Finite Group Databasebeta

← #29748113

Subgroups — #29748113

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#21450160C211yesmaximal, characteristic
2#21447141C211yesmaximal, characteristic, complemented
3#21447012C211yesmaximal, characteristic
4#21446970C211yesmaximal, characteristic, complemented
5#21446827C211yesmaximal, characteristic
6#21446776C211yesmaximal, characteristic, complemented
7#21446494C211yesmaximal, characteristic, complemented
8#21446453C211yesmaximal, characteristic, complemented
9#21446427C211yesmaximal, characteristic
10#21446401C211yesmaximal, characteristic, complemented
11#21446160C211yesmaximal, characteristic, complemented
12#21446126C211yesmaximal, characteristic
13#21445905C211yesmaximal, characteristic
14C4×#14519701C211yesmaximal, characteristic, complemented
15C61×#2360C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839703—21no—
31#16839690—21no—
32#16839668C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839650—21no—
35#16839626—21no—
36#16839622C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C22⋊Q8×C61C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C22×C4⋊C4×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#389—21no—
53C61×#389C2211yescharacteristic
54C61×#387—21no—
55C61×#385—21no—
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C22⋊C4×C61—21no—
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×#11758202—22no—
61C2×#11758202—41no—
62C2×#11758202—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519769—22no—
68#14519769—41no—
69#14519756—22no—
70#14519756—41no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80#14519708—22no—
81C4⋊#92418—41no—
82#14519701—22no—
83#14519701—22no—
84#14519701—41no—
85#14519700—22no—
86#14519700—22no—
87#14519700—41no—
88C22⋊#92418—41no—
89C22⋊#92418—41no—
90C4×#92418—41no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C4×Q8×C61—22no—
103C4×Q8×C61—41no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—41no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C24×C61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—81no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61—81no—
170C4⋊C4×C61#4112yes—
171C4⋊C4×C61#4112yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—42no—
180C42×C61—81no—
181C22⋊Dic61—42no—
182C22⋊Dic61—42no—
183#11758202—44no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61#4112yes—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193C4×Dic61—42no—
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—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61#13512yes—
213C23×C61C22×D411yescharacteristic
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61—41no—
217Q8×C61—41no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
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—21no—
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—41no—
247C2×C4×C61—41no—
248C2×C4×C61—84no—
249C2×C4×C61#13712yes—
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268Dic122—41no—
269Dic122—41no—
270Dic122—41no—
271Dic122—41no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#26725—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61#52511yescharacteristic
285C22×C61#54611yescharacteristic
286C22×C61#54611yescharacteristic
287C22×C61C2×#13511yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61#57711yescharacteristic
290C22×C61#57811yescharacteristic
291C4×C61—22no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61—41no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—81no—
300C4×C61#52911yescharacteristic
301C4×C61#53811yescharacteristic
302C4×C61#55511yescharacteristic
303C4×C61#55911yescharacteristic
304Dic61—42no—
305Dic61—81no—
306Dic61—81no—
307Dic61—81no—
308#2489—611no—
309#2360D6111yescharacteristic
310#2056—611no—
311#2035—611no—
312#2026—611no—
313#2001—611no—
314#1989—611no—
315#1933—611no—
316#1927—611no—
317#1923—611no—
318#1920—611no—
319#1878—611no—
320#1872—611no—
321#1823—611no—
322C4×#103—611no—
323C2×C61—42no—
324C2×C61#245911yescharacteristic
325C2×C61#310811yescharacteristic
326C2×C61#312011yescharacteristic
327C2×C61#329611yescharacteristic
328C2×C61#330011yescharacteristic
329C2×C61#331211yescharacteristic
330C2×C61#333111yescharacteristic
331C22×Q16—611no—
332C2×#125—611no—
333C2×C22⋊Q8—611no—
334C2×C22⋊Q8C2×D6111yescharacteristic
335C2×C4×Q8C2×D6111yescharacteristic
