Finite Group Databasebeta

← #29734262

Subgroups — #29734262

Order 15616.

972 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456212C211yesmaximal, characteristic, complemented
2#21455634C211yesmaximal, characteristic, complemented
3#21451940C211yesmaximal, characteristic, complemented
4#21447045C212yesmaximal, complemented
5#21447040C212yesmaximal, complemented
6#21446813C211yesmaximal, characteristic, complemented
7#21446652C211yesmaximal, characteristic, complemented
8#21446052C212yesmaximal, complemented
9#21446045C212yesmaximal, complemented
10#21445629C211yesmaximal, characteristic, complemented
11C61×#3512C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840121—21no—
14#16840121—21no—
15#16840112—21no—
16#16840108—21no—
17#16840057—21no—
18#16840052—21no—
19#16840047—21no—
20#16840042—21no—
21#16839805C2212yes—
22#16839786C2211yescharacteristic
23C2×#14519704C2212yes—
24C2×#14519704C2212yes—
25C2×#14519704C2212yes—
26#16839763C2212yes—
27#16839763C2212yes—
28#16839763C2212yes—
29C4×#11758194C2211yescharacteristic
30#16839692—22no—
31#16839682—22no—
32#16839631C2212yes—
33#16839627C2212yes—
34#16839608C2211yescharacteristic
35#16839531C2212yes—
36#16839531C2212yes—
37#16839521C2212yes—
38C61×#553—21no—
39C61×#553—21no—
40C61×#553—22no—
41C61×#553—22no—
42C61×#552—21no—
43C61×#552—21no—
44C61×#541—21no—
45C61×#540—21no—
46C61×#537C2211yescharacteristic
47C61×#537C2211yescharacteristic
48C61×#532C2211yescharacteristic
49C61×#529—21no—
50C61×#529—21no—
51C2×#125×C61C2211yescharacteristic
52C61×#520C2211yescharacteristic
53C61×#520C2211yescharacteristic
54C4×#41×C61C2211yescharacteristic
55C22×Dic122—21no—
56C2×#11758194C2312yes—
57C2×#11758194C2312yes—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—24no—
65C2×C4×Dic61C2312yes—
66#14519710—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70#14519704—24no—
71#14519704—24no—
72#14519704—24no—
73#14519701—22no—
74#14519701—22no—
75#14519701—22no—
76#14519698—22no—
77#14519698—22no—
78#14519698—22no—
79#14519688—21no—
80C4×#92418—21no—
81#14519684—21no—
82#14519684—24no—
83#14519684—21no—
84C2×#41×C61C2311yescharacteristic
85C2×#41×C61C2311yescharacteristic
86C2×#41×C61C2311yescharacteristic
87C4⋊Q8×C61D411yescharacteristic
88#127×C61D411yescharacteristic
89#126×C61—22no—
90#126×C61—22no—
91#126×C61—22no—
92#126×C61—22no—
93#125×C61—22no—
94#125×C61—22no—
95#125×C61—21no—
96#125×C61—21no—
97#125×C61—21no—
98#125×C61—21no—
99#124×C61D411yescharacteristic
100#124×C61D411yescharacteristic
101#123×C61—22no—
102#123×C61—22no—
103#123×C61—22no—
104#123×C61—22no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—21no—
109#121×C61—22no—
110#121×C61—22no—
111#121×C61—21no—
112#121×C61—21no—
113#121×C61—21no—
114#121×C61—21no—
115#121×C61—21no—
116#121×C61—21no—
117#121×C61—22no—
118#121×C61—22no—
119C4×Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61D411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61—22no—
124C4×D4×C61—22no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—22no—
130C4×D4×C61—22no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61D411yescharacteristic
134C4×D4×C61D411yescharacteristic
135#117×C61—22no—
136#117×C61—22no—
137#117×C61—21no—
138#117×C61—21no—
139#117×C61—21no—
140#117×C61C2311yescharacteristic
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61D412yes—
144C2×C4⋊C4×C61D412yes—
145C2×C4⋊C4×C61C2311yescharacteristic
146C2×C4⋊C4×C61C2311yescharacteristic
147C2×C42×C61—21no—
148C2×C42×C61C2311yescharacteristic
149#41×C61—22no—
150#41×C61—22no—
151#41×C61—22no—
152#41×C61—22no—
153#41×C61—22no—
154#41×C61—22no—
155C2×Q8×C61C2×D411yescharacteristic
156C2×Q8×C61C2×D411yescharacteristic
157C2×Q8×C61C2×D411yescharacteristic
158C2×D4×C61—21no—
159C2×D4×C61—21no—
160C2×D4×C61—21no—
161C2×D4×C61—22no—
162C2×D4×C61—22no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61C2×D411yescharacteristic
166C2×D4×C61C2×D411yescharacteristic
167C2×D4×C61C2×D411yescharacteristic
168C22×C4×C61—22no—
169C22×C4×C61—22no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61—21no—
174C22×C4×C61—21no—
175C22×C4×C61C2×D412yes—
