristic
14C4×#14519702C211yesmaximal, characteristic, complemented
15C61×#1941C211yesmaximal, characteristic, complemented
16C2×#14519757C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4⋊C4×D61C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C61⋊#401—21no—
29#16839689—21no—
30#16839667C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839658—21no—
33#16839652—21no—
34#16839648—21no—
35#16839645—21no—
36#16839633—21no—
37#16839627C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839619C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×#11758198C2211yescharacteristic
43#16839609—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#106×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×C4×C8×C61C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519757—22no—
65#14519757—22no—
66C4×Dic122—22no—
67C4×Dic122—41no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C4⋊C4×D61—22no—
76C4⋊C4×D61—41no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198D411yescharacteristic
81C2×#11758198D411yescharacteristic
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86#14519710—22no—
87#14519708—22no—
88#14519706—22no—
89#14519702—22no—
90#14519702—22no—
91#14519700—22no—
92#14519700—41no—
93#14519689—22no—
94#14519689—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×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
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#106×C61—22no—
116#106×C61—41no—
117C4×C8×C61—41no—
118#95×C61—21no—
119#95×C61—21no—
120C23×D61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—81no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122#4112yes—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61#4112yes—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—81no—
165C4⋊C4×C61#4112yes—
166C42×C61—21no—
167C42×C61—21no—
168#11758198—24no—
169#11758198—22no—
170#11758198—22no—
171#11758198—42no—
172#11758198—81no—
173C4⋊Dic61—24no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—81no—
180C4⋊Dic61#4112yes—
181#11758196—42no—
182#11758196—42no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—42no—
186C4×Dic61—81no—
187C4×Dic61—81no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C23×C61C22×D411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61#13512yes—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—41no—
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#13712yes—
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C2×Dic61—21no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—84no—
252Dic122—22no—
253Dic122—22no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—22no—
257C8×C61—41no—
258#92418—41no—
259#92418—41no—
260#27037—611nomaximal, Sylow
261C22×C61#52511yescharacteristic
262C22×C61#54611yescharacteristic
263C22×C61#54711yescharacteristic
264C22×C61C2×#13511yescharacteristic
265C22×C61C2×#13711yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57711yescharacteristic
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—41no—
275C4×C61—22no—
276C4×C61—42no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61#52911yescharacteristic
282C4×C61#53911yescharacteristic
283C4×C61#55311yescharacteristic
284C4×C61#55611yescharacteristic
285Dic61—41no—
286Dic61—41no—
287Dic61—42no—
288Dic61—41no—
289Dic61—42no—
290Dic61—81no—
291Dic61—81no—
292#2494—611no—
293#2360—611no—
294#2074—611no—
295#2066—611no—
296#2036—611no—
297#2035—611no—
298#1949—611no—
299#1941D6111yescharacteristic, complemented
300#1923—611no—
301#1921—611no—
302#1920—611no—
303#1865—611no—
304#1853—611no—
305#1823—611no—
306C4×#103—611no—
307C2×C61#246611yescharacteristic
308C2×C61#310811yescharacteristic
309C2×C61#312511yescharacteristic
310C2×C61#329611yescharacteristic
311C2×C61#332011yescharacteristic
312C2×C61#333711yescharacteristic
313C2×C61#335611yescharacteristic
314D61—42no—
315C2×C22⋊Q8—611no—
316C2×C4×Q8—611no—
317C22×C4⋊C4—611no—
318C2×#107—611no—
319C2×#106—611no—
320C2×#106C2×D6111yescharacteristic
321C2×#106C2×D6111yescharacteristic
322C2×C4⋊C8—611no—
323C2×#103—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103—611no—
327C2×C22⋊C8—611no—
328C2×C22⋊C8—611no—
329C2×C4×C8C2×D6111yescharacteristic
330#401—1221no—
331#397—1221no—
332#391—611no—
333#389—611no—
334#389—1221no—
