Finite Group Databasebeta

← #29753285

Subgroups — #29753285

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#21450399C211yesmaximal, characteristic, complemented
2#21448892C211yesmaximal, characteristic, complemented
3#21447350C211yesmaximal, characteristic, complemented
4#21447128C211yesmaximal, characteristic
5#21447058C211yesmaximal, characteristic, complemented
6#21446898C211yesmaximal, characteristic
7#21446673C211yesmaximal, characteristic, complemented
8#21446453C211yesmaximal, characteristic, complemented
9#21446444C211yesmaximal, characteristic, complemented
10#21446403C211yesmaximal, characteristic, complemented
11#21446376C211yesmaximal, characteristic, complemented
12#21446175C211yesmaximal, characteristic, complemented
13#21446167C211yesmaximal, characteristic
14#21445823C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×#14519771—21no—
19C2×#14519764C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30#16839691—21no—
31#16839686—21no—
32#16839668C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839656—21no—
39#16839653—21no—
40#16839650—21no—
41#16839635—21no—
42#16839631C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196C2211yescharacteristic
45C4×C22⋊Dic61—21no—
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C2×#11758190—22no—
62C2×#11758190—22no—
63C2×#11758190—41no—
64C2×#11758199—22no—
65C2×#11758199—41no—
66C2×#11758199—41no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519771—44no—
70#14519764—22no—
71#14519764—22no—
72#14519742—22no—
73#14519742—41no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
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×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519709—22no—
97#14519708—41no—
98#14519706—41no—
99#14519702—41no—
100#14519700—22no—
101#14519700—41no—
102#14519699—41no—
103#14519698—41no—
104#14519697—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C4×D4×C61—22no—
116C4×D4×C61—41no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#102×C61—22no—
124#102×C61—22no—
125C4×C8×C61—41no—
126#95×C61—21no—
127C24×C61—21no—
128C23×D61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61#4112yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—81no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122#4112yes—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61#4112yes—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61#4112yes—
175C22⋊C4×C61—42no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—42no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181C22⋊Dic61—42no—
182C22⋊Dic61—81no—
183#11758199—44no—
184#11758198—42no—
185#11758198—42no—
186#11758198—42no—
187#11758198—42no—
188#11758198—81no—
189#11758198—81no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194#11758196—22no—
195#11758196—22no—
196#11758196—42no—
197#11758196—42no—
198#11758196—42no—
199#11758196—81no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212#11758190—44no—
213C23×C61—22no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61C22×D411yescharacteristic
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—84no—
247C2×C4×C61AGL(1,ℤ/8)12yes—
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—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61#13512yes—
275D244—41no—
276D244—41no—
277D244—41no—
278D244—41no—
279Dic122—22no—
280Dic122—22no—
281Dic122—41no—
282Dic122—41no—
283C8×C61—22no—
284C8×C61—41no—
285#92418—41no—
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#52511yescharacteristic
296C22×C61#54611yescharacteristic
297C22×C61#55011yescharacteristic
298C22×C61C2×#13511yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57711yescharacteristic
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C4×C61—22no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61#52911yescharacteristic
315C4×C61#53811yescharacteristic
316C4×C61#55111yescharacteristic
317C4×C61#55311yescharacteristic
318Dic61—22no—
319Dic61—41no—
320Dic61—41no—
321Dic61—42no—
322Dic61—41no—
323Dic61—81no—
324Dic61—81no—
325#2512—611no—
326#2367—611no—
327#2094—611no—
328#2054—611no—
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×#102D6111yescharacteristic, complemented
340C2×C61—82no—
341C2×C61#246711yescharacteristic
342C2×C61#310611yescharacteristic
343C2×C61#312511yescharacteristic
344C2×C61#330111yescharacteristic
345C2×C61#330611yescharacteristic
346C2×C61#339411yescharacteristic
347C2×C61#339711yescharacteristic
348D61—161no—
349C22×SD16—611no—
350C22×D8—611no—
351C2×#126—1221no—
352C2×#124—611no—
353C2×C4×Q8—611no—
