Finite Group Databasebeta

← #29753295

Subgroups — #29753295

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450414C211yesmaximal, characteristic, complemented
2#21448894C211yesmaximal, characteristic, complemented
3#21447342C211yesmaximal, characteristic, complemented
4#21447127C211yesmaximal, characteristic
5#21447055C211yesmaximal, characteristic, complemented
6#21446675C211yesmaximal, characteristic, complemented
7#21446463C211yesmaximal, characteristic, complemented
8#21446436C211yesmaximal, characteristic, complemented
9#21446398C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446176C211yesmaximal, characteristic, complemented
12#21446169C211yesmaximal, characteristic
13#21445832C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2012C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839690—21no—
29#16839687—21no—
30#16839669C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839655—21no—
36#16839655—21no—
37#16839655—21no—
38#16839650—21no—
39#16839637—21no—
40#16839627C2211yescharacteristic
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×SD16×C61C2211yescharacteristic
53C22×D8×C61C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#390C2211yescharacteristic
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×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519710—41no—
87#14519709—22no—
88#14519706—41no—
89#14519701—22no—
90#14519701—41no—
91#14519698—22no—
92#14519698—22no—
93#14519698—41no—
94#14519698—41no—
95C8×Dic61—41no—
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×D4×C61C2311yescharacteristic
107C22×D4×C61C2311yescharacteristic
108C2×SD16×C61—22no—
109C2×SD16×C61—22no—
110C2×SD16×C61—41no—
111C2×D8×C61—22no—
112C2×D8×C61—41no—
113C2×D8×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×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C42×C61C2311yescharacteristic
124#103×C61—41no—
125#102×C61—22no—
126C24×C61—21no—
127C24×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#4112yes—
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—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61#4112yes—
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
162SD16×C61—44no—
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#4112yes—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—42no—
175C22⋊C4×C61—81no—
176C42×C61—21no—
177C42×C61—21no—
178C22⋊Dic61—42no—
179C22⋊Dic61—42no—
180C22⋊Dic61—42no—
181C22⋊Dic61—81no—
182C22⋊Dic61—81no—
183C22⋊Dic61—81no—
184C22⋊Dic61—81no—
185C4⋊Dic61—21no—
186C4⋊Dic61—22no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—41no—
194C4⋊Dic61—42no—
195C4⋊Dic61#4112yes—
196#11758196—81no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—42no—
202C4×Dic61—81no—
203C4×Dic61—81no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C23×C61—22no—
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×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61C22×D411yescharacteristic
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—41no—
228Q8×C61—41no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
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—41no—
278C2×Dic61—84no—
279C2×Dic61#13512yes—
280C8×C61—22no—
281C8×C61—41no—
282#92418—41no—
283#92418—41no—
284#27072—611nomaximal, Sylow
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
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#52511yescharacteristic
300C22×C61#54611yescharacteristic
301C22×C61#55011yescharacteristic
302C22×C61C2×#13511yescharacteristic
303C22×C61#57511yescharacteristic
304C22×C61#57611yescharacteristic
305C22×C61#57711yescharacteristic
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—42no—
309C4×C61—81no—
310C4×C61#52911yescharacteristic
311C4×C61#53811yescharacteristic
312C4×C61#55111yescharacteristic
313C4×C61#55311yescharacteristic
314Dic61—22no—
315Dic61—41no—
316Dic61—42no—
317Dic61—41no—
318Dic61—41no—
319Dic61—81no—
320Dic61—81no—
321Dic61—81no—
322#2513—611no—
323#2367—611no—
324#2092—611no—
325#2054—611no—
326#2042—611no—
327#2012D6111yescharacteristic
328#1968—611no—
329#1928—611no—
330#1925—611no—
331#1919—611no—
332#1916—611no—
333#1880—611no—
334#1879—611no—
335#1805—611no—
336C4×#102—611no—
337C2×C61—82no—
338C2×C61—161no—
339C2×C61#246711yescharacteristic
340C2×C61#310211yescharacteristic
341C2×C61#311911yescharacteristic
342C2×C61#329711yescharacteristic
343C2×C61#332111yescharacteristic
344C2×C61#339411yescharacteristic
345C2×C61#340311yescharacteristic
346C22×SD16C2×D6111yescharacteristic
347C22×D8C2×D6111yescharacteristic
348C2×#124C2×D6111yescharacteristic
349C2×C4×Q8—611no—
350C2×C4×D4—611no—
