Finite Group Databasebeta

← #29753248

Subgroups — #29753248

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450419C211yesmaximal, characteristic
2#21448879C211yesmaximal, characteristic, complemented
3#21447365C211yesmaximal, characteristic, complemented
4#21447133C211yesmaximal, characteristic
5#21446902C211yesmaximal, characteristic
6#21446670C211yesmaximal, characteristic
7#21446462C211yesmaximal, characteristic, complemented
8#21446440C211yesmaximal, characteristic
9#21446401C211yesmaximal, characteristic, complemented
10#21446377C211yesmaximal, characteristic, complemented
11#21446172C211yesmaximal, characteristic
12#21446172C211yesmaximal, characteristic
13#21445832C211yesmaximal, characteristic, complemented
14#103×Dic61C211yesmaximal, characteristic, complemented
15C61×#2043C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839689—21no—
29#16839688—21no—
30#16839669C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839663C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839661—21no—
36#16839654—21no—
37#16839652—21no—
38#16839648—21no—
39#16839648—21no—
40#16839627C2211yescharacteristic
41C4⋊C4×Dic61—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C2×#11758201—22no—
61C2×#11758201—22no—
62C2×#11758201—22no—
63C2×#11758201—41no—
64C2×#11758201—41no—
65C2×#11758201—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68Q8×Dic61—22no—
69Q8×Dic61—22no—
70Q8×Dic61—41no—
71Q8×Dic61—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519711—41no—
92#14519708—22no—
93#14519706—41no—
94#14519701—22no—
95#14519701—41no—
96#14519699—22no—
97#14519699—41no—
98C8×Dic61—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4⋊Q8×C61—22no—
114C4⋊Q8×C61—22no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#103×C61—22no—
122#103×C61—22no—
123#103×C61—41no—
124#103×C61—41no—
125#95×C61—21no—
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—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61#4112yes—
138C2×Q8×C61#4112yes—
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×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—81no—
172C42×C61—21no—
173C42×C61—21no—
174#11758201—44no—
175#11758201—44no—
176#11758198—42no—
177#11758198—42no—
178#11758198—81no—
179#11758198—81no—
180#11758198—81no—
181#11758198—81no—
182C4⋊Dic61—22no—
183C4⋊Dic61—21no—
184C4⋊Dic61—22no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—42no—
190C4⋊Dic61—42no—
191C4⋊Dic61#4112yes—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C4×Dic61—42no—
199C4×Dic61—81no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418#4112yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61—22no—
217Q8×C61—22no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—41no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
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#13512yes—
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
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#13712yes—
267D244—41no—
268D244—41no—
269D244—41no—
270D244—41no—
271C8×C61—41no—
272C8×C61—41no—
273#92418—22no—
274#92418—41no—
275#27069—611nomaximal, Sylow
276C22×C61#52511yescharacteristic
277C22×C61#54611yescharacteristic
278C22×C61#55011yescharacteristic
279C22×C61C2×#13511yescharacteristic
280C22×C61C2×#13711yescharacteristic
281C22×C61#57711yescharacteristic
282C22×C61#57711yescharacteristic
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C4×C61—22no—
291C4×C61—41no—
292C4×C61—42no—
293C4×C61—42no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61#52911yescharacteristic
297C4×C61#53811yescharacteristic
298C4×C61#55111yescharacteristic
299C4×C61#55311yescharacteristic
300Dic61—22no—
301Dic61—41no—
302Dic61—41no—
303Dic61—42no—
304Dic61—41no—
305Dic61—81no—
306Dic61—81no—
307#2513—611no—
308#2365—611no—
309#2097—611no—
310#2055—611no—
311#2043D6111yescharacteristic, complemented
312#2013—611no—
313#1968—611no—
314#1928—611no—
315#1925—611no—
316#1920—611no—
317#1916—611no—
318#1880—611no—
319#1880—611no—
320#1805—611no—
321C4×#103—611no—
322C2×C61#246711yescharacteristic
323C2×C61#310811yescharacteristic
324C2×C61#312611yescharacteristic
325C2×C61#330311yescharacteristic
326C2×C61#332011yescharacteristic
327C2×C61#339411yescharacteristic
328C2×C61#340011yescharacteristic
329D61—161no—
330C22×SD16—611no—
331C22×SD16—611no—
332C2×C4⋊Q8C2×D6111yescharacteristic
333C2×C4×Q8—611no—
334C2×C4×Q8—611no—
335C2×#107—611no—
336C2×#106—611no—