354C2×C4×D4C2×D6111yescharacteristic
355C2×#106—611no—
356C2×#106—611no—
357C2×C4⋊C8—611no—
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×C8C2×D6111yescharacteristic
367#397—1221no—
368#396—1221no—
369#391—611no—
370#390—611no—
371#390—611no—
372#390—611no—
373#390—611no—
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⋊C4C2×D6111yescharacteristic
383C4×C22⋊C4—1221no—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387#340C2×D6111yescharacteristic
388#336—611no—
389#336—611no—
390#336—611no—
391#336—611no—
392C61#2707111yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×D4—611no—
395C22×D4C22×D6111yescharacteristic
396C23×C4—21no—
397C2×SD16—1222no—
398C2×SD16—1222no—
399C2×SD16—2441no—
400C2×D8—1222no—
401C2×D8—2441no—
402C2×D8—2441no—
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—22no—
413C4×D4—41no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
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—21no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42C22×D6111yescharacteristic
442#106—1222no—
443#106—2441no—
444C4⋊C8—2441no—
445#103—1222no—
446#103—2441no—
447#103—2441no—
448#102—22no—
449#102—22no—
450#102—1222no—
451#102—2441no—
452#102—2441no—
453C4×C8—41no—
454#95—21no—
455#95—611no—
456#95—611no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461C24—21no—
462C24—1221no—
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—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4#1175822212yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497SD16—2444no—
498D8—2444no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8#1175822212yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—4881no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4#1175822212yes—
527C22⋊C4—42no—
528C22⋊C4—1224no—
529C22⋊C4—1224no—
530C22⋊C4—4881no—
531C22⋊C4—4881no—
532C22⋊C4—4881no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C22⋊C4—2442no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—1221no—
542C42—1221no—
543C42—1221no—
544C42—1221no—
545C42—4881no—
546C42—2442no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23C2×D4×D6111yescharacteristic
562Q8—1222no—
563Q8—1222no—
564Q8—2441no—
565Q8—2441no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—2441no—
571D4—2441no—
572D4—2441no—
573D4—2441no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—84no—
589C2×C4—612no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1222no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
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#1451980112yes—
611C2×C4#41×D6111yescharacteristic
612C2×C4C2×#1175822211yescharacteristic
613C2×C4C2×#1175822211yescharacteristic
614C2×C4C2×#1175822211yescharacteristic
615C2×C4#1451982211yescharacteristic
616C2×C4#1451982311yescharacteristic
617C8—22no—
618C8—41no—
619C8—2441no—
620C8—2441no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—4881no—
629C22—4881no—
630C22—4881no—
631C22—4881no—
632C22—4881no—
633C22—4881no—
634C22—4881no—
635C22#1684027111yescharacteristic
636C22#1684040211yescharacteristic
637C22#1684043311yescharacteristic
638C22#135×D6111yescharacteristic
639C22C2×#1451980111yescharacteristic
640C22#1684056811yescharacteristic
641C22#1684056911yescharacteristic
642C4—22no—
643C4—41no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—1222no—
648C4—4881no—
649C4—4881no—
650C4—2441no—
651C4—2441no—
652C4—2442no—
653C4—2441no—
654C4#1684029611yescharacteristic
655C4#1684034511yescharacteristic
656C4#1684044111yescharacteristic
657C4#1684045611yescharacteristic
658C2—82no—
659C2—9761no—
660C2#2144987011yescharacteristic
661C2#2145587511yescharacteristic
662C2#2145609211yescharacteristic
663C2#2145772211yescharacteristic
664C2#2145777611yescharacteristic
665C2#2145881611yescharacteristic
666C2#2145883711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC4×#102D61
hypocenterC61#27071
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. #21450399C2semidirect
2. #21448892C2semidirect
3. #21447350C2semidirect
5. #21447058C2semidirect
7. #21446673C2semidirect
8. #21446453C2semidirect
9. #21446444C2semidirect
10. #21446403C2semidirect
11. #21446376C2semidirect
12. #21446175C2semidirect
14. #21445823C2semidirect
15. C4×#102×C61C2semidirect
339. C4×#102D61semidirect
392. C61#27071semidirect

Maximal subgroups

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

subgroupindex
#214503992
#214488922
#214473502
#214471282
#214470582
#214468982
#214466732
#214464532
#214464442
#214464032
#214463762
#214461752
#214461672
#214458232
C4×#102×C612
#2707161