Finite Group Databasebeta

← #29653327

Subgroups — #29653327

Order 15616.

900 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#21455763C211yesmaximal, characteristic, complemented
2#21455728C211yesmaximal, characteristic
3#21451617C211yesmaximal, characteristic, complemented
4#21444861C212yesmaximal, complemented
5#21444850C212yesmaximal, complemented
6#21444827C212yesmaximal, complemented
7#21444816C212yesmaximal, complemented
8#21444391C211yesmaximal, characteristic, complemented
9#21444362C211yesmaximal, characteristic
10#21444183C211yesmaximal, characteristic, complemented
11C61×#3538C211yesmaximal, characteristic
12C2×#14519736C2211yescharacteristic
13#16840100—21no—
14#16840100—22no—
15#16840095C2211yescharacteristic
16#16840095C2212yes—
17#16840090C2211yescharacteristic
18#16840090C2212yes—
19#16840083—21no—
20#16840078C2211yescharacteristic
21#16840073C2211yescharacteristic
22#16839802—21no—
23#16839798—21no—
24#16839771—21no—
25#16839758—21no—
26#16839726C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28#16839701C2212yes—
29#16839692C2212yes—
30#16839690—22no—
31#16839669—21no—
32#16839658—21no—
33#16839622C2211yescharacteristic
34#16839499C2212yes—
35#16839499C2212yes—
36#16839499C2212yes—
37#16839496C2212yes—
38#16839488C2212yes—
39#16839488C2212yes—
40C61×#563—22no—
41C61×#552—21no—
42C61×#552—21no—
43C61×#551—22no—
44D4×Q8×C61—21no—
45D4×Q8×C61—21no—
46C61×#544—21no—
47C61×#544C2212yes—
48C61×#541—21no—
49C61×#539—21no—
50C61×#539C2212yes—
51C61×#536—21no—
52C61×#525C2211yescharacteristic
53C2×C22⋊Q8×C61C2211yescharacteristic
54C2×C22⋊Q8×C61C2211yescharacteristic
55C2×#11758194—21no—
56C22×#92418—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C2×C22⋊Dic61—21no—
66C2×C22⋊Dic61—21no—
67C2×C4×Dic61—21no—
68#14519710—21no—
69#14519710—21no—
70#14519710—22no—
71#14519708—21no—
72#14519708—21no—
73#14519708—22no—
74#14519701—21no—
75#14519701—21no—
76#14519701—22no—
77#14519701—22no—
78#14519698—21no—
79#14519698—21no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C22⋊#92418C2311yescharacteristic
83C22⋊#92418C2311yescharacteristic
84C22⋊#92418C2312yes—
85#14519684—22no—
86#14519684—22no—
87#14519684—22no—
88#14519684C2312yes—
89C2×#41×C61D411yescharacteristic
90C22×Q8×C61—21no—
91C22×Q8×C61D411yescharacteristic
92C22×Q8×C61D411yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C4⋊Q8×C61—21no—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61—22no—
98#126×C61—22no—
99#126×C61—22no—
100#126×C61—22no—
101#126×C61—22no—
102#125×C61—21no—
103#125×C61—22no—
104#125×C61—22no—
105#125×C61—22no—
106#125×C61—22no—
107#124×C61—22no—
108#123×C61—22no—
109#123×C61—22no—
110#123×C61—22no—
111#123×C61—22no—
112#123×C61—21no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61—21no—
115C22⋊Q8×C61—22no—
116C22⋊Q8×C61—22no—
117C22⋊Q8×C61—22no—
118C22⋊Q8×C61—21no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61—22no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61D411yescharacteristic
126C22⋊Q8×C61D412yes—
127C22⋊Q8×C61C2312yes—
128C22⋊Q8×C61C2311yescharacteristic
129#121×C61D411yescharacteristic
130#121×C61C2311yescharacteristic
131#120×C61—21no—
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134C4×D4×C61—22no—
135C4×D4×C61—22no—
136C4×D4×C61—21no—
137C4×D4×C61—21no—
138#117×C61—21no—
139#117×C61—22no—
140#117×C61—22no—
141#117×C61D411yescharacteristic
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61D411yescharacteristic
144C2×C4⋊C4×C61D411yescharacteristic
145C2×C4⋊C4×C61C2312yes—
146C2×C22⋊C4×C61—21no—
147C2×C22⋊C4×C61D412yes—
148C24×C61C2×D411yescharacteristic
149#41×C61—42no—
150C2×Q8×C61—22no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—42no—
155C2×Q8×C61—42no—
