Finite Group Databasebeta

← #29646227

Subgroups — #29646227

Order 15616.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840456C211yesmaximal, characteristic
2#21454477C211yesmaximal, characteristic, complemented
3#21444189C212yesmaximal, complemented
4#21444186C212yesmaximal, complemented
5#21444180C211yesmaximal, characteristic, complemented
6#21444178C211yesmaximal, characteristic, complemented
7#21444159C212yesmaximal, complemented
8#21444155C211yesmaximal, characteristic, complemented
9#21444139C212yesmaximal, complemented
10#21444135C211yesmaximal, characteristic, complemented
11C61×#2924C211yesmaximal, characteristic, complemented
12#16840456—21no—
13#16840456—21no—
14#16840456—22no—
15#16840456—21no—
16#16840456—21no—
17#16840456—22no—
18C2×#14519771C2212yes—
19C2×#14519768C411yescharacteristic
20C2×#14519759C412yes—
21C2×#14519752C411yescharacteristic
22C2×#14519751C2211yescharacteristic
23C2×#14519750C412yes—
24C2×C4×D244C2212yes—
25C2×#14519739C411yescharacteristic
26C2×C4×C22⋊D61C2211yescharacteristic
27C2×C4⋊C4×D61C411yescharacteristic
28#16839823—22no—
29#16839819C2212yes—
30#16839815C2212yes—
31#16839789—21no—
32#16839726C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34#16839718—21no—
35#16839487C2211yescharacteristic
36#16839487C2211yescharacteristic
37#16839487C2212yes—
38#16839485C2211yescharacteristic
39#16839485C2211yescharacteristic
40#16839485C2212yes—
41#16839485C2212yes—
42#16839485C2212yes—
43#16839485C2212yes—
44#16839485C2212yes—
45C2×#117×C61C2211yescharacteristic
46C61×#433—22no—
47C61×#432C2212yes—
48C61×#431C2212yes—
49C61×#424—21no—
50C61×#407C2211yescharacteristic
51C2×C22⋊C8×C61C2211yescharacteristic
52C61×#405—21no—
53C22×D244C2×C412yes—
54C22×D244C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C22×C22⋊D61C2×C412yes—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61C2×C412yes—
59C22×C4×D61C2×C412yes—
60C2×#11758194—21no—
61C22×#92418—21no—
62#14519771—22no—
63#14519771—22no—
64#14519771—22no—
65#14519771—22no—
66#14519768—22no—
67#14519768—21no—
68#14519768—21no—
69#14519759—22no—
70#14519759—22no—
71#14519759—22no—
72#14519759—22no—
73#14519752—22no—
74#14519752—21no—
75#14519752—21no—
76#14519751—21no—
77#14519751—21no—
78#14519751—21no—
79#14519751—21no—
80#14519750—22no—
81#14519750—22no—
82#14519750—22no—
83#14519750—22no—
84C4×D244—22no—
85C4×D244—22no—
86C4×D244—22no—
87C4×D244—22no—
88#14519739—22no—
89#14519739—21no—
90#14519739—21no—
91C4×C22⋊D61—21no—
92C4×C22⋊D61—21no—
93C4×C22⋊D61—21no—
94C4×C22⋊D61—21no—
95C2×C4⋊Dic61C2×C411yescharacteristic
96C2×C4⋊Dic61C2×C412yes—
97C2×#11758196C2×C412yes—
98C2×#11758196C2×C412yes—
99C2×#11758196C2311yescharacteristic
100C4⋊C4×D61—22no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61—21no—
103C2×C22⋊Dic61C2×C411yescharacteristic
104C2×#11758198C2×C412yes—
105C2×#11758198C2×C412yes—
106C2×#11758198C2311yescharacteristic
107C2×#11758198C2312yes—
108C2×C4×Dic61C2×C411yescharacteristic
109C4⋊#92418—22no—
110C4⋊#92418—22no—
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113C22⋊#92418C2311yescharacteristic
114C22⋊#92418C2311yescharacteristic
115C22⋊#92418C2312yes—
116#14519688—22no—
117C4×#92418—22no—
118C23×C4×C61C2311yescharacteristic
119C2×M16×C61—21no—
