Finite Group Databasebeta

← #29753310

Subgroups — #29753310

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#21450392C211yesmaximal, characteristic, complemented
2#21448894C211yesmaximal, characteristic
3#21447345C211yesmaximal, characteristic
4#21447132C211yesmaximal, characteristic
5#21447055C211yesmaximal, characteristic
6#21446681C211yesmaximal, characteristic, complemented
7#21446463C211yesmaximal, characteristic, complemented
8#21446457C211yesmaximal, characteristic, complemented
9#21446391C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446162C211yesmaximal, characteristic, complemented
12#21446155C211yesmaximal, characteristic
13#21445822C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#2015C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839704—21no—
29#16839703—21no—
30#16839669C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839655—21no—
36#16839655—21no—
37#16839647—21no—
38#16839644—21no—
39#16839637—21no—
40#16839631C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C22⋊C4×Dic61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×Q16×C61C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62Q8×Dic61—22no—
63Q8×Dic61—41no—
64D4×Dic61—22no—
65D4×Dic61—41no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84#14519711—22no—
85#14519710—41no—
86#14519706—41no—
87#14519701—22no—
88#14519701—22no—
89#14519701—41no—
90#14519701—41no—
91#14519698—22no—
92#14519698—41no—
93C8×Dic61—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×Q8×C61C2311yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107C22×D4×C61C2311yescharacteristic
108C2×Q16×C61—22no—
109C2×Q16×C61—22no—
110C2×Q16×C61—41no—
111C2×SD16×C61—22no—
112C2×SD16×C61—41no—
113C2×SD16×C61—41no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116#124×C61—22no—
117#124×C61—22no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#103×C61—41no—
123#102×C61—22no—
124#95×C61—21no—
125#95×C61—21no—
126C24×C61—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—21no—
138C2×Q8×C61#4112yes—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—81no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162Q16×C61—44no—
163SD16×C61—44no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61#4112yes—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C42×C61—21no—
175C42×C61—21no—
176C22⋊Dic61—42no—
177C22⋊Dic61—42no—
178C22⋊Dic61—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189C4⋊Dic61#4112yes—
190#11758196—42no—
191#11758196—81no—
192#11758196—81no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—42no—
198C4×Dic61—81no—
199C4×Dic61—81no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C23×C61—22no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61C22×D411yescharacteristic
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—41no—
225D4×C61—41no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
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×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
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—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—84no—
270C2×Dic61#13512yes—
271C2×Dic61#13712yes—
272C8×C61—22no—
273C8×C61—41no—
274#92418—41no—
275#92418—41no—
276#27073—611nomaximal, Sylow
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61#52511yescharacteristic
285C22×C61#54611yescharacteristic
286C22×C61#55011yescharacteristic
287C22×C61C2×#13511yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61#57611yescharacteristic
290C22×C61#57811yescharacteristic
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61#52911yescharacteristic
297C4×C61#53811yescharacteristic
298C4×C61#55111yescharacteristic
299C4×C61#55311yescharacteristic
300Dic61—22no—
301Dic61—22no—
302Dic61—41no—
303Dic61—41no—
304Dic61—42no—
305Dic61—81no—
306Dic61—81no—
307Dic61—81no—
308#2511—611no—
309#2367—611no—
310#2093—611no—
311#2055—611no—
312#2042—611no—
313#2015D6111yescharacteristic
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—82no—
324C2×C61#246711yescharacteristic
325C2×C61#310711yescharacteristic
326C2×C61#312411yescharacteristic
327C2×C61#330011yescharacteristic
328C2×C61#331411yescharacteristic
329C2×C61#339611yescharacteristic
330C2×C61#339811yescharacteristic
331C22×Q16C2×D6111yescharacteristic
332C22×SD16C2×D6111yescharacteristic
333C2×#124C2×D6111yescharacteristic
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×#103C2×D6111yescharacteristic
344C2×#102—611no—
345C2×#102—611no—
