Finite Group Databasebeta

← #29655432

Subgroups — #29655432

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456645C211yesmaximal, characteristic, complemented
2#21456576C211yesmaximal, characteristic, complemented
3#21453854C211yesmaximal, characteristic, complemented
4#21444856C212yesmaximal, complemented
5#21444850C212yesmaximal, complemented
6#21444788C212yesmaximal, complemented
7#21444782C212yesmaximal, complemented
8#21444375C211yesmaximal, characteristic, complemented
9#21444348C211yesmaximal, characteristic, complemented
10#21444183C211yesmaximal, characteristic, complemented
11C61×#3536C211yesmaximal, characteristic
12C2×#14519756C2211yescharacteristic
13#16840103—21no—
14#16840095C2211yescharacteristic, complemented
15#16840092—21no—
16#16840078C2212yes—
17#16840078C2211yescharacteristic, complemented
18#16840018—21no—
19#16840006—21no—
20#16839989—22no—
21#16839924—21no—
22#16839917C2211yescharacteristic, complemented
23#16839913—21no—
24#16839900C2212yes—
25#16839900C2211yescharacteristic, complemented
26#16839726C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28#16839701C2212yescomplemented
29#16839700—21no—
30#16839690—22no—
31#16839682C2212yescomplemented
32#16839681—21no—
33#16839625C2211yescharacteristic
34#16839499C2212yescomplemented
35#16839496C2212yes—
36#16839496C2212yes—
37#16839496C2212yescomplemented
38#16839488C2212yescomplemented
39#16839488C2212yescomplemented
40C61×#562—22no—
41C61×#554—21no—
42C61×#548—21no—
43C61×#547—21no—
44C61×#546—21no—
45C61×#546—22no—
46C61×#543—21no—
47C61×#542—21no—
48C61×#539C2212yes—
49C61×#538—21no—
50C61×#537—21no—
51C61×#537C2212yes—
52C61×#525C2211yescharacteristic
53C2×C22⋊Q8×C61C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C22×Dic122C2311yescharacteristic
56C2×#11758202—21no—
57C2×#11758200—22no—
58C2×#11758200—21no—
59C2×#11758194—21no—
60C22×#92418—21no—
61#14519756—21no—
62#14519756—21no—
63#14519756—21no—
64#14519756—21no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×#11758196—21no—
67C2×#11758196—22no—
68C2×C22⋊Dic61—21no—
69C2×C22⋊Dic61—21no—
70#14519710—21no—
71#14519708—22no—
72#14519708—21no—
73#14519701—21no—
74#14519700—22no—
75#14519700—21no—
76#14519700—21no—
77#14519698—22no—
78#14519698—21no—
79C22⋊#92418—21no—
80C22⋊#92418—21no—
81C22⋊#92418C2312yes—
82C22⋊#92418C2311yescharacteristic
83C22⋊#92418C2311yescharacteristic
84#14519684—22no—
85#14519684—21no—
86#14519684—22no—
87#14519684—22no—
88#14519684C2312yes—
89C2×#41×C61—21no—
90C2×#41×C61—22no—
91C2×#41×C61D411yescharacteristic
92C22×Q8×C61D411yescharacteristic
93C22×D4×C61—21no—
94C22×D4×C61D411yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C4⋊Q8×C61—21no—
97#126×C61—22no—
98#125×C61—22no—
99#124×C61—22no—
100#124×C61—21no—
101#124×C61—22no—
102#123×C61—22no—
103#123×C61—22no—
104#123×C61—22no—
105#123×C61—22no—
106#123×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—22no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61—21no—
115C22⋊Q8×C61—21no—
116C22⋊Q8×C61—22no—
117C22⋊Q8×C61—22no—
118C22⋊Q8×C61D411yescharacteristic
119C22⋊Q8×C61C2311yescharacteristic
120#121×C61—21no—
121#121×C61—22no—
122#121×C61—22no—
123#121×C61—22no—
124#121×C61—22no—
125#121×C61—21no—
126#121×C61—22no—
127#121×C61—21no—
128#121×C61D411yescharacteristic