176C22×C4×C61C2×D412yes—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2411yescharacteristic
180C22×Dic61—22no—
181C22×Dic61—22no—
182C2×Dic122—41no—
183C2×Dic122—41no—
184C2×Dic122—82no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61C2×D411yescharacteristic
202C4⋊C4×C61C2×D411yescharacteristic
203C22⋊C4×C61—22no—
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—21no—
211C22⋊C4×C61—21no—
212C22⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—21no—
215C22⋊C4×C61—21no—
216C42×C61—22no—
217C42×C61—22no—
218C42×C61—21no—
219C42×C61—21no—
220C42×C61—21no—
221C42×C61C2×D411yescharacteristic
222C42×C61C2×D411yescharacteristic
223C42×C61C2×D411yescharacteristic
224C42×C61C2×D411yescharacteristic
225C4⋊Dic61—22no—
226C4⋊Dic61—22no—
227C4⋊Dic61—41no—
228C4⋊Dic61—41no—
229#11758196—84no—
230C4×Dic61—22no—
231C4×Dic61#4114yes—
232#11758194—22no—
233#11758194—22no—
234#11758194—22no—
235#11758194—22no—
236C2×#92418—22no—
237C2×#92418—22no—
238C23×C61—21no—
239C23×C61—21no—
240C23×C61—21no—
241C23×C61C22×D411yescharacteristic
242Q8×C61—21no—
243Q8×C61—21no—
244Q8×C61—22no—
245Q8×C61—22no—
246D4×C61—21no—
247D4×C61—21no—
248D4×C61—22no—
249D4×C61—22no—
250D4×C61—42no—
251D4×C61—42no—
252D4×C61—42no—
253D4×C61—81no—
254D4×C61—81no—
255D4×C61—82no—
256D4×C61—82no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—42no—
273C2×C4×C61—81no—
274C2×C4×C61—81no—
275C2×C4×C61—82no—
276C2×C4×C61—82no—
277C2×C4×C61#12011yescharacteristic
278C2×C4×C61#12011yescharacteristic
279C2×C4×C61#12011yescharacteristic
280C2×C4×C61#12011yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C2×#4112yes—
284C2×C4×C612−1+412yes—
285C2×Dic61—24no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61—44no—
289C2×Dic61—42no—
290Dic122—82no—
291Dic122—82no—
292#92418—42no—
293#92418—42no—
294#25670—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61C2×#12011yescharacteristic
307C22×C61#54812yes—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—21no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—42no—
316C4×C61—41no—
317C4×C61—42no—
318C4×C61—41no—
319C4×C61#53211yescharacteristic
320C4×C61#53611yescharacteristic
321C4×C61#54712yes—
322Dic61—44no—
323Dic61—82no—
324#3512D6111yescharacteristic
325#3141—611no—
326#3079—611no—
327#2660—611no—
328#2040—612no—
329#2039—612no—
330#1998—611no—
331#1963—611no—
332#1860—612no—
333#1859—612no—
334#1759—611no—
335C2×C61—22no—
336C2×C61—81no—
337C2×C61—81no—
338C2×C61—81no—
339C2×C61#243211yescharacteristic
340C2×C61#302111yescharacteristic
341C2×C61#302211yescharacteristic
342#553—21no—
343#553—21no—
344#553—22no—
345#553—22no—
346#552—21no—
347#552—21no—
348#541—21no—
349#540—21no—
350#537C2×D6111yescharacteristic
351#537C2×D6111yescharacteristic
352#532C2×D6111yescharacteristic
353C2×C4⋊Q8—611no—
354#529—21no—
355#529—21no—
356C2×#125C2×D6111yescharacteristic
357#520C2×D6111yescharacteristic
358#520C2×D6111yescharacteristic
359C4×#41C2×D6111yescharacteristic
360#489—1221no—
361#489—1221no—
362#487—1221no—
363#486—1221no—
364#477—1221no—
365#476—1221no—
366#475—1221no—
367#474—1221no—
368#428—612no—
369#423—611no—
370C2×#104—612no—
371C2×#104—612no—
372C2×#104—612no—
373#417—612no—
374#417—612no—
375#417—612no—
376C4×M16—611no—
377#398—1222no—
378#395—1222no—
379#384—612no—
380#382—612no—
381#376—611no—
382#343—612no—
383#343—612no—
384#339—612no—
385C61#2567011yescharacteristic, complemented, Sylow
386C2×#41C22×D6111yescharacteristic
387C2×#41C22×D6111yescharacteristic
388C2×#41C22×D6111yescharacteristic
389C22×Q8—1221no—
390C2×M16—612no—
391C2×M16—612no—
392C4⋊Q8—1221no—
393C4⋊Q8—1221no—
394C4⋊Q8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8C22⋊D6111yescharacteristic
397#127C22⋊D6111yescharacteristic
398#126—22no—
399#126—22no—
400#126—22no—
401#126—22no—
402#125—22no—
403#125—22no—
404#125—21no—
405#125—21no—
406#125—21no—
407#125—21no—
408#124C22⋊D6111yescharacteristic
409#124C22⋊D6111yescharacteristic
410#123—22no—
411#123—22no—
412#123—22no—
413#123—22no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418#121—22no—
419#121—22no—
420#121—21no—
421#121—21no—
422#121—21no—
423#121—21no—
424#121—21no—
