Finite Group Databasebeta

← #29779378

Subgroups — #29779378

Order 15616.

780 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450407C211yesmaximal, characteristic, complemented
2#21449671C211yesmaximal, characteristic, complemented
3#21447313C211yesmaximal, characteristic, complemented
4#21447222C211yesmaximal, characteristic, complemented
5#21447127C211yesmaximal, characteristic
6#21446955C211yesmaximal, characteristic, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446518C211yesmaximal, characteristic, complemented
9#21446513C211yesmaximal, characteristic, complemented
10#21446513C211yesmaximal, characteristic, complemented
11#21446413C211yesmaximal, characteristic, complemented
12#21446169C211yesmaximal, characteristic
13#21445693C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic, complemented
15C61×#1970C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519770—21no—
18C2×#14519754—21no—
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27#16839687—21no—
28#16839685—21no—
29#16839685—21no—
30#16839674—21no—
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839655—21no—
38#16839637—21no—
39#16839632C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×D8×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C22×Dic122—21no—
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519777—22no—
64#14519777—41no—
65#14519770—44no—
66#14519754—44no—
67D4×Dic61—22no—
68D4×Dic61—41no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196D411yescharacteristic
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
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×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519706—22no—
92#14519698—22no—
93#14519698—22no—
94#14519698—22no—
95#14519698—41no—
96#14519687—22no—
97#14519687—41no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C22×D4×C61C2311yescharacteristic
106C2×D8×C61—22no—
107C2×D8×C61—22no—
108C2×D8×C61—41no—
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61C2311yescharacteristic
118#107×C61—22no—
119#102×C61—22no—
120#102×C61—22no—
121C4×C8×C61—41no—
122C24×C61—21no—
123C24×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×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#4112yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61—81no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—81no—
155D8×C61—44no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61#4112yes—
161C2×C8×C61#4112yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C22⋊C4×C61—42no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—42no—
170C42×C61—21no—
171C42×C61—21no—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176C22⋊Dic61—81no—
177C22⋊Dic61—81no—
178C22⋊Dic61—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188C4⋊Dic61—42no—
189C4⋊Dic61—81no—
190C4⋊Dic61#4112yes—
191#11758196—22no—
192#11758196—22no—
193#11758196—42no—
194#11758196—42no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—42no—
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×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61C22×D411yescharacteristic
221D4×C61—22no—
222D4×C61—22no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
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×C61AGL(1,ℤ/8)12yes—
240C2×C4×C61AGL(1,ℤ/8)12yes—
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
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—21no—
259C2×Dic61—21no—
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—41no—
270C2×Dic61—41no—
271C2×Dic61—84no—
272Dic122—84no—
273C8×C61—22no—
274C8×C61—22no—
275#92418—41no—
276#92418—41no—
277#29095—611nomaximal, Sylow
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61#53211yescharacteristic
293C22×C61#54611yescharacteristic
294C22×C61#54811yescharacteristic
295C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57611yescharacteristic
298C22×C61#57811yescharacteristic
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61#52911yescharacteristic
304C4×C61#53911yescharacteristic
305C4×C61#55111yescharacteristic
306C4×C61#55611yescharacteristic
307Dic61—41no—
308Dic61—41no—
309Dic61—41no—
310Dic61—42no—
311Dic61—41no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
315#2512—611no—
316#2452—611no—
317#2087—611no—
318#2071—611no—
319#2054—611no—
320#2023—611no—
321#1990—611no—
322#1970D6111yescharacteristic
323#1936—611no—
324#1936—611no—
325#1936—611no—
326#1921—611no—
327#1879—611no—
328#1771—611no—
329#1762—611no—
330C2×C61—82no—
331C2×C61—161no—
332C2×C61#256011yescharacteristic
333C2×C61#317511yescharacteristic
334C2×C61#318411yescharacteristic
335C2×C61#330811yescharacteristic
336C2×C61#331811yescharacteristic
337C2×C61#336611yescharacteristic