335#389C2×D6111yescharacteristic
336#389C2×D6111yescharacteristic
337#388—1221no—
338#387—1221no—
339#387—1221no—
340#385—1221no—
341#382—611no—
342#380—611no—
343#380—611no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4—611no—
346C4×C22⋊C4—611no—
347C4×C22⋊C4—1221no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340C2×D6111yescharacteristic
355#340C2×D6111yescharacteristic
356#336—611no—
357#336—611no—
358C61#2703711yescharacteristic, complemented, Sylow
359C22×Q8—611no—
360C23×C4—611no—
361C23×C4—1221no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8C22×D6111yescharacteristic
365C22×C8C22×D6111yescharacteristic
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C4×Q8—1222no—
369C4×Q8—2441no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—2441no—
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C22⋊C4—611no—
386C2×C22⋊C4—611no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—611no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395#107—1222no—
396#106—22no—
397#106—41no—
398#106—1222no—
399C4⋊C8—1222no—
400#103—1222no—
401#103—1222no—
402#103—1222no—
403#103—2441no—
404C22⋊C8—1222no—
405C22⋊C8—2441no—
406C4×C8—41no—
407#95—21no—
408#95—21no—
409#95—611no—
410#95—611no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—611no—
421C2×Q8—612no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—4881no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4D4×D6111yescharacteristic
448C22×C4D4×D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8#1175822612yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—612no—
472C4⋊C4—1224no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—4881no—
479C4⋊C4—2442no—
480C4⋊C4—2442no—
481C4⋊C4#1175822212yes—
482C22⋊C4—1222no—
483C22⋊C4—1224no—
484C22⋊C4—1222no—
485C22⋊C4—4881no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—1221no—
490C42—1221no—
491C42—4881no—
492C42—4881no—
493C42—2442no—
494C23—612no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23—1221no—
501C23C2×D4×D6111yescharacteristic
502Q8—1222no—
503Q8—1222no—
504Q8—2441no—
505Q8—2441no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
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—41no—
522C2×C4—41no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1222no—
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—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4#1451980612yes—
550C2×C4#41×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C2×C4C2×#1175822211yescharacteristic
553C2×C4C2×#1175822611yescharacteristic
554C2×C4#1451982211yescharacteristic
555C2×C4#1451982511yescharacteristic
556C8—22no—
557C8—41no—
558C8—2441no—
559C8—2441no—
560C22—1222no—
561C22—1222no—
562C22—1222no—
563C22—2441no—
564C22—2441no—
565C22—2441no—
566C22—2441no—
567C22#1684027111yescharacteristic
568C22#1684040111yescharacteristic
569C22#1684041511yescharacteristic
570C22#135×D6111yescharacteristic
571C22C2×#1451980611yescharacteristic
572C22#1684056811yescharacteristic
573C22#1684056911yescharacteristic
574C4—22no—
575C4—42no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—4881no—
581C4—4881no—
582C4—2441no—
583C4—2441no—
584C4—2441no—
585C4—2442no—
586C4—2442no—
587C4#1684029711yescharacteristic
588C4#1684035811yescharacteristic
589C4#1684046011yescharacteristic
590C4#1684048411yescharacteristic
591C2—2442no—
592C2#2144986111yescharacteristic
593C2#2145589711yescharacteristic
594C2#2145609211yescharacteristic
595C2#2145764911yescharacteristic
596C2#2145799611yescharacteristic
597C2#2145819711yescharacteristic
598C2#2145843311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1941C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1941D61
hypocenterC61#27037
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
3. #21447241C2semidirect
4. #21447195C2semidirect
7. #21446588C2semidirect
9. #21446408C2semidirect
10. #21446407C2semidirect
11. #21446086C2semidirect
14. C4×#14519702C2semidirect
15. C61×#1941C2semidirect
299. #1941D61semidirect
358. C61#27037semidirect

Maximal subgroups

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

subgroupindex
#214502202
#214488232
#214472412
#214471952
#214470252
#214470172
#214465882
#214464252
#214464082
#214464072
#214460862
#214460102
#214459042
C4×#145197022
C61×#19412
#2703761