Finite Group Databasebeta

← #29752726

Subgroups — #29752726

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448822C211yesmaximal, characteristic
2#21447228C211yesmaximal, characteristic, complemented
3#21447197C211yesmaximal, characteristic, complemented
4#21447015C211yesmaximal, characteristic
5#21447015C211yesmaximal, characteristic
6#21446580C211yesmaximal, characteristic, complemented
7#21446544C211yesmaximal, characteristic, complemented
8#21446427C211yesmaximal, characteristic
9#21446404C211yesmaximal, characteristic, complemented
10#21446394C211yesmaximal, characteristic, complemented
11#21446086C211yesmaximal, characteristic, complemented
12#21446011C211yesmaximal, characteristic
13#21445907C211yesmaximal, characteristic
14C4×#14519700C211yesmaximal, characteristic, complemented
15C61×#2496C211yesmaximal, characteristic
16C2×#14519756C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839659—21no—
34#16839649—21no—
35#16839634—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37C22⋊C4×Dic61—21no—
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×#126×C61—21no—
49C61×#400—21no—
50C61×#397—21no—
51C61×#389C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#387—21no—
54C61×#384C2211yescharacteristic
55C61×#380C2211yescharacteristic
56C61×#380C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519756—22no—
66#14519756—22no—
67C4×Dic122—22no—
68C4×Dic122—41no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—41no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519708—22no—
82#14519708—22no—
83#14519708—22no—
84#14519708—41no—
85C4⋊#92418—22no—
86#14519700—22no—
87#14519700—22no—
88#14519700—22no—
89#14519700—41no—
90C22⋊#92418—22no—
91C22⋊#92418—41no—
92C4×#92418—41no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C23×C4×C61C2311yescharacteristic
98#126×C61—44no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61D411yescharacteristic
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—21no—
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61D411yescharacteristic
120#95×C61D411yescharacteristic
121C24×C61C2×D411yescharacteristic
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—81no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122#4112yes—
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—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61—42no—
163C22⋊C4×C61—41no—
164C22⋊C4×C61—42no—
165C22⋊C4×C61—42no—
166C22⋊C4×C61SD1614yes—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—42no—
172C22⋊Dic61—24no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—42no—
178C4⋊Dic61#4112yes—
179C4⋊Dic61#4112yes—
180#11758196—24no—
181#11758196—42no—
182#11758196—81no—
183C4×Dic61—42no—
184C4×Dic61—81no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418#4112yes—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61C2×SD1612yes—
204C23×C61C22×D411yescharacteristic
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
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—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
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×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C2×SD1612yes—
235C2×C4×C61C2×SD1612yes—
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
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—41no—
254C2×Dic61—84no—
255Dic122—22no—
256Dic122—22no—
257Dic122—41no—
258Dic122—41no—
259#92418—22no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#27034—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61#52511yescharacteristic
272C22×C61#54611yescharacteristic
273C22×C61#54711yescharacteristic
274C22×C61C22×SD1611yescharacteristic
275C22×C61#57511yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57811yescharacteristic
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61#52911yescharacteristic
288C4×C61#53911yescharacteristic
289C4×C61#55311yescharacteristic
290C4×C61#55611yescharacteristic
291Dic61—42no—
292Dic61—42no—
293Dic61—42no—
294Dic61—81no—
295#2496D6111yescharacteristic
296#2360—611no—
297#2072—611no—
298#2066—611no—
299#2035—611no—
300#2035—611no—
301#1948—611no—
302#1941—611no—
303#1923—611no—
304#1920—611no—
305#1919—611no—
306#1865—611no—
307#1853—611no—
308#1824—611no—
309C4×#103—611no—
310C2×C61—42no—
311C2×C61#246611yescharacteristic
312C2×C61#310311yescharacteristic