425#121—21no—
426#121—22no—
427#121—22no—
428C4×Q8—21no—
429C4×Q8—21no—
430C4×Q8C22⋊D6111yescharacteristic
431C4×Q8C22⋊D6111yescharacteristic
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—22no—
439C4×D4—22no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4C22⋊D6111yescharacteristic
443C4×D4C22⋊D6111yescharacteristic
444#117—22no—
445#117—22no—
446#117—21no—
447#117—21no—
448#117—21no—
449#117C22×D6111yescharacteristic
450C2×C4⋊C4—21no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—612no—
453C2×C4⋊C4—1224no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4C22⋊D6112yes—
456C2×C4⋊C4C22⋊D6112yes—
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C42—21no—
460C2×C42—612no—
461C2×C42C22×D6111yescharacteristic
462#107—1222no—
463#106—1222no—
464C4⋊C8—1221no—
465C4⋊C8—1221no—
466#104—1224no—
467#104—1224no—
468#104—1224no—
469#103—1222no—
470#103—1222no—
471#103—1222no—
472#102—1222no—
473#102—1222no—
474#102—1222no—
475#97—1221no—
476C4×C8—1221no—
477#95—1221no—
478#95—1221no—
479#95—1224no—
480#41—22no—
481#41—22no—
482#41—22no—
483#41—22no—
484#41—22no—
485#41—22no—
486C2×Q8—4882no—
487C2×Q8—2441no—
488C2×Q8—2441no—
489C2×Q8C2×C22⋊D6111yescharacteristic
490C2×Q8C2×C22⋊D6111yescharacteristic
491C2×Q8C2×C22⋊D6111yescharacteristic
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4C2×C22⋊D6111yescharacteristic
500C2×D4C2×C22⋊D6111yescharacteristic
501C2×D4C2×C22⋊D6111yescharacteristic
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4C2×C22⋊D6112yes—
512C22×C4C2×C22⋊D6112yes—
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516M16—1222no—
517M16—1222no—
518M16—1222no—
519M16—1222no—
520C2×C8—1222no—
521C2×C8—1222no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—4884no—
541C4⋊C4—2441no—
542C4⋊C4—2441no—
543C4⋊C4C2×C22⋊D6111yescharacteristic
544C4⋊C4C2×C22⋊D6111yescharacteristic
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—21no—
557C22⋊C4—21no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—614no—
564C42—1222no—
565C42C2×C22⋊D6111yescharacteristic
566C42C2×C22⋊D6111yescharacteristic
567C42C2×C22⋊D6111yescharacteristic
568C42C2×C22⋊D6111yescharacteristic
569C23—21no—
570C23—21no—
571C23—21no—
572C23C22×C22⋊D6111yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8—22no—
576Q8—22no—
577Q8—4882no—
578Q8—4882no—
579D4—21no—
580D4—21no—
581D4—22no—
582D4—22no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—81no—
587D4—81no—
588D4—82no—
589D4—82no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—1224no—
611C2×C4—2444no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4#1451974611yescharacteristic
616C2×C4#1451974611yescharacteristic
617C2×C4#1451974611yescharacteristic
618C2×C4#1451974611yescharacteristic
619C2×C4C22×C22⋊D6111yescharacteristic
620C2×C4C22×C22⋊D6111yescharacteristic
621C2×C4#41×D6112yes—
622C2×C4#1451982412yes—
623C8—2442no—
624C8—2442no—
625C22—21no—
626C22—21no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22C2×#1451974611yescharacteristic
637C22#1684041912yes—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—41no—
644C4—42no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—4882no—
650C4—2444no—
651C4#1684030811yescharacteristic
652C4#1684033311yescharacteristic
653C4#1684041112yes—
654C2—22no—
655C2—81no—
656C2—81no—
657C2—81no—
658C2#2144944311yescharacteristic
659C2#2145519111yescharacteristic
660C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3512C2
socleC22×C61C2×#120
hypercenter#3512D61
hypocenterC61#25670
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3512

Complements

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

NH ≅ G/Ntype
1. #21456212C2semidirect
2. #21455634C2semidirect
3. #21451940C2semidirect
4. #21447045C2semidirect
5. #21447040C2semidirect
6. #21446813C2semidirect
7. #21446652C2semidirect
8. #21446052C2semidirect
9. #21446045C2semidirect
10. #21445629C2semidirect
385. C61#25670semidirect

Maximal subgroups

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

subgroupindex
#214562122
#214556342
#214519402
#214470452
#214470402
#214468132
#214466522
#214460522
#214460452
#214456292
C61×#35122
#2567061