Finite Group Databasebeta

← #29748218

Subgroups — #29748218

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450160C211yesmaximal, characteristic
2#21447141C211yesmaximal, characteristic, complemented
3#21447019C211yesmaximal, characteristic
4#21446970C211yesmaximal, characteristic, complemented
5#21446827C211yesmaximal, characteristic
6#21446776C211yesmaximal, characteristic, complemented
7#21446494C211yesmaximal, characteristic, complemented
8#21446460C211yesmaximal, characteristic, complemented
9#21446427C211yesmaximal, characteristic
10#21446387C211yesmaximal, characteristic, complemented
11#21446160C211yesmaximal, characteristic, complemented
12#21446131C211yesmaximal, characteristic
13#21445914C211yesmaximal, characteristic
14C4×#14519701C211yesmaximal, characteristic, complemented
15C61×#2362C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
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
50C2×#117×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#388—21no—
54C61×#388—21no—
55C61×#387—21no—
56C4×C4⋊C4×C61—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
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#14519710—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×C61C2311yescharacteristic
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—22no—
101C4×Q8×C61—22no—
102C4×Q8×C61—41no—
103#117×C61—22no—
104#117×C61—41no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C42×C61—21no—
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×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61#4112yes—
167C4⋊C4×C61#4112yes—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—41no—
171C22⋊C4×C61—42no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—42no—
177C42×C61—42no—
178C42×C61—42no—
179C22⋊Dic61—42no—
180C22⋊Dic61—42no—
181#11758202—44no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61#4112yes—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191C4×Dic61—42no—
192C2×#92418—21no—
193C2×#92418—21no—
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—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61C22×D411yescharacteristic
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
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—41no—
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×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265Dic122—41no—
266Dic122—41no—
267Dic122—41no—
268Dic122—41no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#92418—41no—
273#26732—611nomaximal, Sylow
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61#52511yescharacteristic
282C22×C61#54611yescharacteristic
283C22×C61#54611yescharacteristic
284C22×C61C2×#13511yescharacteristic
285C22×C61#57511yescharacteristic
286C22×C61#57811yescharacteristic
287C22×C61#57811yescharacteristic
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—41no—
291C4×C61—42no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61#52911yescharacteristic
298C4×C61#53811yescharacteristic
299C4×C61#55511yescharacteristic
300C4×C61#55911yescharacteristic
301Dic61—42no—
302Dic61—81no—
303Dic61—81no—
304Dic61—81no—
305#2489—611no—
306#2362D6111yescharacteristic
307#2056—611no—
308#2036—611no—
309#2026—611no—
310#2001—611no—
311#1989—611no—
312#1933—611no—
313#1928—611no—
314#1923—611no—
315#1918—611no—
316#1878—611no—
317#1873—611no—
318#1825—611no—
319C4×#103—611no—
320C2×C61—81no—
321C2×C61#245911yescharacteristic
322C2×C61#310211yescharacteristic
323C2×C61#312711yescharacteristic
324C2×C61#330011yescharacteristic
325C2×C61#330211yescharacteristic
326C2×C61#331511yescharacteristic
327C2×C61#332711yescharacteristic
328C22×Q16—611no—
329C2×#125—611no—
330C2×C22⋊Q8—611no—
331C2×C22⋊Q8C2×D6111yescharacteristic
332C2×C4×Q8C2×D6111yescharacteristic
333C2×#117C2×D6111yescharacteristic
334C2×#107—611no—
335C2×C4⋊C8—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344C2×C4×C8—611no—
345#400—1221no—
346#397—1221no—
347#391—611no—
348#391—611no—
349#391C2×D6111yescharacteristic
350#389C2×D6111yescharacteristic
351#388—21no—
352#388—21no—
353#388—1221no—
354#387—21no—
355#382—1221no—
356#380—611no—
357C4×C4⋊C4—21no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4C2×D6111yescharacteristic
360C4×C22⋊C4C2×D6111yescharacteristic
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#336—611no—
370#336—611no—
371C61#2673211yescharacteristic, complemented, Sylow
372C22×Q8—611no—
373C22×Q8C22×D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375C2×Q16—1222no—
376C2×Q16—2441no—
377C2×Q16—2441no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8—611no—
382#125—1222no—
383#125—2441no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—1222no—
387C22⋊Q8—2441no—
388C4×Q8—22no—
389C4×Q8—41no—
390#117—22no—
391#117—41no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4C22⋊D6111yescharacteristic
409C2×C22⋊C4C22⋊D6111yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415C2×C42C22×D6111yescharacteristic
416#107—1222no—
417C4⋊C8—2441no—
418#103—1222no—
419#103—1222no—
420#103—1222no—
421#103—1222no—
422#103—2441no—
423#103—2441no—
424C22⋊C8—2441no—
425C22⋊C8—2441no—
426C4×C8—2441no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95C22⋊D6111yescharacteristic
438#95C22⋊D6111yescharacteristic
439C24C2×C22⋊D6111yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8#1175822212yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472Q16—2444no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—612no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4#1175822212yes—
502C4⋊C4#1175822412yes—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C22⋊C4—2442no—
508C22⋊C4—2442no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—2442no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C22×C22⋊D6111yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—2441no—
530Q8—2441no—
531Q8—2441no—
532Q8—2441no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
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—1222no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4#1451979712yes—
576C2×C4#41×D6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C2×C4C2×#1175822211yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C2×C4#1451982211yescharacteristic
581C2×C4#1451982311yescharacteristic
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C8—2441no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1684027311yescharacteristic
594C22#1684040011yescharacteristic
595C22#1684040611yescharacteristic
596C22C2×#1451979711yescharacteristic
597C22#1684056011yescharacteristic
598C22#1684057711yescharacteristic
599C22#1684057711yescharacteristic
600C4—22no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—4881no—
610C4—4881no—
611C4—4881no—
612C4—2442no—
613C4#1684029211yescharacteristic
614C4#1684034511yescharacteristic
615C4#1684047711yescharacteristic
616C4#1684049511yescharacteristic
617C2—81no—
618C2#2144975011yescharacteristic
619C2#2145583711yescharacteristic
620C2#2145611911yescharacteristic
621C2#2145770611yescharacteristic
622C2#2145773011yescharacteristic
623C2#2145791511yescharacteristic
624C2#2145808111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2362C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2362D61
hypocenterC61#26732
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #21447141C2semidirect
4. #21446970C2semidirect
6. #21446776C2semidirect
7. #21446494C2semidirect
8. #21446460C2semidirect
10. #21446387C2semidirect
11. #21446160C2semidirect
14. C4×#14519701C2semidirect
371. C61#26732semidirect

Maximal subgroups

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

subgroupindex
#214501602
#214471412
#214470192
#214469702
#214468272
#214467762
#214464942
#214464602
#214464272
#214463872
#214461602
#214461312
#214459142
C4×#145197012
C61×#23622
#2673261