129#121×C61D412yes—
130#121×C61C2311yescharacteristic
131#121×C61C2312yes—
132#120×C61—22no—
133#120×C61—21no—
134C4×Q8×C61—21no—
135C4×D4×C61—22no—
136C4×D4×C61—21no—
137C4×D4×C61—22no—
138C4×D4×C61—21no—
139C4×D4×C61—21no—
140#117×C61D411yescharacteristic
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61D411yescharacteristic
143C2×C4⋊C4×C61D411yescharacteristic
144C2×C4⋊C4×C61C2312yes—
145C2×C22⋊C4×C61—22no—
146C2×C22⋊C4×C61—21no—
147C2×C22⋊C4×C61D412yes—
148C24×C61—21no—
149C24×C61C2×D411yescharacteristic
150#41×C61—84no—
151#41×C61—81no—
152#41×C61—82no—
153C2×Q8×C61—22no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—81no—
157C2×Q8×C61C2×D411yescharacteristic
158C2×D4×C61—21no—
159C2×D4×C61—21no—
160C2×D4×C61—22no—
161C2×D4×C61—22no—
162C2×D4×C61—22no—
163C2×D4×C61—21no—
164C2×D4×C61—22no—
165C2×D4×C61—21no—
166C2×D4×C61—81no—
167C2×D4×C61—82no—
168C2×D4×C61C2×D411yescharacteristic
169C2×D4×C61C2×D412yes—
170C22×C4×C61—22no—
171C22×C4×C61—22no—
172C22×C4×C61—22no—
173C22×C4×C61—21no—
174C22×C4×C61—21no—
175C22×C4×C61—81no—
176C22×C4×C61C2×D412yes—
177C22×C4×C61C2×D412yes—
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2411yescharacteristic
184C22×Dic61—21no—
185C22×Dic61—22no—
186C22×Dic61—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—81no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61C2×D412yes—
202C4⋊C4×C61C2×D411yescharacteristic
203C4⋊C4×C61C2×D411yescharacteristic
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C22⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—21no—
216C22⋊C4×C61—22no—
217C22⋊C4×C61—21no—
218C42×C61—22no—
219C42×C61—21no—
220C22⋊Dic61—41no—
221C22⋊Dic61—41no—
222C22⋊Dic61—81no—
223#11758202—81no—
224#11758200—82no—
225#11758200—81no—
226C4⋊Dic61—21no—
227C4⋊Dic61—21no—
228#11758196—41no—
229#11758196—41no—
230#11758196—82no—
231#11758194—81no—
232C2×#92418—22no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418—81no—
236C23×C61—22no—
237C23×C61—22no—
238C23×C61—21no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61—42no—
242C23×C61—81no—
243C23×C61—81no—
244C23×C61#12011yescharacteristic
245C23×C61#12011yescharacteristic
246C23×C61C22×D411yescharacteristic
247Q8×C61—42no—
248Q8×C61—41no—
249Q8×C61—81no—
250Q8×C61—82no—
251D4×C61—42no—
252D4×C61—41no—
253D4×C61—42no—
254D4×C61—42no—
255D4×C61—44no—
256D4×C61—42no—
257D4×C61—82no—
258D4×C61—81no—
259D4×C61—82no—
260D4×C61—82no—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—21no—
264C2×C4×C61—22no—
265C2×C4×C61—21no—
266C2×C4×C61—22no—
267C2×C4×C61—24no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61—42no—
275C2×C4×C61—42no—
276C2×C4×C61—81no—
277C2×C4×C61—82no—
278C2×C4×C61—82no—
279C2×C4×C61—82no—
280C2×C4×C61#12011yescharacteristic
281C2×C4×C61#12011yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—42no—
291Dic122—42no—
292Dic122—41no—
293Dic122—41no—
294#92418—42no—
295#92418—41no—
296#92418—41no—
297#18457—611nomaximal, Sylow
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—41no—
307C22×C61—42no—
308C22×C61—41no—
309C22×C61—42no—
310C22×C61—41no—
311C22×C61—42no—
312C22×C61—82no—
313C22×C61—162no—
314C22×C61C2×#12011yescharacteristic
315C22×C61#57712yes—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—42no—
321C4×C61—42no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—41no—
