Finite Group Databasebeta

← #29753286

Subgroups — #29753286

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450407C211yesmaximal, characteristic, complemented
2#21448889C211yesmaximal, characteristic, complemented
3#21447352C211yesmaximal, characteristic, complemented
4#21447058C211yesmaximal, characteristic, complemented
5#21446894C211yesmaximal, characteristic
6#21446674C211yesmaximal, characteristic, complemented
7#21446451C211yesmaximal, characteristic, complemented
8#21446449C211yesmaximal, characteristic, complemented
9#21446402C211yesmaximal, characteristic, complemented
10#21446373C211yesmaximal, characteristic, complemented
11#21446177C211yesmaximal, characteristic, complemented
12#21446169C211yesmaximal, characteristic
13#21445825C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2054C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519770—21no—
18C2×#14519764C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839687—21no—
31#16839685—21no—
32#16839667C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839656—21no—
37#16839655—21no—
38#16839650—21no—
39#16839634—21no—
40#16839632C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44C22⋊C4×Dic61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×D8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C23×Dic61—21no—
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200—41no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519770—44no—
67#14519764—22no—
68#14519764—22no—
69C4×Dic122—22no—
70C4×Dic122—41no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519708—22no—
95#14519708—41no—
96#14519702—41no—
97#14519702—41no—
98#14519700—22no—
99#14519698—22no—
100#14519698—22no—
101#14519698—41no—
102C8×Dic61—41no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C2×D8×C61—22no—
113C2×D8×C61—41no—
114C2×D8×C61—41no—
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C4⋊C8×C61—41no—
125#102×C61—22no—
126#102×C61—41no—
127C24×C61—21no—
128C24×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61#4112yes—
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×Dic61—21no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—81no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122#4112yes—
163D8×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—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C22⋊C4×C61—42no—
175C22⋊C4×C61—42no—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—81no—
178C42×C61—21no—
179C42×C61—21no—
180C22⋊Dic61—24no—
181C22⋊Dic61—24no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185C22⋊Dic61—81no—
186C22⋊Dic61—81no—
187#11758200—44no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—81no—
195C4⋊Dic61#4112yes—
196#11758196—22no—
197#11758196—22no—
198#11758196—42no—
199#11758196—42no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—81no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418#4112yes—
213C23×C61—22no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61C22×D411yescharacteristic
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
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#13512yes—
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—84no—
278C2×Dic61AGL(1,ℤ/8)12yes—
279Dic122—22no—
280Dic122—22no—
281Dic122—41no—
282Dic122—41no—
283C8×C61—41no—
284C8×C61—41no—
285#92418—22no—
286#92418—41no—
287#27071—611nomaximal, Sylow
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61#52511yescharacteristic
303C22×C61#54611yescharacteristic
304C22×C61#55011yescharacteristic
305C22×C61C2×#13511yescharacteristic
306C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C61#57711yescharacteristic
308C22×C61#57711yescharacteristic
309C4×C61—22no—
310C4×C61—41no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61#52911yescharacteristic
314C4×C61#53811yescharacteristic
315C4×C61#55111yescharacteristic
316C4×C61#55311yescharacteristic
317Dic61—22no—
318Dic61—41no—
319Dic61—41no—
320Dic61—42no—
321Dic61—41no—
322Dic61—42no—
323Dic61—81no—
324Dic61—81no—
325#2512—611no—
326#2367—611no—
327#2094—611no—
328#2054D6111yescharacteristic
329#2042—611no—
330#2012—611no—
331#1968—611no—
332#1927—611no—
333#1926—611no—
334#1920—611no—
335#1916—611no—
336#1880—611no—
337#1879—611no—
338#1803—611no—
339C4×#102—611no—
340C2×C61—82no—
341C2×C61—161no—
342C2×C61#246711yescharacteristic
343C2×C61#310611yescharacteristic
344C2×C61#312511yescharacteristic
345C2×C61#330111yescharacteristic
346C2×C61#330611yescharacteristic
347C2×C61#339411yescharacteristic
348C2×C61#339711yescharacteristic
349C22×SD16—611no—
350C22×D8C2×D6111yescharacteristic
351C2×#126—1221no—
352C2×#124—611no—
353C2×C4×Q8—611no—