346C2×#102C2×D6111yescharacteristic
347C2×C4×C8—611no—
348#400—1221no—
349#400—1221no—
350#391—611no—
351#391—611no—
352#391—611no—
353#391C2×D6111yescharacteristic
354#390—611no—
355#389—611no—
356#388—1221no—
357#388—1221no—
358#387—1221no—
359#387—1221no—
360#385—1221no—
361#384—611no—
362C4×C4⋊C4—611no—
363C4×C4⋊C4—611no—
364C4×C4⋊C4—611no—
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#336C2×D6111yescharacteristic
374C61#2707311yescharacteristic, complemented, Sylow
375C22×Q8C22×D6111yescharacteristic
376C22×Q8C22×D6111yescharacteristic
377C22×D4C22×D6111yescharacteristic
378C23×C4—1221no—
379C2×Q16—22no—
380C2×Q16—22no—
381C2×Q16—41no—
382C2×SD16—22no—
383C2×SD16—41no—
384C2×SD16—41no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8C22×D6111yescharacteristic
388C22×C8C22×D6111yescharacteristic
389#124—22no—
390#124—22no—
391C4×Q8—1222no—
392C4×Q8—2441no—
393C4×D4—1222no—
394C4×D4—2441no—
395C2×C4⋊C4—611no—
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⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
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×C42C22×D6111yescharacteristic
417#107—1222no—
418#107—2441no—
419C4⋊C8—2441no—
420#103—41no—
421#103—1222no—
422#103—1222no—
423#103—2441no—
424#103—2441no—
425#102—22no—
426#102—1222no—
427#102—2441no—
428C4×C8—2441no—
429#95—21no—
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—
441#95—1221no—
442C24—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—21no—
453C2×Q8—21no—
454C2×Q8#1175822412yes—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4#1175822412yes—
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×C4—1221no—
472C22×C4—4881no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478Q16—44no—
479SD16—44no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8#1175822412yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—612no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—4881no—
505C4⋊C4—4881no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—4881no—
511C22⋊C4—2442no—
512C22⋊C4—2442no—
513C42—21no—
514C42—21no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42—1221no—
519C42—4881no—
520C42—4881no—
521C42—2442no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D6111yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—41no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
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—612no—
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—1222no—
570C2×C4—4884no—
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—2441no—
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822411yescharacteristic
584C2×C4C2×#1175822411yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4#1451982211yescharacteristic
587C2×C4#1451982211yescharacteristic
588C8—22no—
589C8—41no—
590C8—2441no—
591C8—2441no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#1684026811yescharacteristic
600C22#1684040911yescharacteristic
601C22#1684043211yescharacteristic
602C22#135×D6111yescharacteristic
603C22#137×D6111yescharacteristic
604C22#1684056411yescharacteristic
605C22#1684057811yescharacteristic
606C4—41no—
607C4—42no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—1222no—
612C4—1222no—
613C4—4881no—
614C4—4881no—
615C4—4881no—
616C4—2441no—
617C4—2441no—
618C4—2442no—
619C4#1684029311yescharacteristic
620C4#1684034711yescharacteristic
621C4#1684044311yescharacteristic
622C4#1684045511yescharacteristic
623C2—82no—
624C2#2144986511yescharacteristic
625C2#2145588311yescharacteristic
626C2#2145608211yescharacteristic
627C2#2145771311yescharacteristic
628C2#2145790711yescharacteristic
629C2#2145882811yescharacteristic
630C2#2145884711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2015C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2015D61
hypocenterC61#27073
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2015

Complements

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

NH ≅ G/Ntype
1. #21450392C2semidirect
6. #21446681C2semidirect
7. #21446463C2semidirect
8. #21446457C2semidirect
9. #21446391C2semidirect
10. #21446375C2semidirect
11. #21446162C2semidirect
13. #21445822C2semidirect
374. C61#27073semidirect

Maximal subgroups

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

subgroupindex
#214503922
#214488942
#214473452
#214471322
#214470552
#214466812
#214464632
#214464572
#214463912
#214463752
#214461622
#214461552
#214458222
#102×Dic612
C61×#20152
#2707361