325C4×C61—42no—
326Dic61—82no—
327Dic61—81no—
328Dic61—81no—
329#3536D6111yescharacteristic
330#3196—611no—
331#3187—611no—
332#2817—611no—
333#1616—612no—
334#1615—612no—
335#1604—612no—
336#1603—612no—
337#1516—611no—
338#1510—611no—
339#1472—611no—
340C2×C61—22no—
341C2×C61—41no—
342C2×C61—41no—
343C2×C61—81no—
344C2×C61—82no—
345C2×C61#300911yescharacteristic
346C2×C61#302011yescharacteristic
347C2×C61#302611yescharacteristic
348#562—22no—
349#554—21no—
350#548—21no—
351#547—21no—
352#546—21no—
353#546—22no—
354#543—21no—
355#542—21no—
356#539C2×D6112yes—
357#538—21no—
358#537—21no—
359#537C2×D6112yes—
360#525C2×D6111yescharacteristic
361C2×C22⋊Q8—611no—
362C2×C22⋊Q8C2×D6111yescharacteristic
363C2×#121C2×D6111yescharacteristic
364#485—1221no—
365#484—611no—
366#483—1221no—
367#481—612no—
368#481—611no—
369#470—1221no—
370#468—1221no—
371#465—1222no—
372#452—1221no—
373#451—611no—
374#450—1221no—
375#448—612no—
376#448—611no—
377#407—611no—
378C2×C22⋊C8—611no—
379#400—612no—
380#400—1221no—
381#397—1222no—
382#395—612no—
383#395—1221no—
384#381—611no—
385#328—612no—
386#327—612no—
387#327—612no—
388#327—612no—
389#324—612no—
390#324—612no—
391C61#1845711yescharacteristic, complemented, Sylow
392C2×#41—21no—
393C2×#41—22no—
394C2×#41C22⋊D6111yescharacteristic
395C22×Q8—611no—
396C22×Q8C22⋊D6111yescharacteristic
397C22×D4—21no—
398C22×D4C22⋊D6111yescharacteristic
399C23×C4C22×D6111yescharacteristic
400C2×Q16—1221no—
401C2×SD16—1222no—
402C2×SD16—1221no—
403C2×M16—1221no—
404C22×C8—1221no—
405C4⋊Q8—21no—
406#126—22no—
407#125—22no—
408#124—22no—
409#124—21no—
410#124—22no—
411#123—22no—
412#123—22no—
413#123—22no—
414#123—21no—
415#123—22no—
416C22⋊Q8—21no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—21no—
424C22⋊Q8—21no—
425C22⋊Q8—22no—
426C22⋊Q8—22no—
427C22⋊Q8—1221no—
428C22⋊Q8—1221no—
429C22⋊Q8—1221no—
430C22⋊Q8—1221no—
431C22⋊Q8C22⋊D6111yescharacteristic
432C22⋊Q8C22×D6111yescharacteristic
433#121—21no—
434#121—22no—
435#121—22no—
436#121—22no—
437#121—22no—
438#121—21no—
439#121—21no—
440#121—22no—
441#121C22⋊D6112yes—
442#121C22⋊D6111yescharacteristic
443#121C22×D6112yes—
444#121C22×D6111yescharacteristic
445#120—22no—
446#120—21no—
447C4×Q8—21no—
448C4×D4—22no—
449C4×D4—21no—
450C4×D4—21no—
451C4×D4—21no—
452C4×D4—22no—
453#117C22⋊D6111yescharacteristic
454C2×C4⋊C4—21no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—1222no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4C22⋊D6111yescharacteristic
459C2×C4⋊C4C22⋊D6111yescharacteristic
460C2×C4⋊C4C22×D6112yes—
461C2×C22⋊C4—22no—
462C2×C22⋊C4—21no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C22⋊C4C22⋊D6112yes—
466#107—1221no—
467#106—1222no—
468#106—1221no—
469#103—1221no—
470#103—1222no—
471#103—1221no—
472#103—1221no—
473#102—1222no—
474#102—1221no—
475C22⋊C8—611no—
476C22⋊C8—611no—
477C22⋊C8—612no—
478C22⋊C8—1221no—
479C22⋊C8—1221no—
480#95—612no—
481#95—1221no—
482#95—1222no—
483#95—1222no—
484#95—1222no—
485C24—21no—
486C24C2×C22⋊D6111yescharacteristic
487#41—84no—
488#41—81no—
489#41—82no—
490C2×Q8—22no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—81no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—4881no—
497C2×Q8C2×C22⋊D6111yescharacteristic
498C2×D4—21no—
499C2×D4—22no—