156C2×Q8×C61—82no—
157C2×Q8×C61C2×D412yes—
158C2×Q8×C61C2×D411yescharacteristic
159C2×D4×C61—21no—
160C2×D4×C61—21no—
161C2×D4×C61—21no—
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—22no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—42no—
168C22×C4×C61C2×D412yes—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D412yes—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C22×Dic61—21no—
177C22×Dic61—22no—
178C22×Dic61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61C2×D412yes—
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C22⋊C4×C61—22no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61—21no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—21no—
211C42×C61—22no—
212C42×C61—22no—
213C42×C61—22no—
214C42×C61—21no—
215C22⋊Dic61—41no—
216C22⋊Dic61—41no—
217C22⋊Dic61—81no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61—42no—
223#11758196—82no—
224C4×Dic61—41no—
225C4×Dic61—41no—
226#11758194—42no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—22no—
230C2×#92418—42no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61#12011yescharacteristic
237C23×C61#12011yescharacteristic
238C23×C61C22×D411yescharacteristic
239Q8×C61—42no—
240Q8×C61—41no—
241Q8×C61—42no—
242Q8×C61—42no—
243Q8×C61—81no—
244Q8×C61—82no—
245D4×C61—41no—
246D4×C61—42no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61—81no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—81no—
269C2×C4×C61—82no—
270C2×C4×C61#12011yescharacteristic
271C2×C4×C61#12011yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—42no—
281C2×Dic61—41no—
282#92418—41no—
283#92418—41no—
284#92418—42no—
285#18238—611nomaximal, Sylow
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61C2×#12011yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57811yescharacteristic
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61#45812yes—
313Dic61—82no—
314Dic61—81no—
315Dic61—81no—
316#3538D6111yescharacteristic
317#3095—611no—
318#3091—611no—
319#2625—611no—
320#1617—612no—
321#1615—612no—
322#1611—612no—
323#1609—612no—
324#1520—611no—
325#1513—611no—
326#1472—611no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—41no—
330C2×C61—41no—
331C2×C61—81no—
332C2×C61#301311yescharacteristic
333C2×C61#302011yescharacteristic
334C2×C61C2×#45811yescharacteristic
335#563—22no—
336#552—21no—
337#552—21no—
338#551—22no—
339D4×Q8—21no—
340D4×Q8—21no—
341#544—21no—
342#544C2×D6112yes—
343#541—21no—
344#539—21no—
345#539C2×D6112yes—
346#536—21no—
347#525C2×D6111yescharacteristic
348C2×C22⋊Q8C2×D6111yescharacteristic
349C2×C22⋊Q8C2×D6111yescharacteristic
350C2×#117—611no—
351#485—1221no—
352#485—1222no—
353#484—611no—
354#484—612no—
355#483—611no—
356#483—612no—
357#482—1221no—
358#481—611no—
359#480—611no—
360#428—1221no—
361#427—1221no—
362#419—1221no—
363#416—1221no—
364#407—611no—
365C2×C22⋊C8—611no—
366#400—612no—
367#398—612no—
368#397—1222no—
369#391—1221no—
370#389—1221no—
371#380—611no—
372#328—612no—
373#328—612no—
374#328—612no—
375#327—612no—
376#324—612no—
377#324—612no—
378C61#1823811yescharacteristic, complemented, Sylow
379C2×#41C22⋊D6111yescharacteristic
380C22×Q8—21no—
381C22×Q8C22⋊D6111yescharacteristic
382C22×Q8C22⋊D6111yescharacteristic
383C23×C4C22×D6111yescharacteristic
384C2×M16—1221no—
385C22×C8—1221no—
386C4⋊Q8—21no—
387C4⋊Q8—22no—
388C4⋊Q8—22no—
389C4⋊Q8—22no—
390#126—22no—
391#126—22no—
392#126—22no—
393#126—22no—
394#125—21no—
395#125—22no—
396#125—22no—
397#125—22no—
398#125—22no—
399#124—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403#123—21no—
404#123—22no—
405C22⋊Q8—21no—
406C22⋊Q8—21no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—22no—
417C22⋊Q8C22⋊D6112yes—
418C22⋊Q8C22⋊D6111yescharacteristic