120C22×C8×C61—21no—
121#117×C61D411yescharacteristic
122#117×C61D411yescharacteristic
123#117×C61D412yes—
124#117×C61D411yescharacteristic
125#117×C61D411yescharacteristic
126#117×C61D412yes—
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C22⋊C4×C61C2×C412yes—
130C2×C42×C61C2312yes—
131C4⋊C8×C61—22no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61—21no—
134C22⋊C8×C61—21no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2311yescharacteristic
137C22⋊C8×C61C2312yes—
138#97×C61—22no—
139C4×C8×C61—22no—
140C24×C61C22×C411yescharacteristic
141C23×D61C22×C412yes—
142C23×D61C22×C411yescharacteristic
143C22×C4×C61—42no—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C412yes—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—82no—
159C22×Dic61C22×C411yescharacteristic
160C22×Dic61C22×C411yescharacteristic
161C22×Dic61C22×C412yes—
162C22×Dic61C22×C411yescharacteristic
163C2×D244—24no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—42no—
169C2×D244—42no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—42no—
177C2×C4×D61—82no—
178C2×C4×D61—82no—
179M16×C61—42no—
180C2×C8×C61—21no—
181C2×C8×C61—22no—
182C2×C8×C61—21no—
183C2×C8×C61—42no—
184C4⋊C4×C61C22⋊C412yes—
185C4⋊C4×C61C22⋊C411yescharacteristic
186C4⋊C4×C61C22⋊C411yescharacteristic
187C4⋊C4×C61C22⋊C412yes—
188C4⋊C4×C61C22⋊C411yescharacteristic
189C4⋊C4×C61C22⋊C411yescharacteristic
190C22⋊C4×C61C22⋊C412yes—
191C22⋊C4×C61C22⋊C412yes—
192C22⋊C4×C61C22⋊C412yes—
193C22⋊C4×C61C22⋊C412yes—
194C42×C61C2×D412yes—
195C42×C61C2×D412yes—
196C42×C61C2×D412yes—
197C42×C61C2×D412yes—
198C22⋊Dic61—21no—
199C22⋊Dic61—21no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—21no—
205#11758198—21no—
206#11758198—22no—
207#11758198—22no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—22no—
211C4⋊Dic61—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—21no—
217#11758196—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220#11758194—42no—
221C2×#92418—21no—
222C2×#92418—22no—
223C2×#92418—21no—
224C2×#92418—42no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61C2×C22⋊C411yescharacteristic
230C23×C61C2×C22⋊C411yescharacteristic
231C23×C61C23×C411yescharacteristic
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—42no—
240C22×D61—42no—
241C22×D61#13112yes—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C22⋊D61—42no—
260C22⋊D61—42no—
261C22⋊D61—82no—
262C22⋊D61—82no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272D244—44no—
273D244—44no—
274D244—42no—
275D244—42no—
276C4×D61—42no—
277C4×D61—42no—
278C4×D61—82no—
279C4×D61—82no—
280C4×D61—82no—
281C4×D61—82no—
282C8×C61—41no—
283C8×C61—41no—
284C8×C61—42no—
285#92418—41no—
286#92418—41no—
287#92418—42no—
288#17415—611nomaximal, Sylow
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61C2×#13111yescharacteristic
301C22×C61#56811yescharacteristic
302C2×D61—22no—
303C2×D61—42no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—82no—
311C2×D61—82no—
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—22no—
318C4×C61—22no—
319C4×C61#41012yes—
320Dic61—41no—
321Dic61—41no—
322Dic61—41no—
323Dic61—42no—
324C2×#553—611no—
325#2924—611no—
326#2924D6111yescharacteristic, complemented
327#1473—612no—
328#1473—612no—
329#1471—611no—
330#1471—611no—