337C2×C4⋊C8—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103C2×D6111yescharacteristic
341C2×#103C2×D6111yescharacteristic
342C2×#103C2×D6111yescharacteristic
343C2×#103C2×D6111yescharacteristic
344C2×#102—611no—
345C2×#102—611no—
346C2×C4×C8—611no—
347#397—1221no—
348#397—1221no—
349#391—611no—
350#391—611no—
351#390—611no—
352#390—611no—
353#390—611no—
354#389—1221no—
355#389C2×D6111yescharacteristic
356#388—1221no—
357#388—1221no—
358#387—1221no—
359#387—1221no—
360#382—611no—
361C4×C4⋊C4—611no—
362C4×C4⋊C4—611no—
363C4×C4⋊C4—611no—
364C4×C4⋊C4—1221no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#336—611no—
370#336—611no—
371#336—611no—
372#336C2×D6111yescharacteristic
373C61#2706911yescharacteristic, complemented, Sylow
374C22×Q8C22×D6111yescharacteristic
375C22×Q8C22×D6111yescharacteristic
376C22×D4—611no—
377C2×SD16—1222no—
378C2×SD16—1222no—
379C2×SD16—1222no—
380C2×SD16—2441no—
381C2×SD16—2441no—
382C2×SD16—2441no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8C22×D6111yescharacteristic
386C22×C8C22×D6111yescharacteristic
387C4⋊Q8—22no—
388C4⋊Q8—22no—
389C4×Q8—1222no—
390C4×Q8—1222no—
391C4×Q8—2441no—
392C4×Q8—2441no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418#107—2441no—
419#106—1222no—
420C4⋊C8—2441no—
421#103—22no—
422#103—22no—
423#103—41no—
424#103—41no—
425#103—1222no—
426#103—2441no—
427#102—1222no—
428#102—2441no—
429C4×C8—2441no—
430#95—21no—
431#95—611no—
432#95—611no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441C24—1221no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8#1175822412yes—
453C2×Q8#1175822412yes—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4C23×D6111yescharacteristic
474SD16—2444no—
475SD16—2444no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—612no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—81no—
495C4⋊C4—612no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C22⋊C4—4881no—
506C22⋊C4—4881no—
507C22⋊C4—4881no—
508C22⋊C4—4881no—
509C22⋊C4—2442no—
510C22⋊C4—2442no—
511C42—21no—
512C42—21no—
513C42—1221no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C42—4881no—
518C42—2442no—
519C42—2442no—
520C42—2442no—
521C23—2441no—
522C23—2441no—
523C23—2441no—
524C23—2441no—
525C23—2441no—
526C23—2441no—
527C23—2441no—
528C23C2×D4×D6111yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537D4—2441no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
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—612no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1222no—
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#1451980012yes—
580C2×C4#41×D6111yescharacteristic
581C2×C4#41×D6111yescharacteristic
582C2×C4C2×#1175822411yescharacteristic
583C2×C4C2×#1175822411yescharacteristic
584C2×C4#1451982211yescharacteristic
585C2×C4#1451982311yescharacteristic
586C8—41no—
587C8—41no—
588C8—1222no—
589C8—2441no—
590C22—4881no—
591C22—4881no—
592C22—4881no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—4881no—
597C22#1684026811yescharacteristic
598C22#1684040511yescharacteristic
599C22#1684043011yescharacteristic
600C22C2×#1451980011yescharacteristic
601C22#137×D6111yescharacteristic
602C22#1684056811yescharacteristic
603C22#1684057111yescharacteristic
604C4—22no—
605C4—42no—
606C4—42no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—1222no—
611C4—4881no—
612C4—4881no—
613C4—2441no—
614C4—2441no—
615C4—2441no—
616C4—2442no—
617C4#1684028911yescharacteristic
618C4#1684034911yescharacteristic
619C4#1684043511yescharacteristic
620C4#1684045511yescharacteristic
621C2—9761no—
622C2#2144986311yescharacteristic
623C2#2145590011yescharacteristic
624C2#2145610411yescharacteristic
625C2#2145774611yescharacteristic
626C2#2145798911yescharacteristic
627C2#2145881411yescharacteristic
628C2#2145886711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2043C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2043D61
hypocenterC61#27069
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
2. #21448879C2semidirect
3. #21447365C2semidirect
7. #21446462C2semidirect
9. #21446401C2semidirect
10. #21446377C2semidirect
13. #21445832C2semidirect
14. #103×Dic61C2semidirect
15. C61×#2043C2semidirect
311. #2043D61semidirect
373. C61#27069semidirect

Maximal subgroups

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

subgroupindex
#214504192
#214488792
#214473652
#214471332
#214469022
#214466702
#214464622
#214464402
#214464012
#214463772
#214461722
#214458322
#103×Dic612
C61×#20432
#2706961