419C22⋊Q8C22×D6112yes—
420C22⋊Q8C22×D6111yescharacteristic
421#121C22⋊D6111yescharacteristic
422#121C22×D6111yescharacteristic
423#120—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×D4—22no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—22no—
430#117—21no—
431#117—22no—
432#117—22no—
433#117—1221no—
434#117—1221no—
435#117—1221no—
436#117—1221no—
437#117C22⋊D6111yescharacteristic
438C2×C4⋊C4—21no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1222no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4C22⋊D6111yescharacteristic
444C2×C4⋊C4C22⋊D6111yescharacteristic
445C2×C4⋊C4C22×D6112yes—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4C22⋊D6112yes—
450C2×C42—1221no—
451#107—1221no—
452#107—1221no—
453#107—1222no—
454#106—1221no—
455#106—1221no—
456#106—1222no—
457#103—1221no—
458#103—1221no—
459#103—1222no—
460#103—1222no—
461#102—1221no—
462#102—1221no—
463C22⋊C8—612no—
464C22⋊C8—611no—
465C22⋊C8—611no—
466C22⋊C8—1221no—
467C22⋊C8—1221no—
468#95—612no—
469#95—1222no—
470#95—1222no—
471#95—1222no—
472C24C2×C22⋊D6111yescharacteristic
473#41—42no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—82no—
481C2×Q8C2×C22⋊D6112yes—
482C2×Q8C2×C22⋊D6111yescharacteristic
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C22⋊D6111yescharacteristic
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—42no—
492C22×C4—1222no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4C2×C22⋊D6112yes—
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6112yes—
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—2442no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1222no—
507C2×C8—2442no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—4882no—
530C4⋊C4—2442no—
531C4⋊C4C2×C22⋊D6112yes—
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C4⋊C4C2×C22⋊D6111yescharacteristic
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—4881no—
547C22⋊C4—2441no—
548C22⋊C4—2441no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—21no—
553C42—2441no—
554C42—2441no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23#1451974611yescharacteristic
561C23#1451974611yescharacteristic
562C23C22×C22⋊D6111yescharacteristic
563Q8—42no—
564Q8—41no—
565Q8—42no—
566Q8—42no—
567Q8—81no—
568Q8—82no—
569D4—41no—
570D4—42no—
571D4—41no—
572D4—41no—
573D4—81no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2442no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4#1451974611yescharacteristic
603C2×C4#1451974611yescharacteristic
604C2×C4C22×C22⋊D6111yescharacteristic
605C2×C4C22×C22⋊D6111yescharacteristic
606C8—2441no—
607C8—2441no—
608C8—2442no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—81no—
621C22—81no—
622C22C2×#1451974611yescharacteristic
623C22#1684057011yescharacteristic
624C22#1684057711yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—4882no—
636C4—4881no—
637C4—4881no—
638C4#1683995212yes—
639C2—21no—
640C2—21no—
641C2—41no—
642C2—41no—
643C2—81no—
644C2#2145513411yescharacteristic
645C2#2145518311yescharacteristic
646C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3538C2
socleC22×C61C2×#120
prime radicalC61×#525C22
hypercenter#3538D61
hypocenterC61#18238
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3538

Complements

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

NH ≅ G/Ntype
1. #21455763C2semidirect
3. #21451617C2semidirect
4. #21444861C2semidirect
5. #21444850C2semidirect
6. #21444827C2semidirect
7. #21444816C2semidirect
8. #21444391C2semidirect
10. #21444183C2semidirect
378. C61#18238semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214557282
#214516172
#214448612
#214448502
#214448272
#214448162
#214443912
#214443622
#214441832
C61×#35382
#1823861