331#1466—612no—
332#1465—611no—
333#1460—612no—
334#1459—611no—
335C2×C61—21no—
336C2×C61—21no—
337C2×C61—41no—
338C2×C61—41no—
339C2×C61#288011yescharacteristic
340C2×C61C2×#41011yescharacteristic
341C2×C61#288811yescharacteristic
342D61—42no—
343D61—82no—
344#553—1221no—
345#553—1221no—
346#553—1222no—
347#553—1221no—
348#553—1221no—
349#553—1222no—
350C2×#126—612no—
351C2×#125—611no—
352C2×#123—612no—
353C2×#121—611no—
354C2×#121—611no—
355C2×#121—612no—
356C2×C4×D4—611no—
357C2×C4×D4—611no—
358C2×C4×D4—612no—
359C2×#117C2×D6111yescharacteristic
360C22×C4⋊C4—611no—
361#433—22no—
362#433—1222no—
363#432—612no—
364#432C2×D6112yes—
365#431—612no—
366#431C2×D6112yes—
367#424—21no—
368#424—1221no—
369#407—611no—
370#407C2×D6111yescharacteristic
371C2×C22⋊C8—611no—
372C2×C22⋊C8C2×D6111yescharacteristic
373#405—21no—
374#405—1221no—
375#324—611no—
376#324—611no—
377#324—612no—
378#323—611no—
379#323—611no—
380#323—612no—
381#323—612no—
382#323—612no—
383#323—612no—
384#323—612no—
385C61#1741511yescharacteristic, complemented, Sylow
386C22×D4—611no—
387C22×D4—612no—
388C22×D4—612no—
389C22×D4—611no—
390C23×C4—612no—
391C23×C4—612no—
392C23×C4—611no—
393C23×C4C22×D6111yescharacteristic
394C2×M16—21no—
395C2×M16—1221no—
396C22×C8—21no—
397C22×C8—1221no—
398#126—1222no—
399#126—1222no—
400#126—1222no—
401#126—1222no—
402#125—1222no—
403#125—1221no—
404#125—1221no—
405#123—1222no—
406#123—1222no—
407#123—1222no—
408#123—1222no—
409#121—1221no—
410#121—1221no—
411#121—1221no—
412#121—1221no—
413#121—1222no—
414#121—1221no—
415#121—1221no—
416#121—1222no—
417#121—1222no—
418#121—1222no—
419#121—1222no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4—1221no—
423C4×D4—1221no—
424C4×D4—1222no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431#117D24411yescharacteristic
432#117D24411yescharacteristic
433#117D24412yes—
434#117C22⋊D6111yescharacteristic
435#117C22⋊D6111yescharacteristic
436#117C22⋊D6112yes—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—612no—
440C2×C4⋊C4—612no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1222no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4C4×D6111yescharacteristic
446C2×C4⋊C4C4×D6111yescharacteristic
447C2×C22⋊C4—611no—
448C2×C22⋊C4—612no—
449C2×C22⋊C4—612no—
450C2×C22⋊C4—611no—
451C2×C22⋊C4—612no—
452C2×C22⋊C4C4×D6112yes—
453C2×C42—611no—
454C2×C42C22×D6112yes—
455C4⋊C8—22no—
456C4⋊C8—22no—
457C4⋊C8—1222no—
458C4⋊C8—1222no—
459C22⋊C8—21no—
460C22⋊C8—21no—
461C22⋊C8—611no—
462C22⋊C8—611no—
463C22⋊C8—612no—
464C22⋊C8—1221no—
465C22⋊C8—1221no—
466C22⋊C8C22×D6111yescharacteristic
467C22⋊C8C22×D6111yescharacteristic
468C22⋊C8C22×D6112yes—
469#97—22no—
470#97—1222no—
471C4×C8—22no—
472C4×C8—1222no—
473C24—611no—
474C24—612no—
475C24C2×C4×D6111yescharacteristic
476C2×D4—1224no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1221no—
486C2×D4—4882no—
487C2×D4—2442no—
488C2×D4—2442no—
489C2×D4—2441no—
490C22×C4—42no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—612no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—4882no—
502C22×C4—4882no—
503C22×C4—2442no—
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4C2×C4×D6112yes—
506C22×C4C2×C4×D6112yes—
507C22×C4C2×D24411yescharacteristic
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512M16—42no—
