Finite Group Databasebeta

← #29752684

Subgroups — #29752684

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450226C211yesmaximal, characteristic, complemented
2#21448835C211yesmaximal, characteristic
3#21447236C211yesmaximal, characteristic, complemented
4#21447194C211yesmaximal, characteristic, complemented
5#21447008C211yesmaximal, characteristic
6#21447002C211yesmaximal, characteristic
7#21446590C211yesmaximal, characteristic, complemented
8#21446546C211yesmaximal, characteristic, complemented
9#21446422C211yesmaximal, characteristic, complemented
10#21446395C211yesmaximal, characteristic, complemented
11#21446389C211yesmaximal, characteristic, complemented
12#21446081C211yesmaximal, characteristic, complemented
13#21446012C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#1824C211yesmaximal, characteristic, complemented
16C2×#14519751C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×C4×C22⋊D61—21no—
19C2×#14519711C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839704—21no—
31#16839687—21no—
32#16839663C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839656—21no—
37#16839652—21no—
38#16839641—21no—
39#16839633—21no—
40#16839629—21no—
41#16839627C2211yescharacteristic
42#16839622C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×C22⋊Dic61C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C61×#380C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C22⋊D61—21no—
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519751—22no—
65#14519751—22no—
66#14519739—22no—
67#14519739—41no—
68C4×C22⋊D61—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61D411yescharacteristic
77C2×C22⋊Dic61D411yescharacteristic
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519708—22no—
88#14519708—22no—
89#14519708—41no—
90#14519706—22no—
91#14519699—22no—
92#14519699—41no—
93#14519697—22no—
94#14519697—22no—
95C22⋊#92418—22no—
96C8×Dic61—41no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684D411yescharacteristic
106#14519684D411yescharacteristic
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—41no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C22⋊C8×C61—41no—
120#95×C61—21no—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61—81no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—24no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244#4112yes—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—81no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61#4112yes—
172C22⋊C4×C61—81no—
173C22⋊Dic61—22no—
174C22⋊Dic61—22no—
175C22⋊Dic61—42no—
176#11758198—24no—
177#11758198—42no—
178#11758198—42no—
179#11758198—81no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—81no—
186C4⋊Dic61#4112yes—
187#11758196—42no—
188#11758196—81no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—42no—
194C4×Dic61—42no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418#4112yes—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61#13512yes—
208C23×C61C22×D411yescharacteristic
209C22×D61—22no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
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—84no—
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C22⋊D61—48no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
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×Dic61AGL(1,ℤ/8)12yes—
264D244—22no—
265D244—22no—
266D244—41no—
267D244—41no—
268C4×D61—84no—
269C8×C61—41no—
270C8×C61—41no—
271#92418—22no—
272#92418—41no—
273#27031—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61#52511yescharacteristic
282C22×C61#54611yescharacteristic
283C22×C61#54711yescharacteristic
284C22×C61C2×#13511yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61#57711yescharacteristic
287C22×C61#57711yescharacteristic
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—81no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61—81no—
300C4×C61#52911yescharacteristic
301C4×C61#53911yescharacteristic
302C4×C61#55311yescharacteristic
303C4×C61#55611yescharacteristic
304Dic61—22no—
305Dic61—41no—
306Dic61—41no—
307Dic61—42no—
308Dic61—41no—
309Dic61—81no—
310Dic61—81no—
311#2495—611no—
312#2361—611no—
313#2073—611no—
314#2065—611no—
315#2034—611no—
316#2033—611no—
317#1949—611no—
318#1941—611no—
319#1923—611no—
320#1919—611no—
321#1918—611no—
322#1864—611no—
323#1853—611no—
324#1824D6111yescharacteristic, complemented
325C4×#102—611no—
326C2×C61—42no—
327C2×C61#246611yescharacteristic
328C2×C61#310511yescharacteristic
329C2×C61#312511yescharacteristic
330C2×C61#330111yescharacteristic
331C2×C61#331011yescharacteristic
332C2×C61#333711yescharacteristic
333C2×C61#335411yescharacteristic
334D61—82no—
335C2×#121—611no—
336C2×C4×D4—611no—
337C2×C4×D4—1221no—
338C22×C4⋊C4C2×D6111yescharacteristic
339C2×#107—611no—
340C2×#106—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×C4⋊C8—611no—