500C2×D4—21no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—21no—
504C2×D4—22no—
505C2×D4—21no—
506C2×D4—81no—
507C2×D4—82no—
508C2×D4C2×C22⋊D6111yescharacteristic
509C2×D4C2×C22⋊D6112yes—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—81no—
516C22×C4—1222no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4C2×C22⋊D6112yes—
520C22×C4C2×C22⋊D6112yes—
521C22×C4C2×C22⋊D6111yescharacteristic
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C2×C22⋊D6111yescharacteristic
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C23×D6111yescharacteristic
527Q16—4881no—
528SD16—4882no—
529SD16—4881no—
530M16—4881no—
531C2×C8—1222no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—4881no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—4882no—
549C4⋊C4—2441no—
550C4⋊C4—2441no—
551C4⋊C4C2×C22⋊D6112yes—
552C4⋊C4C2×C22⋊D6111yescharacteristic
553C4⋊C4C2×C22⋊D6111yescharacteristic
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—21no—
564C22⋊C4—22no—
565C22⋊C4—21no—
566C22⋊C4—22no—
567C22⋊C4—21no—
568C22⋊C4—4881no—
569C22⋊C4—2441no—
570C22⋊C4—2441no—
571C42—22no—
572C42—21no—
573C23—22no—
574C23—22no—
575C23—21no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—81no—
580C23—81no—
581C23#1451974611yescharacteristic
582C23#1451974611yescharacteristic
583C23C22×C22⋊D6111yescharacteristic
584Q8—41no—
585Q8—42no—
586Q8—81no—
587Q8—82no—
588Q8—2442no—
589Q8—2441no—
590Q8—2441no—
591D4—42no—
592D4—41no—
593D4—42no—
594D4—42no—
595D4—44no—
596D4—42no—
597D4—82no—
598D4—81no—
599D4—82no—
600D4—82no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2442no—
627C2×C4#1451974611yescharacteristic
628C2×C4#1451974611yescharacteristic
629C2×C4C22×C22⋊D6111yescharacteristic
630C2×C4C22×C22⋊D6111yescharacteristic
631C8—2442no—
632C8—2441no—
633C8—2441no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—41no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—82no—
649C22—162no—
650C22C2×#1451974611yescharacteristic
651C22#1684057312yes—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—41no—
657C4—41no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—4882no—
663C4—4881no—
664C4—4881no—
665C2—22no—
666C2—41no—
667C2—41no—
668C2—82no—
669C2—81no—
670C2#2145509611yescharacteristic
671C2#2145518311yescharacteristic
672C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3536C2
socleC22×C61C2×#120
hypercenter#3536D61
hypocenterC61#18457
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3536

Complements

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

NH ≅ G/Ntype
1. #21456645C2semidirect
2. #21456576C2semidirect
3. #21453854C2semidirect
4. #21444856C2semidirect
5. #21444850C2semidirect
6. #21444788C2semidirect
7. #21444782C2semidirect
8. #21444375C2semidirect
9. #21444348C2semidirect
10. #21444183C2semidirect
14. #16840095C22semidirect
17. #16840078C22semidirect
22. #16839917C22semidirect
25. #16839900C22semidirect
28. #16839701C22semidirect
31. #16839682C22semidirect
34. #16839499C22semidirect
37. #16839496C22semidirect
38. #16839488C22semidirect
39. #16839488C22semidirect
391. C61#18457semidirect

Maximal subgroups

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

subgroupindex
#214566452
#214565762
#214538542
#214448562
#214448502
#214447882
#214447822
#214443752
#214443482
#214441832
C61×#35362
#1845761