313C2×C61#312011yescharacteristic
314C2×C61#329811yescharacteristic
315C2×C61#331111yescharacteristic
316C2×C61#333511yescharacteristic
317C2×C61#335211yescharacteristic
318C2×#126—21no—
319C2×C22⋊Q8—611no—
320C2×C4×Q8—611no—
321C22×C4⋊C4—611no—
322C2×#106—611no—
323C2×#106—611no—
324C2×#106—611no—
325C2×#106—611no—
326C2×C4⋊C8—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103—611no—
330C2×#103—611no—
331C2×C22⋊C8—611no—
332C2×C22⋊C8—611no—
333C2×C4×C8—611no—
334#400—21no—
335#397—21no—
336#391—611no—
337#389—611no—
338#389—1221no—
339#389C2×D6111yescharacteristic
340#389C2×D6111yescharacteristic
341#387—21no—
342#387—1221no—
343#385—1221no—
344#384C2×D6111yescharacteristic
345#380C2×D6111yescharacteristic
346#380C2×D6111yescharacteristic
347C4×C4⋊C4—611no—
348C4×C4⋊C4C2×D6111yescharacteristic
349C4×C22⋊C4—1221no—
350C4×C22⋊C4C2×D6111yescharacteristic
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#336—611no—
360#336—611no—
361C61#2703411yescharacteristic, complemented, Sylow
362C22×Q8—611no—
363C23×C4—1221no—
364C23×C4C22×D6111yescharacteristic
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—611no—
369#126—44no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C4×Q8—1222no—
373C4×Q8—2441no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—2441no—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4C22⋊D6111yescharacteristic
395C2×C22⋊C4C22⋊D6111yescharacteristic
396C2×C42—21no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42C22×D6111yescharacteristic
400C2×C42C22×D6111yescharacteristic
401#106—1222no—
402#106—1222no—
403#106—1222no—
404#106—2441no—
405C4⋊C8—1222no—
406#103—1222no—
407#103—1222no—
408#103—1222no—
409#103—2441no—
410C22⋊C8—1222no—
411C22⋊C8—2441no—
412C4×C8—2441no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95C22⋊D6111yescharacteristic
423#95C22⋊D6111yescharacteristic
424C24C2×C22⋊D6111yescharacteristic
425C2×Q8—612no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—4881no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454C2×C8—612no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—612no—
479C4⋊C4—612no—
480C4⋊C4—1224no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—4881no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—1224no—
492C22⋊C4#1175820014yes—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—4881no—
499C42—2442no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23C2×#1175820012yes—
507C23C22×C22⋊D6111yescharacteristic
508Q8—1222no—
509Q8—1222no—
510Q8—2441no—
511Q8—2441no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—4884no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4C2×#1175820012yes—
555C2×C4C2×#1175820012yes—
556C2×C4#41×D6111yescharacteristic
557C2×C4#41×D6111yescharacteristic
558C2×C4#41×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4#1451982311yescharacteristic
561C2×C4#1451982411yescharacteristic
562C8—1222no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1684026911yescharacteristic
574C22#1684040011yescharacteristic
575C22#1684041111yescharacteristic
576C22C22×#1175820011yescharacteristic
577C22#1684056011yescharacteristic
578C22#1684057011yescharacteristic
579C22#1684057711yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—4881no—
590C4—2442no—
591C4—2442no—
592C4—2442no—
593C4#1684029511yescharacteristic
594C4#1684035311yescharacteristic
595C4#1684046311yescharacteristic
596C4#1684047911yescharacteristic
597C2—42no—
598C2#2144985111yescharacteristic
599C2#2145584611yescharacteristic
600C2#2145602711yescharacteristic
601C2#2145767811yescharacteristic
602C2#2145784911yescharacteristic
603C2#2145817611yescharacteristic
604C2#2145838811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2496C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2496D61
hypocenterC61#27034
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
2. #21447228C2semidirect
3. #21447197C2semidirect
6. #21446580C2semidirect
7. #21446544C2semidirect
9. #21446404C2semidirect
10. #21446394C2semidirect
11. #21446086C2semidirect
14. C4×#14519700C2semidirect
361. C61#27034semidirect

Maximal subgroups

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

subgroupindex
#214488222
#214472282
#214471972
#214470152
#214465802
#214465442
#214464272
#214464042
#214463942
#214460862
#214460112
#214459072
C4×#145197002
C61×#24962
#2703461