344C2×#102—611no—
345C2×#102—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×C22⋊C8—611no—
349C2×C22⋊C8C2×D6111yescharacteristic
350C2×C4×C8—611no—
351#400—1221no—
352#397—1221no—
353#390—611no—
354#389—611no—
355#389—611no—
356#389—611no—
357#388—1221no—
358#388—1221no—
359#386—1221no—
360#385—1221no—
361#383—1221no—
362#382—611no—
363#380—611no—
364#380C2×D6111yescharacteristic
365C4×C4⋊C4—611no—
366C4×C4⋊C4—611no—
367C4×C22⋊C4—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340C2×D6111yescharacteristic
373#340C2×D6111yescharacteristic
374#340C2×D6111yescharacteristic
375#340C2×D6111yescharacteristic
376#336—611no—
377#336—611no—
378C61#2703111yescharacteristic, complemented, Sylow
379C22×D4—611no—
380C22×D4—1221no—
381C23×C4—21no—
382C23×C4—1221no—
383C23×C4C22×D6111yescharacteristic
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388#121—1222no—
389#121—1222no—
390C4×D4—1222no—
391C4×D4—2441no—
392C4×D4—2444no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—41no—
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—611no—
408C2×C22⋊C4—611no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1221no—
417C2×C42—1221no—
418#107—1222no—
419#106—1222no—
420#106—1222no—
421#106—2441no—
422C4⋊C8—1222no—
423#102—1222no—
424#102—1222no—
425#102—1222no—
426#102—2441no—
427C22⋊C8—41no—
428C22⋊C8—1222no—
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—
442C24D4×D6111yescharacteristic
443C2×D4—612no—
444C2×D4—1224no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—2442no—
451C2×D4—2442no—
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—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—4881no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4D4×D6111yescharacteristic
476C22×C4C23×D6111yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—612no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—612no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—4881no—
504C4⋊C4—4881no—
505C4⋊C4—2442no—
506C4⋊C4#1175822212yes—
507C22⋊C4—81no—
508C22⋊C4—1224no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—4881no—
512C22⋊C4—2442no—
513C22⋊C4—2442no—
514C22⋊C4—2442no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42—1221no—
519C42—2442no—
520C42—2442no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—1222no—
528C23—2441no—
529C23—2441no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23#1451980012yes—
535C23C2×D4×D6111yescharacteristic
536D4—1222no—
537D4—1222no—
538D4—2448no—
539D4—2441no—
540D4—2441no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
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—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—612no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—4884no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4#41×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4#41×D6111yescharacteristic
586C2×C4C2×#1175822211yescharacteristic
587C2×C4#1451982211yescharacteristic
588C2×C4#1451982411yescharacteristic
589C8—41no—
590C8—41no—
591C8—1222no—
592C8—2441no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—4881no—
601C22—4881no—
602C22—4881no—
603C22—4881no—
604C22—2442no—
605C22—2442no—
606C22—2442no—
607C22#1684027011yescharacteristic
608C22#1684040211yescharacteristic
609C22#1684041011yescharacteristic
610C22C2×#1451980011yescharacteristic
611C22AGL(1,ℤ/8)×D6111yescharacteristic
612C22#1684056811yescharacteristic
613C22#1684057211yescharacteristic
614C4—41no—
615C4—42no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—1222no—
620C4—4881no—
621C4—4881no—
622C4—2441no—
623C4—2441no—
624C4—2441no—
625C4—2442no—
626C4#1684029011yescharacteristic
627C4#1684035511yescharacteristic
628C4#1684045711yescharacteristic
629C4#1684048011yescharacteristic
630C2—42no—
631C2—4882no—
632C2#2144985411yescharacteristic
633C2#2145586911yescharacteristic
634C2#2145609911yescharacteristic
635C2#2145771711yescharacteristic
636C2#2145784111yescharacteristic
637C2#2145819611yescharacteristic
638C2#2145840711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1824C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1824D61
hypocenterC61#27031
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
1. #21450226C2semidirect
3. #21447236C2semidirect
4. #21447194C2semidirect
7. #21446590C2semidirect
8. #21446546C2semidirect
9. #21446422C2semidirect
10. #21446395C2semidirect
11. #21446389C2semidirect
12. #21446081C2semidirect
13. #21446012C2semidirect
14. #21445633C2semidirect
15. C61×#1824C2semidirect
324. #1824D61semidirect
378. C61#27031semidirect

Maximal subgroups

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

subgroupindex
#214502262
#214488352
#214472362
#214471942
#214470082
#214470022
#214465902
#214465462
#214464222
#214463952
#214463892
#214460812
#214460122
#214456332
C61×#18242
#2703161