Finite Group Databasebeta

← #29748091

Subgroups — #29748091

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450172C211yesmaximal, characteristic, complemented
2#21448831C211yesmaximal, characteristic, complemented
3#21447127C211yesmaximal, characteristic, complemented
4#21447007C211yesmaximal, characteristic, complemented
5#21446963C211yesmaximal, characteristic, complemented
6#21446814C211yesmaximal, characteristic
7#21446773C211yesmaximal, characteristic, complemented
8#21446505C211yesmaximal, characteristic, complemented
9#21446448C211yesmaximal, characteristic, complemented
10#21446423C211yesmaximal, characteristic, complemented
11#21446398C211yesmaximal, characteristic, complemented
12#21446133C211yesmaximal, characteristic, complemented
13#21445908C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic, complemented
15C61×#1879C211yesmaximal, characteristic, complemented
16C2×#14519768C2211yescharacteristic
17C2×#14519748C2211yescharacteristic
18C2×#14519748C2211yescharacteristic
19C2×C4×C22⋊D61—21no—
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839690—21no—
31#16839678—21no—
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839662—21no—
35#16839660C2211yescharacteristic
36#16839656—21no—
37#16839655—21no—
38#16839638—21no—
39#16839635—21no—
40#16839619C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×#11758198C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×D8×C61C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×C22⋊D61—21no—
60C22×C22⋊D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519768—22no—
66#14519768—41no—
67#14519748—22no—
68#14519748—22no—
69#14519748—22no—
70#14519748—41no—
71C4×C22⋊D61—44no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C4⋊C4×D61—22no—
81C4⋊C4×D61—41no—
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×#11758198—21no—
89C2×#11758198D411yescharacteristic
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519709—22no—
95#14519706—41no—
96#14519698—22no—
97#14519698—22no—
98#14519698—22no—
99#14519698—41no—
100#14519689—41no—
101#14519689—41no—
102C4×#92418—41no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C2×D8×C61—22no—
113C2×D8×C61—41no—
114C2×D8×C61—41no—
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C4×D4×C61—22no—
118C4×D4×C61—41no—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61C2311yescharacteristic
125#102×C61—22no—
126#102×C61—41no—
127C24×C61—21no—
128C24×C61—21no—
129C23×D61C2×D411yescharacteristic
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—21no—
141C2×D4×C61#4112yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61—81no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—81no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—81no—
172D8×C61—44no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61#4112yes—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—81no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191C22⋊Dic61—81no—
192C22⋊Dic61—81no—
193C22⋊Dic61—81no—
194#11758198—22no—
195#11758198—22no—
196#11758198—81no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204C4⋊Dic61#4112yes—
205C4⋊Dic61#4112yes—
206#11758196—42no—
207#11758196—42no—
208#11758196—42no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—42no—
212C4×Dic61—81no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61C22×D411yescharacteristic
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61#13512yes—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—41no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—84no—
261C2×C4×C61AGL(1,ℤ/8)12yes—
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C22⋊D61—48no—
269C22⋊D61—84no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—22no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C4×D61—41no—
292C4×D61—41no—
293C4×D61—41no—
294C4×D61—41no—
295C4×D61—84no—
296C8×C61—41no—
297C8×C61—41no—
298#92418—41no—
299#92418—41no—
300#26723—611nomaximal, Sylow
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61#52511yescharacteristic
316C22×C61#54611yescharacteristic
317C22×C61#54611yescharacteristic
318C22×C61C2×#13511yescharacteristic
319C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
320C22×C61#57611yescharacteristic
321C22×C61#57711yescharacteristic
322C2×D61—22no—
323C2×D61—22no—
324C2×D61—22no—
325C2×D61—41no—
326C2×D61—41no—
327C2×D61—41no—
328C2×D61—41no—
329C4×C61—22no—
330C4×C61—41no—
331C4×C61—42no—
332C4×C61—81no—
333C4×C61#52911yescharacteristic
334C4×C61#53811yescharacteristic
335C4×C61#55511yescharacteristic
336C4×C61#55911yescharacteristic
337Dic61—41no—
338Dic61—42no—
339Dic61—41no—
340Dic61—41no—
341Dic61—42no—
342Dic61—81no—
343Dic61—81no—
344#2490—611no—
345#2361—611no—
346#2054—611no—
347#2034—611no—
348#2025—611no—
349#1999—611no—
350#1989—611no—
351#1934—611no—
352#1926—611no—
353#1923—611no—
354#1919—611no—
355#1879D6111yescharacteristic, complemented
356#1874—611no—
357#1824—611no—
358C4×#102—611no—
359C2×C61—82no—
360C2×C61—161no—
361C2×C61#245911yescharacteristic
362C2×C61#310511yescharacteristic
363C2×C61#311911yescharacteristic
364C2×C61#329711yescharacteristic
365C2×C61#330111yescharacteristic
366C2×C61#331811yescharacteristic
367C2×C61#333211yescharacteristic
368D61—42no—
369C22×D8C2×D6111yescharacteristic
370C2×#125—611no—
371C2×#121—611no—
372C2×#121—611no—
