Finite Group Databasebeta

← #29646302

Subgroups — #29646302

Order 15616.

1022 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
1C2×#16840462C211yesmaximal, characteristic
2#21454854C211yesmaximal, characteristic, complemented
3#21444176C211yesmaximal, characteristic, complemented
4#21444174C211yesmaximal, characteristic, complemented
5#21444168C212yesmaximal, complemented
6#21444165C212yesmaximal, complemented
7#21444159C211yesmaximal, characteristic, complemented
8#21444155C212yesmaximal, complemented
9#21444139C211yesmaximal, characteristic, complemented
10#21444135C212yesmaximal, complemented
11C61×#2974C211yesmaximal, characteristic, complemented
12#16840462—21no—
13#16840462—22no—
14#16840462—21no—
15#16840462—21no—
16#16840462—22no—
17#16840462—21no—
18C2×#14519770C412yes—
19C2×#14519769C2211yescharacteristic
20C2×#14519760C2212yes—
21C2×#14519750C412yes—
22C2×#14519748C411yescharacteristic
23C2×#14519748C411yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×C4×C22⋊D61C2212yes—
26C2×#14519735C411yescharacteristic
27C2×C4⋊C4×D61C411yescharacteristic
28#16839840—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2211yescharacteristic
32#16839815C2211yescharacteristic
33#16839726C2212yes—
34C2×C22⋊#92418C2212yes—
35#16839487C2212yes—
36#16839487C2212yes—
37#16839487C2212yes—
38#16839487C2212yes—
39#16839485C2211yescharacteristic
40#16839485C2212yes—
41#16839485C2211yescharacteristic
42#16839485C2211yescharacteristic
43#16839485C2211yescharacteristic
44#16839485C2212yes—
45C2×C4×D4×C61C2211yescharacteristic
46C61×#436—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2211yescharacteristic
50C61×#431C2211yescharacteristic
51C61×#407C2212yes—
52C2×C22⋊C8×C61C2212yes—
53C22×D244C2×C411yescharacteristic
54C22×C22⋊D61C2×C412yes—
55C22×C22⋊D61C2×C412yes—
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61C2×C412yes—
59C2×#11758194—22no—
60C22×#92418—22no—
61#14519770—22no—
62#14519770—22no—
63#14519770—22no—
64#14519770—22no—
65#14519769—21no—
66#14519769—21no—
67#14519769—21no—
68#14519769—21no—
69#14519760—22no—
70#14519760—22no—
71#14519760—22no—
72#14519760—22no—
73#14519750—22no—
74#14519750—22no—
75#14519750—22no—
76#14519750—22no—
77#14519748—21no—
78#14519748—21no—
79#14519748—22no—
80#14519748—21no—
81#14519748—21no—
82#14519748—22no—
83C4×D244—21no—
84C4×D244—21no—
85C4×D244—21no—
86C4×D244—21no—
87C4×C22⋊D61—22no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—22no—
90C4×C22⋊D61—22no—
91#14519735—22no—
92#14519735—21no—
93#14519735—21no—
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×C4⋊Dic61C2×C411yescharacteristic
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758196C2×C412yes—
98C2×#11758196C2×C412yes—
99C2×#11758196C2312yes—
100C4⋊C4×D61—22no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61—21no—
103C2×C22⋊Dic61C2×C412yes—
104C2×C22⋊Dic61C2×C412yes—
105C2×#11758198C2×C412yes—
106C2×#11758198C2311yescharacteristic
107C2×#11758198C2311yescharacteristic
108C2×C4×Dic61C2×C412yes—
109C4⋊#92418—21no—
110C4⋊#92418—21no—
111C22⋊#92418—22no—
112C22⋊#92418—22no—
113C22⋊#92418C2311yescharacteristic
114C22⋊#92418C2312yes—
115C22⋊#92418C2311yescharacteristic
116#14519688—21no—
117C4×#92418—21no—
118C22×D4×C61C2×C411yescharacteristic
119C23×C4×C61C2312yes—
120C2×M16×C61—22no—
121C22×C8×C61—22no—
122C4×D4×C61D411yescharacteristic
123C4×D4×C61D412yes—
124C4×D4×C61D411yescharacteristic
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127C4×D4×C61D411yescharacteristic
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C22⋊C4×C61C2×C412yes—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—21no—
132C4⋊C8×C61—21no—
133C22⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2312yes—
137C22⋊C8×C61C2311yescharacteristic
138#97×C61—21no—
139C4×C8×C61—21no—
140C24×C61C22×C412yes—
141C23×D61C22×C411yescharacteristic
142C23×D61C22×C411yescharacteristic
143C2×D4×C61—42no—
144C2×D4×C61C22⋊C411yescharacteristic
145C2×D4×C61C22⋊C411yescharacteristic
146C2×D4×C61C22⋊C412yes—
