Finite Group Databasebeta

← #29753309

Subgroups — #29753309

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#21450394C211yesmaximal, characteristic, complemented
2#21447347C211yesmaximal, characteristic
3#21447136C211yesmaximal, characteristic
4#21447055C211yesmaximal, characteristic
5#21446913C211yesmaximal, characteristic
6#21446679C211yesmaximal, characteristic, complemented
7#21446460C211yesmaximal, characteristic, complemented
8#21446457C211yesmaximal, characteristic, complemented
9#21446390C211yesmaximal, characteristic, complemented
10#21446378C211yesmaximal, characteristic, complemented
11#21446160C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic
13#21445824C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic
15C61×#2367C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839704—21no—
31#16839701—21no—
32#16839668C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839668C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839648—21no—
37#16839645—21no—
38#16839632C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×#124×C61C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#390C2211yescharacteristic
53C61×#388—21no—
54C61×#388—21no—
55C61×#385—21no—
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61—21no—
59C22×Dic122C2311yescharacteristic
60C2×#11758202—22no—
61C2×#11758202—22no—
62C2×#11758202—41no—
63C2×#11758200—22no—
64C2×#11758200—41no—
65C2×#11758200—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×C4×Dic61—21no—
77#14519710—22no—
78#14519710—41no—
79C4⋊#92418—41no—
80#14519701—22no—
81#14519701—41no—
82#14519701—41no—
83#14519700—22no—
84#14519700—41no—
85#14519698—22no—
86#14519698—22no—
87#14519698—41no—
88C4×#92418—41no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100#124×C61—22no—
101#124×C61—22no—
102C4×Q8×C61—22no—
103C4×Q8×C61—41no—
104C4×D4×C61—22no—
105C4×D4×C61—41no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
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#4112yes—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—81no—
145C22×C4×C61C2×D411yescharacteristic
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—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61#4112yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—42no—
172C22⋊C4×C61—42no—
173C22⋊C4×C61—81no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—42no—
181C42×C61—81no—
182#11758202—44no—
183#11758200—44no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—42no—
191C4⋊Dic61—81no—
192#11758196—81no—
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×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—
218D4×C61—22no—
219D4×C61—22no—
220D4×C61—41no—
221D4×C61—41no—
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—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
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—84no—
248C2×C4×C61#13512yes—
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—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—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268Dic122—41no—
269Dic122—41no—
270Dic122—41no—
271Dic122—41no—
272#92418—22no—
273#92418—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—22no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—81no—
300C4×C61#52911yescharacteristic
301C4×C61#53811yescharacteristic
302C4×C61#55111yescharacteristic
303C4×C61#55311yescharacteristic
304Dic61—81no—
305Dic61—81no—
306Dic61—81no—
307Dic61—81no—
308#2511—611no—
309#2367D6111yescharacteristic
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—82no—
324C2×C61#246711yescharacteristic
325C2×C61#310711yescharacteristic
326C2×C61#312411yescharacteristic
327C2×C61#330011yescharacteristic
328C2×C61#331411yescharacteristic
329C2×C61#339611yescharacteristic
330C2×C61#339811yescharacteristic
331C22×Q16—611no—
332C22×SD16—611no—
333C2×#124C2×D6111yescharacteristic
334C2×C4×Q8C2×D6111yescharacteristic
335C2×C4×D4C2×D6111yescharacteristic
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—1221no—
349#400—1221no—
350#391—611no—
351#391—611no—
352#391—611no—
353#391C2×D6111yescharacteristic
354#390C2×D6111yescharacteristic
355#389—611no—
356#388—21no—
357#388—21no—
358#387—1221no—
359#387—1221no—
360#385—21no—
361#384—611no—
362C4×C4⋊C4—611no—
363C4×C4⋊C4C2×D6111yescharacteristic
364C4×C4⋊C4C2×D6111yescharacteristic
365C4×C22⋊C4—21no—
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×D4C22×D6111yescharacteristic
378C23×C4—21no—
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—22no—
390#124—22no—
391C4×Q8—22no—
392C4×Q8—41no—
393C4×D4—22no—
394C4×D4—41no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C4⋊C4C22⋊D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C42—21no—
412C2×C42—21no—
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—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95C22⋊D6111yescharacteristic
441#95C22⋊D6111yescharacteristic
442C24—21no—
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—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4#1175822212yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
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—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
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#1175822412yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—81no—
521C42—4881no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C22×C22⋊D6111yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—2441no—
535Q8—2441no—
536Q8—2441no—
537Q8—2441no—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—41no—
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—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—2441no—
569C2×C4—2441no—
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#1451979712yes—
581C2×C4#1451980712yes—
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822411yescharacteristic
584C2×C4C2×#1175822211yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4#1451982211yescharacteristic
587C2×C4#1451982311yescharacteristic
588C8—1222no—
589C8—2441no—
590C8—2441no—
591C8—2441no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#1684027311yescharacteristic
600C22#1684040611yescharacteristic
601C22#1684043111yescharacteristic
602C22C2×#1451979711yescharacteristic
603C22C2×#1451980711yescharacteristic
604C22#1684056511yescharacteristic
605C22#1684057711yescharacteristic
606C4—22no—
607C4—22no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—4881no—
616C4—4881no—
617C4—4881no—
618C4—4881no—
619C4#1684029211yescharacteristic
620C4#1684034511yescharacteristic
621C4#1684044111yescharacteristic
622C4#1684045811yescharacteristic
623C2—82no—
624C2#2144986411yescharacteristic
625C2#2145589011yescharacteristic
626C2#2145608111yescharacteristic
627C2#2145770611yescharacteristic
628C2#2145790011yescharacteristic
629C2#2145882711yescharacteristic
630C2#2145885011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2367C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2367D61
hypocenterC61#27073
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #21450394C2semidirect
6. #21446679C2semidirect
7. #21446460C2semidirect
8. #21446457C2semidirect
9. #21446390C2semidirect
10. #21446378C2semidirect
11. #21446160C2semidirect
13. #21445824C2semidirect
374. C61#27073semidirect

Maximal subgroups

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

subgroupindex
#214503942
#214473472
#214471362
#214470552
#214469132
#214466792
#214464602
#214464572
#214463902
#214463782
#214461602
#214461532
#214458242
C4×#145196982
C61×#23672
#2707361