Finite Group Databasebeta

← #29753311

Subgroups — #29753311

Order 15616.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448888C211yesmaximal, characteristic
2#21447348C211yesmaximal, characteristic
3#21447134C211yesmaximal, characteristic
4#21447060C211yesmaximal, characteristic
5#21446912C211yesmaximal, characteristic
6#21446679C211yesmaximal, characteristic, complemented
7#21446460C211yesmaximal, characteristic, complemented
8#21446457C211yesmaximal, characteristic, complemented
9#21446387C211yesmaximal, characteristic, complemented
10#21446378C211yesmaximal, characteristic, complemented
11#21446163C211yesmaximal, characteristic, complemented
12#21446156C211yesmaximal, characteristic
13#21445821C211yesmaximal, characteristic, complemented
14C4×#14519697C211yesmaximal, characteristic
15C61×#2511C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×C4×#92418C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839652—21no—
36#16839651—21no—
37#16839633—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39#16839609—21no—
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C61×#400—21no—
49C61×#400—21no—
50C61×#391C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#391C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#387—21no—
55C61×#387—21no—
56C61×#384C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202—41no—
65C2×#11758201—22no—
66C2×#11758201—41no—
67C2×#11758201—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519765—22no—
73#14519765—22no—
74C4×Dic122—22no—
75C4×Dic122—41no—
76C4×D244—22no—
77C4×D244—41no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88#14519710—22no—
89#14519710—41no—
90C4⋊#92418—41no—
91#14519701—22no—
92#14519701—41no—
93#14519700—22no—
94#14519700—41no—
95#14519700—41no—
96#14519697—22no—
97#14519697—22no—
98#14519697—41no—
99C4×#92418—41no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×Q8×C61C2311yescharacteristic
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C42×C61—21no—
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116#95×C61—21no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C23×D61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244#4112yes—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—81no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122#4112yes—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61—81no—
172C4⋊C4×C61—81no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—81no—
180#11758202—44no—
181#11758201—44no—
182#11758198—24no—
183#11758198—24no—
184#11758198—42no—
185#11758198—42no—
186#11758198—81no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61#4112yes—
191#11758196—42no—
192C4×Dic61—42no—
193C4×Dic61—81no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418#4112yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61#13512yes—
247C2×C4×C61#13712yes—
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263D244—22no—
264D244—22no—
265D244—41no—
266D244—41no—
267C4×D61—84no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272#92418—22no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#27073—611nomaximal, Sylow
277C22×C61#52511yescharacteristic
278C22×C61#54611yescharacteristic
279C22×C61#55011yescharacteristic
280C22×C61C2×#13511yescharacteristic
281C22×C61C2×#13711yescharacteristic
282C22×C61#57611yescharacteristic
283C22×C61#57811yescharacteristic
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61#52911yescharacteristic
302C4×C61#53811yescharacteristic
303C4×C61#55111yescharacteristic
304C4×C61#55311yescharacteristic
305Dic61—42no—
306Dic61—42no—
307Dic61—81no—
308#2511D6111yescharacteristic, complemented
309#2367—611no—
310#2093—611no—
311#2055—611no—
312#2042—611no—
313#2015—611no—
314#1969—611no—
315#1928—611no—
316#1928—611no—
317#1918—611no—
318#1916—611no—
319#1878—611no—
320#1877—611no—
321#1803—611no—
322C4×#102—611no—
323C2×C61#246711yescharacteristic
324C2×C61#310711yescharacteristic
325C2×C61#312411yescharacteristic
326C2×C61#330011yescharacteristic
327C2×C61#331411yescharacteristic
328C2×C61#339611yescharacteristic
329C2×C61#339811yescharacteristic
330D61—82no—
331C22×Q16—611no—
332C22×SD16—611no—
333C2×#124—611no—
334C2×C4×Q8—611no—
335C2×C4×D4—611no—
336C2×#107—611no—
337C2×#107—611no—
338C2×C4⋊C8—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#102—611no—
345C2×#102—611no—
346C2×#102—611no—
347C2×C4×C8—611no—
348#400—21no—
349#400—21no—
350#391—611no—