147C22×C4×C61—82no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C22×C412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—24no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61—42no—
162C2×C22⋊D61—82no—
163C22×Dic61C22×C412yes—
164C22×Dic61C22×C412yes—
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C411yescharacteristic
167C2×D244—22no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—41no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—42no—
178C2×C4×D61—42no—
179C2×C4×D61—82no—
180M16×C61—82no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—22no—
184C2×C8×C61—82no—
185C4⋊C4×C61C22⋊C412yes—
186C4⋊C4×C61C22⋊C411yescharacteristic
187C4⋊C4×C61C22⋊C411yescharacteristic
188C22⋊C4×C61C22⋊C412yes—
189C22⋊C4×C61C22⋊C412yes—
190C22⋊C4×C61C22⋊C412yes—
191C22⋊C4×C61C22⋊C412yes—
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C22⋊Dic61—22no—
197C22⋊Dic61—22no—
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—21no—
203#11758198—21no—
204#11758198—21no—
205#11758198—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217#11758196—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220#11758194—82no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—22no—
224C2×#92418—82no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—42no—
228C23×C61—42no—
229C23×C61C2×C22⋊C412yes—
230C23×C61C2×C22⋊C412yes—
231C23×C61C23×C411yescharacteristic
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61#13112yes—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—42no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—41no—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C22⋊D61—44no—
263C22⋊D61—44no—
264C22⋊D61—82no—
265C22⋊D61—82no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—42no—
275D244—42no—
276D244—42no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—42no—
280C4×D61—42no—
281C4×D61—82no—
282C4×D61—82no—
283C8×C61—41no—
284C8×C61—42no—
285C8×C61—41no—
286#92418—41no—
287#92418—42no—
288#92418—41no—
289#17424—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—82no—
299C22×C61—82no—
300C22×C61C22×C22⋊C411yescharacteristic
301C22×C61C2×#13111yescharacteristic
302C22×C61#56811yescharacteristic
303C2×D61—22no—
304C2×D61—42no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—42no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—81no—
312C2×D61—81no—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—22no—
318C4×C61—22no—
319C4×C61—21no—
320C4×C61#41312yes—
321Dic61—42no—
322Dic61—42no—
323Dic61—41no—
324Dic61—41no—
325C2×#553—611no—
326#2974—611no—
327#2974D6111yescharacteristic, complemented
328#1470—611no—
329#1470—611no—
330#1468—612no—
331#1468—612no—
332#1466—611no—
333#1465—612no—
334#1460—611no—
335#1459—612no—
336C2×C61—21no—
337C2×C61—21no—
338C2×C61—42no—
339C2×C61—42no—
340C2×C61#288011yescharacteristic
341C2×C61#288311yescharacteristic
342C2×C61C2×#41311yescharacteristic
343D61—42no—
344D61—81no—
345#553—1221no—
346#553—1222no—
347#553—1221no—
348#553—1221no—
349#553—1222no—
350#553—1221no—
351C2×#126—612no—
352C2×#125—611no—
353C2×#123—612no—
354C2×#121—611no—
355C2×#121—611no—
356C2×#121—612no—
357C2×C4×D4—612no—
358C2×C4×D4—611no—
359C2×C4×D4C2×D6111yescharacteristic
360C2×#117—611no—
361C22×C4⋊C4—611no—
362#436—21no—
363#436—1221no—
364#435—22no—
365#435—1222no—
366C8×D4—21no—
367C8×D4—1221no—
368#432—611no—
369#432C2×D6111yescharacteristic
370#431—611no—
371#431C2×D6111yescharacteristic
372#407—612no—
373#407C2×D6112yes—
374C2×C22⋊C8—612no—
375C2×C22⋊C8C2×D6112yes—
376#324—612no—
377#324—612no—
378#324—612no—
379#324—612no—
380#323—611no—
381#323—612no—
382#323—611no—
383#323—611no—
384#323—611no—
385#323—612no—
386C61#1742411yescharacteristic, complemented, Sylow
387C22×D4—612no—
388C22×D4—612no—
389C22×D4—611no—
390C22×D4C4×D6111yescharacteristic
391C23×C4—612no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4C22×D6112yes—
395C2×M16—22no—
396C2×M16—1222no—
397C22×C8—22no—
398C22×C8—1222no—
399#126—1222no—
400#126—1222no—
401#126—1222no—