354C2×C4×D4—611no—
355C2×#106—611no—
356C2×#106—611no—
357C2×C4⋊C8C2×D6111yescharacteristic
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102C2×D6111yescharacteristic
365C2×#102C2×D6111yescharacteristic
366C2×C4×C8—611no—
367#397—1221no—
368#396—1221no—
369#391—611no—
370#390—611no—
371#390—611no—
372#390C2×D6111yescharacteristic
373#390C2×D6111yescharacteristic
374#389—611no—
375#388—1221no—
376#388—1221no—
377#388—1221no—
378#385—1221no—
379#384—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—611no—
382C4×C4⋊C4—611no—
383C4×C22⋊C4—1221no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340C2×D6111yescharacteristic
388#336—611no—
389#336—611no—
390#336—611no—
391#336—611no—
392C61#2707111yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×D4C22×D6111yescharacteristic
395C22×D4C22×D6111yescharacteristic
396C23×C4—1221no—
397C2×SD16—1222no—
398C2×SD16—1222no—
399C2×SD16—2441no—
400C2×D8—22no—
401C2×D8—41no—
402C2×D8—41no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407#126—2444no—
408#124—1222no—
409#124—1222no—
410C4×Q8—1222no—
411C4×Q8—2441no—
412C4×D4—1222no—
413C4×D4—2441no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42C22×D6111yescharacteristic
442#106—1222no—
443#106—2441no—
444C4⋊C8—41no—
445#103—1222no—
446#103—2441no—
447#103—2441no—
448#102—22no—
449#102—41no—
450#102—1222no—
451#102—1222no—
452#102—2441no—
453C4×C8—2441no—
454#95—611no—
455#95—611no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461C24—21no—
462C24—21no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4#1175822412yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—4881no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497SD16—2444no—
498D8—44no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—612no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—612no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1222no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4881no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—81no—
531C22⋊C4—1224no—
532C22⋊C4—1224no—
533C22⋊C4—4881no—
534C22⋊C4—4881no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C22⋊C4—2442no—
538C42—21no—
539C42—21no—
540C42—1221no—
541C42—1221no—
542C42—1221no—
543C42—1221no—
544C42—4881no—
545C42—2442no—
546C42—2442no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C2×D4×D6111yescharacteristic
562Q8—1222no—
563Q8—1222no—
564Q8—2441no—
565Q8—2441no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—41no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—612no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—4884no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4#1451979612yes—
611C2×C4#41×D6111yescharacteristic
612C2×C4#41×D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C2×C4#1451982211yescharacteristic
616C2×C4#1451982211yescharacteristic
617C8—41no—
618C8—41no—
619C8—1222no—
620C8—2441no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22#1684027211yescharacteristic
636C22#1684040111yescharacteristic
637C22#1684043011yescharacteristic
638C22C2×#1451979611yescharacteristic
639C22AGL(1,ℤ/8)×D6111yescharacteristic
640C22#1684056911yescharacteristic
641C22#1684057211yescharacteristic
642C4—22no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—1222no—
647C4—4881no—
648C4—4881no—
649C4—2441no—
650C4—2441no—
651C4—2442no—
652C4—2442no—
653C4—2441no—
654C4#1684028911yescharacteristic
655C4#1684034711yescharacteristic
656C4#1684044211yescharacteristic
657C4#1684045711yescharacteristic
658C2—82no—
659C2—161no—
660C2#2144987111yescharacteristic
661C2#2145587611yescharacteristic
662C2#2145609311yescharacteristic
663C2#2145772711yescharacteristic
664C2#2145777511yescharacteristic
665C2#2145881911yescharacteristic
666C2#2145884211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2054C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenter#2054D61
hypocenterC61#27071
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2054

Complements

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

NH ≅ G/Ntype
1. #21450407C2semidirect
2. #21448889C2semidirect
3. #21447352C2semidirect
4. #21447058C2semidirect
6. #21446674C2semidirect
7. #21446451C2semidirect
8. #21446449C2semidirect
9. #21446402C2semidirect
10. #21446373C2semidirect
11. #21446177C2semidirect
13. #21445825C2semidirect
14. #102×Dic61C2semidirect
392. C61#27071semidirect

Maximal subgroups

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

subgroupindex
#214504072
#214488892
#214473522
#214470582
#214468942
#214466742
#214464512
#214464492
#214464022
#214463732
#214461772
#214461692
#214458252
#102×Dic612
C61×#20542
#2707161