513M16—2442no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—21no—
517C2×C8—42no—
518C2×C8—1221no—
519C2×C8—1222no—
520C2×C8—1221no—
521C2×C8—2442no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4#1175819812yes—
533C4⋊C4#1175819811yescharacteristic
534C4⋊C4#1175819811yescharacteristic
535C4⋊C4#1175819812yes—
536C4⋊C4#1175819811yescharacteristic
537C4⋊C4#1175819811yescharacteristic
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—1222no—
542C22⋊C4—1221no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4#1175819812yes—
549C22⋊C4#1175819812yes—
550C22⋊C4#1175819812yes—
551C22⋊C4#1175819812yes—
552C42—1221no—
553C42—1221no—
554C42C2×D24412yes—
555C42C2×D24412yes—
556C42C2×C22⋊D6112yes—
557C42C2×C22⋊D6112yes—
558C23—21no—
559C23—21no—
560C23—41no—
561C23—41no—
562C23—612no—
563C23—1221no—
564C23—1221no—
565C23—1222no—
566C23—1222no—
567C23—1222no—
568C23—2442no—
569C23—2442no—
570C23—2441no—
571C23—2441no—
572C23C2×#1175819811yescharacteristic
573C23C2×#1175819811yescharacteristic
574C23C22×C4×D6111yescharacteristic
575D4—4882no—
576D4—4882no—
577D4—2444no—
578D4—2444no—
579D4—2442no—
580D4—2442no—
581D4—2442no—
582D4—2442no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—4882no—
599C2×C4—4882no—
600C2×C4—4882no—
601C2×C4—4882no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2441no—
605C2×C4—2442no—
606C2×C4—2441no—
607C2×C4C2×#1175819811yescharacteristic
608C2×C4C2×#1175819812yes—
609C2×C4C2×#1175819812yes—
610C2×C4C2×#1175819812yes—
611C2×C4C2×#1175819812yes—
612C2×C4C2×#1175819811yescharacteristic
613C2×C4C22×C22⋊D6111yescharacteristic
614C2×C4C22×D24411yescharacteristic
615C8—41no—
616C8—41no—
617C8—42no—
618C8—2441no—
619C8—2441no—
620C8—2442no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—81no—
630C22—81no—
631C22—1222no—
632C22—4882no—
633C22—4882no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22—2442no—
638C22—2442no—
639C22—2441no—
640C22—2441no—
641C22C22×#1175819811yescharacteristic
642C22#131×D6111yescharacteristic
643C22#1684052711yescharacteristic
644C4—22no—
645C4—22no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—22no—
650C4—22no—
651C4—2441no—
652C4—2441no—
653C4—2441no—
654C4—2442no—
655C4#1683973512yes—
656C2—21no—
657C2—21no—
658C2—41no—
659C2—41no—
660C2—4882no—
661C2—2442no—
662C2#2145426611yescharacteristic
663C2C2×#1683973511yescharacteristic
664C2#2145429711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2924C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840456C2
cosocleC22×C4×C61C24
hypercenter#2924D61
hypocenterC61#17415
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2924

Complements

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

NH ≅ G/Ntype
2. #21454477C2semidirect
3. #21444189C2semidirect
4. #21444186C2semidirect
5. #21444180C2semidirect
6. #21444178C2semidirect
7. #21444159C2semidirect
8. #21444155C2semidirect
9. #21444139C2semidirect
10. #21444135C2semidirect
11. C61×#2924C2semidirect
326. #2924D61semidirect
385. C61#17415semidirect

Maximal subgroups

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

subgroupindex
C2×#168404562
#214544772
#214441892
#214441862
#214441802
#214441782
#214441592
#214441552
#214441392
#214441352
C61×#29242
#1741561