Finite Group Databasebeta

← #29748144

Subgroups — #29748144

Order 15616.

790 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#21450173C211yesmaximal, characteristic
2#21448828C211yesmaximal, characteristic, complemented
3#21447132C211yesmaximal, characteristic, complemented
4#21447019C211yesmaximal, characteristic, complemented
5#21446968C211yesmaximal, characteristic, complemented
6#21446775C211yesmaximal, characteristic, complemented
7#21446504C211yesmaximal, characteristic, complemented
8#21446442C211yesmaximal, characteristic, complemented
9#21446422C211yesmaximal, characteristic, complemented
10#21446414C211yesmaximal, characteristic, complemented
11#21446176C211yesmaximal, characteristic, complemented
12#21446137C211yesmaximal, characteristic
13#21445906C211yesmaximal, characteristic, complemented
14#103×Dic61C211yesmaximal, characteristic, complemented
15C61×#2000C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839687—21no—
28#16839676—21no—
29#16839669C2211yescharacteristic
30#16839665—21no—
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661—21no—
34#16839661C2211yescharacteristic
35#16839656—21no—
36#16839648—21no—
37#16839637—21no—
38#16839622C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C22⋊C4×Dic61—21no—
42C4×C22⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×C22⋊Q8×C61C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519768—22no—
63#14519768—41no—
64Q8×Dic61—22no—
65Q8×Dic61—41no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—41no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61D411yescharacteristic
83C2×C22⋊Dic61D411yescharacteristic
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519710—22no—
90#14519706—41no—
91#14519701—22no—
92#14519701—41no—
93#14519698—22no—
94#14519698—41no—
95C22⋊#92418—41no—
96C8×Dic61—41no—
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×C61—21no—
107C22×D4×C61C2311yescharacteristic
108C23×C4×C61C2311yescharacteristic
109C2×SD16×C61—22no—
110C2×SD16×C61—41no—
111C2×SD16×C61—41no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61—22no—
116#121×C61—22no—
117#121×C61—41no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123#103×C61—22no—
124#102×C61—22no—
125C22⋊C8×C61—41no—
126C24×C61—21no—
127C24×C61C2×D411yescharacteristic
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—42no—
141C2×D4×C61—42no—
142C2×D4×C61—81no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—81no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165SD16×C61—44no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61#4112yes—
176C4⋊C4×C61#4112yes—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—42no—
179C22⋊Dic61—22no—
180C22⋊Dic61—22no—
181C22⋊Dic61—42no—
182C22⋊Dic61—81no—
183C22⋊Dic61—81no—
184C22⋊Dic61—81no—
185C22⋊Dic61—81no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4⋊Dic61—81no—
197C4⋊Dic61#4112yes—
198#11758196—42no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—42no—
204C4×Dic61—81no—
205C4×Dic61—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61#13512yes—
226C23×C61C22×D411yescharacteristic
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—41no—
230Q8×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—84no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
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—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—84no—
282C2×Dic61#13712yes—
283C8×C61—41no—
284C8×C61—41no—
285#92418—41no—
286#92418—41no—
287#26727—611nomaximal, Sylow
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
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#54611yescharacteristic
305C22×C61C2×#13511yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57611yescharacteristic
308C22×C61#57711yescharacteristic
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61#52911yescharacteristic
314C4×C61#53811yescharacteristic
315C4×C61#55511yescharacteristic
316C4×C61#55911yescharacteristic
317Dic61—22no—
318Dic61—41no—
319Dic61—41no—
320Dic61—41no—
321Dic61—42no—
322Dic61—81no—
323Dic61—81no—
324Dic61—81no—
325#2490—611no—
326#2360—611no—
327#2055—611no—
328#2036—611no—
329#2026—611no—
330#2000D6111yescharacteristic
331#1989—611no—
332#1934—611no—
333#1925—611no—
334#1923—611no—
335#1921—611no—
336#1880—611no—
337#1874—611no—
338#1823—611no—
339C4×#103—611no—
340C2×C61—42no—
341C2×C61—161no—
342C2×C61#245911yescharacteristic
343C2×C61#310811yescharacteristic
344C2×C61#311911yescharacteristic
345C2×C61#329611yescharacteristic
346C2×C61#329711yescharacteristic
347C2×C61#332011yescharacteristic
348C2×C61#333311yescharacteristic
349C22×SD16C2×D6111yescharacteristic
350C2×#125—611no—
351C2×C22⋊Q8C2×D6111yescharacteristic
352C2×#121C2×D6111yescharacteristic