351C2×#107—611no—
352C2×#106—611no—
353C2×C4⋊C8—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103C2×D6111yescharacteristic
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102C2×D6111yescharacteristic
362C2×C4×C8—611no—
363#397—1221no—
364#397—1221no—
365#391—611no—
366#390—611no—
367#390—611no—
368#390—611no—
369#390C2×D6111yescharacteristic
370#389—611no—
371#388—1221no—
372#388—1221no—
373#388—1221no—
374#388—1221no—
375#385—1221no—
376#382—611no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C22⋊C4—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#336—611no—
386#336—611no—
387#336—611no—
388#336C2×D6111yescharacteristic
389C61#2707211yescharacteristic, complemented, Sylow
390C22×Q8C22×D6111yescharacteristic
391C22×D4C22×D6111yescharacteristic
392C22×D4C22×D6111yescharacteristic
393C23×C4—1221no—
394C2×SD16—22no—
395C2×SD16—22no—
396C2×SD16—41no—
397C2×D8—22no—
398C2×D8—41no—
399C2×D8—41no—
400C22×C8—611no—
401C22×C8—611no—
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404#124—22no—
405#124—22no—
406C4×Q8—1222no—
407C4×Q8—2441no—
408C4×D4—1222no—
409C4×D4—2441no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42C22×D6111yescharacteristic
436#107—2441no—
437#106—1222no—
438C4⋊C8—2441no—
439#103—41no—
440#103—1222no—
441#103—2441no—
442#102—22no—
443#102—1222no—
444#102—1222no—
445#102—2441no—
446#102—2441no—
447C4×C8—2441no—
448#95—611no—
449#95—611no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457C24—21no—
458C24—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8#1175822412yes—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4#1175822412yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—4881no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4D4×D6111yescharacteristic
491C22×C4D4×D6111yescharacteristic
492C22×C4C23×D6111yescharacteristic
493SD16—44no—
494D8—44no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8#1175822412yes—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—612no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—4881no—
519C4⋊C4—2441no—
520C4⋊C4—2442no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—4881no—
526C22⋊C4—4881no—
527C22⋊C4—4881no—
528C22⋊C4—4881no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C42—21no—
533C42—21no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—4881no—
539C42—4881no—
540C42—2442no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×D4×D6111yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—612no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—4884no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4#41×D6111yescharacteristic
606C2×C4C2×#1175822411yescharacteristic
607C2×C4C2×#1175822411yescharacteristic
608C2×C4C2×#1175822411yescharacteristic
609C2×C4#1451982211yescharacteristic
610C2×C4#1451982211yescharacteristic
611C8—22no—
612C8—41no—
613C8—2441no—
614C8—2441no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22#1684026811yescharacteristic
630C22#1684040911yescharacteristic
631C22#1684043211yescharacteristic
632C22#135×D6111yescharacteristic
633C22#1684055811yescharacteristic
634C22#1684056611yescharacteristic
635C22#1684056911yescharacteristic
636C4—42no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—1222no—
641C4—4881no—
642C4—4881no—
643C4—4881no—
644C4—2441no—
645C4—2442no—
646C4—2441no—
647C4—2441no—
648C4#1684029311yescharacteristic
649C4#1684034711yescharacteristic
650C4#1684044311yescharacteristic
651C4#1684045511yescharacteristic
652C2—82no—
653C2—161no—
654C2#2144986511yescharacteristic
655C2#2145583011yescharacteristic
656C2#2145601911yescharacteristic
657C2#2145767611yescharacteristic
658C2#2145801211yescharacteristic
659C2#2145881211yescharacteristic
660C2#2145890411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2012C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenter#2012D61
hypocenterC61#27072
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2012

Complements

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

NH ≅ G/Ntype
1. #21450414C2semidirect
2. #21448894C2semidirect
3. #21447342C2semidirect
5. #21447055C2semidirect
6. #21446675C2semidirect
7. #21446463C2semidirect
8. #21446436C2semidirect
9. #21446398C2semidirect
10. #21446375C2semidirect
11. #21446176C2semidirect
13. #21445832C2semidirect
14. #102×Dic61C2semidirect
389. C61#27072semidirect

Maximal subgroups

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

subgroupindex
#214504142
#214488942
#214473422
#214471272
#214470552
#214466752
#214464632
#214464362
#214463982
#214463752
#214461762
#214461692
#214458322
#102×Dic612
C61×#20122
#2707261