338C2×C61#336711yescharacteristic
339C22×D8C2×D6111yescharacteristic
340C2×C4⋊Q8—611no—
341C2×#126—1221no—
342C2×C22⋊Q8—1221no—
343C2×C4×D4—611no—
344C2×#107C2×D6111yescharacteristic
345C2×C4⋊C8—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×#102—611no—
349C2×#102—611no—
350C2×#102C2×D6111yescharacteristic
351C2×#102C2×D6111yescharacteristic
352C2×#97—611no—
353C2×#97—611no—
354C2×C4×C8C2×D6111yescharacteristic
355#397—1221no—
356#396—1221no—
357#396—1221no—
358#393—1221no—
359#390—611no—
360#390—611no—
361#390—611no—
362#390C2×D6111yescharacteristic
363#390C2×D6111yescharacteristic
364#389—611no—
365#389—611no—
366#389—611no—
367#388—1221no—
368#385—1221no—
369#384—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#336—611no—
381#336—611no—
382C61#2909511yescharacteristic, complemented, Sylow
383C22×Q8—1221no—
384C22×D4C22×D6111yescharacteristic
385C22×D4C22×D6111yescharacteristic
386C23×C4—1221no—
387C2×D8—22no—
388C2×D8—22no—
389C2×D8—41no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—1222no—
395C4⋊Q8—2441no—
396#126—2444no—
397C22⋊Q8—2444no—
398C4×D4—1222no—
399C4×D4—2441no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429#107—22no—
430C4⋊C8—1222no—
431#102—22no—
432#102—22no—
433#102—1222no—
434#102—1222no—
435#102—1222no—
436#102—2441no—
437#97—1222no—
438#97—2441no—
439C4×C8—41no—
440#95—611no—
441#95—611no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446C24—21no—
447C24—21no—
448C2×Q8—4881no—
449C2×Q8—2442no—
450C2×Q8—2442no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4#1175822212yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—4881no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4D4×D6111yescharacteristic
477C22×C4D4×D6111yescharacteristic
478C22×C4C23×D6111yescharacteristic
479D8—44no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8#1175822412yes—
491C2×C8#1175822412yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—612no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—4881no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4—2442no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—4881no—
517C22⋊C4—4881no—
518C22⋊C4—4881no—
519C22⋊C4—2442no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C42—21no—
524C42—21no—
525C42—1221no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—2442no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×D4×D6111yescharacteristic
545Q8—4884no—
546D4—22no—
547D4—22no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—1221no—
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—1221no—
575C2×C4—1221no—
576C2×C4—1222no—
577C2×C4—4884no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4#1451980112yes—
590C2×C4#1451980112yes—
591C2×C4#41×D6111yescharacteristic
592C2×C4C2×#1175822411yescharacteristic
593C2×C4C2×#1175822411yescharacteristic
594C2×C4C2×#1175822211yescharacteristic
595C2×C4#1451982211yescharacteristic
596C2×C4#1451982411yescharacteristic
597C8—22no—
598C8—22no—
599C8—2441no—
600C8—2441no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#1684031011yescharacteristic
616C22#1684040111yescharacteristic
617C22#1684042111yescharacteristic
618C22C2×#1451980111yescharacteristic
619C22C2×#1451980111yescharacteristic
620C22#1684056411yescharacteristic
621C22#1684057611yescharacteristic
622C4—22no—
623C4—22no—
624C4—81no—
625C4—81no—
626C4—4881no—
627C4—4881no—
628C4—4881no—
629C4—2441no—
630C4—2442no—
631C4—2441no—
632C4—2441no—
633C4—2441no—
634C4#1684029211yescharacteristic
635C4#1684035711yescharacteristic
636C4#1684043811yescharacteristic
637C4#1684048111yescharacteristic
638C2—82no—
639C2—161no—
640C2#2145096811yescharacteristic
641C2#2145645511yescharacteristic
642C2#2145655511yescharacteristic
643C2#2145781611yescharacteristic
644C2#2145796611yescharacteristic
645C2#2145852411yescharacteristic
646C2#2145853511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1970C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenter#1970D61
hypocenterC61#29095
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1970

Complements

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

NH ≅ G/Ntype
1. #21450407C2semidirect
2. #21449671C2semidirect
3. #21447313C2semidirect
4. #21447222C2semidirect
6. #21446955C2semidirect
7. #21446780C2semidirect
8. #21446518C2semidirect
9. #21446513C2semidirect
10. #21446513C2semidirect
11. #21446413C2semidirect
13. #21445693C2semidirect
14. #21445646C2semidirect
382. C61#29095semidirect

Maximal subgroups

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

subgroupindex
#214504072
#214496712
#214473132
#214472222
#214471272
#214469552
#214467802
#214465182
#214465132
#214464132
#214461692
#214456932
#214456462
C61×#19702
#2909561