351#391C2×D6111yescharacteristic
352#391C2×D6111yescharacteristic
353#391C2×D6111yescharacteristic
354#390—611no—
355#389C2×D6111yescharacteristic
356#388—1221no—
357#388—1221no—
358#387—21no—
359#387—21no—
360#385—1221no—
361#384C2×D6111yescharacteristic
362C4×C4⋊C4—611no—
363C4×C4⋊C4C2×D6111yescharacteristic
364C4×C4⋊C4C2×D6111yescharacteristic
365C4×C22⋊C4—1221no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#336—611no—
371#336—611no—
372#336—611no—
373#336—611no—
374C61#2707311yescharacteristic, complemented, Sylow
375C22×Q8—611no—
376C22×Q8C22×D6111yescharacteristic
377C22×D4—611no—
378C23×C4—1221no—
379C2×Q16—1222no—
380C2×Q16—1222no—
381C2×Q16—2441no—
382C2×SD16—1222no—
383C2×SD16—2441no—
384C2×SD16—2441no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8—611no—
389#124—1222no—
390#124—1222no—
391C4×Q8—1222no—
392C4×Q8—2441no—
393C4×D4—1222no—
394C4×D4—2441no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22⋊D6111yescharacteristic
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—21no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415C2×C42C22×D6111yescharacteristic
416C2×C42C22×D6111yescharacteristic
417#107—1222no—
418#107—2441no—
419C4⋊C8—2441no—
420#103—1222no—
421#103—1222no—
422#103—2441no—
423#103—2441no—
424#103—2441no—
425#102—1222no—
426#102—1222no—
427#102—2441no—
428C4×C8—2441no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95C22⋊D6111yescharacteristic
441#95C22⋊D6111yescharacteristic
442C24—1221no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—612no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×D4—612no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—4881no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478Q16—2444no—
479SD16—2444no—
480C2×C8—612no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—612no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—2442no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—4881no—
511C22⋊C4—2442no—
512C22⋊C4—2442no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—81no—
520C42—4881no—
521C42—2442no—
522C23—1222no—
523C23—2441no—
524C23—2441no—
525C23—2441no—
526C23—2441no—
527C23—2441no—
528C23—2441no—
529C23C22×C22⋊D6111yescharacteristic
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—1222no—
535Q8—1222no—
536Q8—2441no—
537Q8—2441no—
538D4—1222no—
539D4—1222no—
540D4—2441no—
541D4—2441no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—4884no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4#1451979712yes—
581C2×C4#1451980812yes—
582C2×C4#41×D6111yescharacteristic
583C2×C4#41×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4#41×D6111yescharacteristic
586C2×C4#1451982311yescharacteristic
587C2×C4#1451982311yescharacteristic
588C8—1222no—
589C8—2441no—
590C8—2441no—
591C8—2441no—
592C22—4881no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—2442no—
597C22—2442no—
598C22—2442no—
599C22#1684026911yescharacteristic
600C22#1684040011yescharacteristic
601C22#1684043111yescharacteristic
602C22C2×#1451979711yescharacteristic
603C22C2×#1451980811yescharacteristic
604C22#1684056511yescharacteristic
605C22#1684057711yescharacteristic
606C4—22no—
607C4—22no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—4881no—
617C4—2442no—
618C4—2442no—
619C4#1684029511yescharacteristic
620C4#1684035111yescharacteristic
621C4#1684043711yescharacteristic
622C4#1684046311yescharacteristic
623C2—4882no—
624C2#2144986611yescharacteristic
625C2#2145588411yescharacteristic
626C2#2145607511yescharacteristic
627C2#2145770011yescharacteristic
628C2#2145789411yescharacteristic
629C2#2145882711yescharacteristic
630C2#2145884611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2511C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2511D61
hypocenterC61#27073
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2511

Complements

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

NH ≅ G/Ntype
6. #21446679C2semidirect
7. #21446460C2semidirect
8. #21446457C2semidirect
9. #21446387C2semidirect
10. #21446378C2semidirect
11. #21446163C2semidirect
13. #21445821C2semidirect
15. C61×#2511C2semidirect
308. #2511D61semidirect
374. C61#27073semidirect

Maximal subgroups

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

subgroupindex
#214488882
#214473482
#214471342
#214470602
#214469122
#214466792
#214464602
#214464572
#214463872
#214463782
#214461632
#214461562
#214458212
C4×#145196972
C61×#25112
#2707361