353C2×C4×Q8—611no—
354C22×C4⋊C4—611no—
355C2×#107—611no—
356C2×C4⋊C8—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103C2×D6111yescharacteristic
360C2×#102—611no—
361C2×#102—611no—
362C2×#102C2×D6111yescharacteristic
363C2×C22⋊C8—611no—
364C2×C22⋊C8C2×D6111yescharacteristic
365C2×C4×C8—611no—
366#397—1221no—
367#394—1221no—
368#391—611no—
369#390—611no—
370#390—611no—
371#390—1221no—
372#389—611no—
373#389—1221no—
374#388—1221no—
375#387—1221no—
376#385—1221no—
377#380—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C22⋊C4—611no—
381C4×C22⋊C4—1221no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340C2×D6111yescharacteristic
390#336—611no—
391#336—611no—
392C61#2672711yescharacteristic, complemented, Sylow
393C22×Q8C22×D6111yescharacteristic
394C22×D4—21no—
395C22×D4C22×D6111yescharacteristic
396C23×C4—1221no—
397C23×C4C22×D6111yescharacteristic
398C2×SD16—22no—
399C2×SD16—41no—
400C2×SD16—41no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405#125—1222no—
406#125—2441no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409#121—22no—
410#121—41no—
411C4×Q8—1222no—
412C4×Q8—2441no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—2441no—
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
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—
441#107—1222no—
442C4⋊C8—2441no—
443#103—22no—
444#103—1222no—
445#103—2441no—
446#102—22no—
447#102—1222no—
448#102—2441no—
449C22⋊C8—41no—
450C22⋊C8—2441no—
451C4×C8—2441no—
452#95—611no—
453#95—611no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460C24—21no—
461C24D4×D6111yescharacteristic
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8#1175822412yes—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—81no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4881no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4D4×D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499SD16—44no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—612no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4881no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4#1175822412yes—
529C4⋊C4#1175822412yes—
530C22⋊C4—24no—
531C22⋊C4—42no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—4881no—
535C22⋊C4—4881no—
536C22⋊C4—4881no—
537C22⋊C4—4881no—
538C22⋊C4—2442no—
539C42—1221no—
540C42—1221no—
541C42—1221no—
542C42—1221no—
543C42—4881no—
544C42—4881no—
545C42—2442no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23#1451979812yes—
560C23C2×D4×D6111yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—84no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—612no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
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—2441no—
611C2×C4#41×D6111yescharacteristic
612C2×C4C2×#1175822411yescharacteristic
613C2×C4C2×#1175822411yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C2×C4#1451982211yescharacteristic
616C2×C4#1451982211yescharacteristic
617C8—41no—
618C8—41no—
619C8—2441no—
620C8—2441no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22#1684026811yescharacteristic
636C22#1684039911yescharacteristic
637C22#1684040111yescharacteristic
638C22C2×#1451979811yescharacteristic
639C22#137×D6111yescharacteristic
640C22#1684056411yescharacteristic
641C22#1684057211yescharacteristic
642C4—41no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—1222no—
647C4—4881no—
648C4—4881no—
649C4—4881no—
650C4—2441no—
651C4—2442no—
652C4—2441no—
653C4—2441no—
654C4#1684029311yescharacteristic
655C4#1684034711yescharacteristic
656C4#1684047611yescharacteristic
657C4#1684049811yescharacteristic
658C2—42no—
659C2—161no—
660C2#2144975111yescharacteristic
661C2#2145589411yescharacteristic
662C2#2145601111yescharacteristic
663C2#2145765811yescharacteristic
664C2#2145767511yescharacteristic
665C2#2145798511yescharacteristic
666C2#2145815011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2000C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2000D61
hypocenterC61#26727
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2000

Complements

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

NH ≅ G/Ntype
2. #21448828C2semidirect
3. #21447132C2semidirect
4. #21447019C2semidirect
5. #21446968C2semidirect
6. #21446775C2semidirect
7. #21446504C2semidirect
8. #21446442C2semidirect
9. #21446422C2semidirect
10. #21446414C2semidirect
11. #21446176C2semidirect
13. #21445906C2semidirect
14. #103×Dic61C2semidirect
392. C61#26727semidirect

Maximal subgroups

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

subgroupindex
#214501732
#214488282
#214471322
#214470192
#214469682
#214467752
#214465042
#214464422
#214464222
#214464142
#214461762
#214461372
#214459062
#103×Dic612
C61×#20002
#2672761