336C22×C4⋊C4C2×D6111yescharacteristic
337C2×#106—611no—
338C2×C4⋊C8—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×C22⋊C8—611no—
346C2×C22⋊C8—611no—
347C2×C4×C8—611no—
348#400—1221no—
349#397—1221no—
350#391—611no—
351#391—611no—
352#391C2×D6111yescharacteristic
353#389—21no—
354#389C2×D6111yescharacteristic
355#388—1221no—
356#387—21no—
357#385—21no—
358#382—1221no—
359#380—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4C2×D6111yescharacteristic
362C4×C22⋊C4—21no—
363C4×C22⋊C4C2×D6111yescharacteristic
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#336—611no—
373#336—611no—
374C61#2672511yescharacteristic, complemented, Sylow
375C22×Q8—611no—
376C22×Q8C22×D6111yescharacteristic
377C23×C4—21no—
378C23×C4C22×D6111yescharacteristic
379C2×Q16—1222no—
380C2×Q16—2441no—
381C2×Q16—2441no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—611no—
386#125—1222no—
387#125—2441no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—1222no—
391C22⋊Q8—2441no—
392C4×Q8—22no—
393C4×Q8—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4C22⋊D6111yescharacteristic
413C2×C22⋊C4C22⋊D6111yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418C2×C42C22×D6111yescharacteristic
419#106—1222no—
420C4⋊C8—2441no—
421#103—1222no—
422#103—1222no—
423#103—1222no—
424#103—1222no—
425#103—2441no—
426#103—2441no—
427C22⋊C8—2441no—
428C22⋊C8—2441no—
429C4×C8—2441no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95C22⋊D6111yescharacteristic
441#95C22⋊D6111yescharacteristic
442C24C2×C22⋊D6111yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8#1175822212yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
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—
479C2×C8—1221no—
480C2×C8—1221no—
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—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—612no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—2442no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C4⋊C4#1175822212yes—
509C4⋊C4#1175822412yes—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—2442no—
514C22⋊C4—2442no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—81no—
521C42—2442no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1451979712yes—
529C23C22×C22⋊D6111yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—2441no—
535Q8—2441no—
536Q8—2441no—
537Q8—2441no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
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—22no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
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—84no—
569C2×C4—1222no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4#1451980812yes—
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822411yescharacteristic
584C2×C4C2×#1175822211yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4#1451982211yescharacteristic
587C2×C4#1451982311yescharacteristic
588C8—2441no—
589C8—2441no—
590C8—2441no—
591C8—2441no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1684027311yescharacteristic
600C22#1684040011yescharacteristic
601C22#1684040611yescharacteristic
602C22C2×#1451979711yescharacteristic
603C22C2×#1451980811yescharacteristic
604C22#1684057311yescharacteristic
605C22#1684057711yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—4881no—
616C4—4881no—
617C4—4881no—
618C4—2442no—
619C4#1684029211yescharacteristic
620C4#1684034511yescharacteristic
621C4#1684047711yescharacteristic
622C4#1684049511yescharacteristic
623C2—42no—
624C2#2144975011yescharacteristic
625C2#2145589911yescharacteristic
626C2#2145603311yescharacteristic
627C2#2145765411yescharacteristic
628C2#2145770011yescharacteristic
629C2#2145787011yescharacteristic
630C2#2145812911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2360C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2360D61
hypocenterC61#26725
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #21447141C2semidirect
4. #21446970C2semidirect
6. #21446776C2semidirect
7. #21446494C2semidirect
8. #21446453C2semidirect
10. #21446401C2semidirect
11. #21446160C2semidirect
14. C4×#14519701C2semidirect
374. C61#26725semidirect

Maximal subgroups

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

subgroupindex
#214501602
#214471412
#214470122
#214469702
#214468272
#214467762
#214464942
#214464532
#214464272
#214464012
#214461602
#214461262
#214459052
C4×#145197012
C61×#23602
#2672561