373C2×C4×D4—1221no—
374C2×C4×D4C2×D6111yescharacteristic
375C22×C4⋊C4—611no—
376C2×#106—611no—
377C2×C4⋊C8—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#102C2×D6111yescharacteristic
383C2×#102C2×D6111yescharacteristic
384C2×C22⋊C8—611no—
385C2×C22⋊C8—611no—
386C2×C4×C8—611no—
387#397—1221no—
388#394—1221no—
389#390—611no—
390#390—611no—
391#390—1221no—
392#390C2×D6111yescharacteristic
393#389—611no—
394#388—1221no—
395#388—1221no—
396#386—1221no—
397#385—1221no—
398#380—611no—
399C4×C4⋊C4—611no—
400C4×C4⋊C4—611no—
401C4×C22⋊C4—611no—
402#340—611no—
403#340—611no—
404#340—611no—
405#340—611no—
406#340—611no—
407#340—611no—
408#340—611no—
409#340C2×D6111yescharacteristic
410#336—611no—
411#336C2×D6111yescharacteristic
412C61#2672311yescharacteristic, complemented, Sylow
413C22×D4—1221no—
414C22×D4—1221no—
415C22×D4C22×D6111yescharacteristic
416C22×D4C22×D6111yescharacteristic
417C23×C4—21no—
418C23×C4—611no—
419C23×C4—1221no—
420C2×D8—22no—
421C2×D8—41no—
422C2×D8—41no—
423C22×C8—611no—
424C22×C8—611no—
425C22×C8C22×D6111yescharacteristic
426C22×C8C22×D6111yescharacteristic
427#125—1222no—
428#125—2441no—
429#121—1222no—
430#121—1222no—
431#121—1222no—
432#121—2441no—
433C4×D4—22no—
434C4×D4—41no—
435C4×D4—2444no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—2441no—
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—611no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C42—611no—
461C2×C42—1221no—
462C2×C42—1221no—
463C2×C42C22×D6111yescharacteristic
464#106—1222no—
465C4⋊C8—2441no—
466#102—22no—
467#102—41no—
468#102—1222no—
469#102—1222no—
470#102—1222no—
471#102—2441no—
472C22⋊C8—2441no—
473C22⋊C8—2441no—
474C4×C8—2441no—
475#95—611no—
476#95—611no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481C24—21no—
482C24—21no—
483C24—611no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—1224no—
496C2×D4—4881no—
497C2×D4—2442no—
498C2×D4—2442no—
499C2×D4—2442no—
500C2×D4—2442no—
501C2×D4#1175822212yes—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—81no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—4881no—
521C22×C4—2442no—
522C22×C4—2442no—
523C22×C4D4×D6111yescharacteristic
524C22×C4D4×D6111yescharacteristic
525C22×C4C23×D6111yescharacteristic
526D8—44no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—612no—
543C4⋊C4—612no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—2442no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4#1175822412yes—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—81no—
558C22⋊C4—1222no—
559C22⋊C4—1224no—
560C22⋊C4—1222no—
561C22⋊C4—4881no—
562C22⋊C4—4881no—
563C22⋊C4—4881no—
564C22⋊C4—4881no—
565C22⋊C4—2442no—
566C22⋊C4—2442no—
567C42—21no—
568C42—21no—
569C42—1221no—
570C42—1221no—
571C42—4881no—
572C42—2442no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—612no—
588C23—1221no—
589C23—1221no—
590C23—1221no—
591C23—1221no—
592C23—1221no—
593C23—1221no—
594C23C2×D4×D6111yescharacteristic
595D4—22no—
596D4—22no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—41no—
603D4—4884no—
604D4—2448no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1222no—
628C2×C4—4884no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4#1451980112yes—
644C2×C4#41×D6111yescharacteristic
645C2×C4#41×D6111yescharacteristic
646C2×C4C2×#1175822411yescharacteristic
647C2×C4C2×#1175822211yescharacteristic
648C2×C4#1451982211yescharacteristic
649C2×C4#1451982211yescharacteristic
650C8—41no—
651C8—41no—
652C8—2441no—
653C8—2441no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—1222no—
669C22—1222no—
670C22—1222no—
671C22—2441no—
672C22—2441no—
673C22—2441no—
674C22—2441no—
675C22#1684027111yescharacteristic
676C22#1684039911yescharacteristic
677C22#1684040911yescharacteristic
678C22#135×D6111yescharacteristic
679C22C2×#1451980111yescharacteristic
680C22#1684056611yescharacteristic
681C22#1684057211yescharacteristic
682C4—22no—
683C4—42no—
684C4—41no—
685C4—81no—
686C4—4881no—
687C4—4881no—
688C4—2441no—
689C4—2442no—
690C4—2441no—
691C4—2441no—
692C4—2442no—
693C4#1684029211yescharacteristic
694C4#1684035011yescharacteristic
695C4#1684047611yescharacteristic
696C4#1684049611yescharacteristic
697C2—82no—
698C2—161no—
699C2—2442no—
700C2#2144975811yescharacteristic
701C2#2145586411yescharacteristic
702C2#2145602211yescharacteristic
703C2#2145767611yescharacteristic
704C2#2145771411yescharacteristic
705C2#2145796411yescharacteristic
706C2#2145814911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1879C2
socleC23×C61C22×D4
prime radical#21446814C2
hypercenter#1879D61
hypocenterC61#26723
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
1. #21450172C2semidirect
2. #21448831C2semidirect
3. #21447127C2semidirect
4. #21447007C2semidirect
5. #21446963C2semidirect
7. #21446773C2semidirect
8. #21446505C2semidirect
9. #21446448C2semidirect
10. #21446423C2semidirect
11. #21446398C2semidirect
12. #21446133C2semidirect
13. #21445908C2semidirect
14. C4×#14519698C2semidirect
15. C61×#1879C2semidirect
355. #1879D61semidirect
412. C61#26723semidirect

Maximal subgroups

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

subgroupindex
#214501722
#214488312
#214471272
#214470072
#214469632
#214468142
#214467732
#214465052
#214464482
#214464232
#214463982
#214461332
#214459082
C4×#145196982
C61×#18792
#2672361