402#126—1222no—
403#125—1221no—
404#125—1221no—
405#125—1221no—
406#125—1221no—
407#123—1222no—
408#123—1222no—
409#123—1222no—
410#123—1222no—
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—
420#121—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4D24411yescharacteristic
430C4×D4D24412yes—
431C4×D4D24411yescharacteristic
432C4×D4C22⋊D6111yescharacteristic
433C4×D4C22⋊D6112yes—
434C4×D4C22⋊D6111yescharacteristic
435#117—1222no—
436#117—1221no—
437#117—1221no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—612no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4C4×D6111yescharacteristic
448C2×C22⋊C4—612no—
449C2×C22⋊C4—612no—
450C2×C22⋊C4—612no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—611no—
453C2×C22⋊C4C4×D6112yes—
454C2×C42—612no—
455C2×C42C22×D6111yescharacteristic
456C4⋊C8—21no—
457C4⋊C8—21no—
458C4⋊C8—1221no—
459C4⋊C8—1221no—
460C22⋊C8—22no—
461C22⋊C8—22no—
462C22⋊C8—611no—
463C22⋊C8—612no—
464C22⋊C8—611no—
465C22⋊C8—1222no—
466C22⋊C8—1222no—
467C22⋊C8C22×D6111yescharacteristic
468C22⋊C8C22×D6112yes—
469C22⋊C8C22×D6111yescharacteristic
470#97—21no—
471#97—1221no—
472C4×C8—21no—
473C4×C8—1221no—
474C24—611no—
475C24—611no—
476C24C2×C4×D6112yes—
477C2×D4—42no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—1224no—
482C2×D4—1222no—
483C2×D4—1222no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—4882no—
487C2×D4—2441no—
488C2×D4—2442no—
489C2×D4#1175819811yescharacteristic
490C2×D4#1175819811yescharacteristic
491C2×D4#1175819812yes—
492C22×C4—82no—
493C22×C4—612no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—612no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—4882no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4C2×C4×D6111yescharacteristic
507C22×C4C2×C4×D6111yescharacteristic
508C22×C4C2×C4×D6112yes—
509C22×C4C2×D24412yes—
510C22×C4C2×D24412yes—
511C22×C4C2×C22⋊D6112yes—
512C22×C4C2×C22⋊D6112yes—
513C22×C4C23×D6111yescharacteristic
514M16—82no—
515M16—4882no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—22no—
519C2×C8—82no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1222no—
523C2×C8—4882no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4#1175819812yes—
537C4⋊C4#1175819811yescharacteristic
538C4⋊C4#1175819811yescharacteristic
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1221no—
544C22⋊C4—1221no—
545C22⋊C4—1221no—
546C22⋊C4—1221no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4#1175819812yes—
550C22⋊C4#1175819812yes—
551C22⋊C4#1175819812yes—
552C22⋊C4#1175819812yes—
553C42—1222no—
554C42—1222no—
555C42C2×D24411yescharacteristic
556C42C2×D24411yescharacteristic
557C42C2×C22⋊D6111yescharacteristic
558C42C2×C22⋊D6111yescharacteristic
559C23—22no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23—612no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23—2441no—
573C23C2×#1175819812yes—
574C23C2×#1175819812yes—
575C23C22×C4×D6111yescharacteristic
576D4—22no—
577D4—22no—
578D4—42no—
579D4—4882no—
580D4—4882no—
581D4—2442no—
582D4—2442no—
583D4—2444no—
584D4—2444no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—4882no—
601C2×C4—4882no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4C2×#1175819812yes—
610C2×C4C2×#1175819812yes—
611C2×C4C2×#1175819811yescharacteristic
612C2×C4C2×#1175819811yescharacteristic
613C2×C4C2×#1175819811yescharacteristic
614C2×C4C2×#1175819811yescharacteristic
615C2×C4C22×C22⋊D6111yescharacteristic
616C2×C4C22×D24411yescharacteristic
617C8—41no—
618C8—42no—
619C8—41no—
620C8—2441no—
621C8—2442no—
622C8—2441no—
623C22—21no—
624C22—21no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—82no—
632C22—82no—
633C22—1222no—
634C22—4881no—
635C22—4881no—
636C22—2442no—
637C22—2442no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22—2441no—
642C22—2441no—
643C22C22×#1175819811yescharacteristic
644C22#131×D6111yescharacteristic
645C22#1684052711yescharacteristic
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—22no—
651C4—22no—
652C4—21no—
653C4—2442no—
654C4—2442no—
655C4—2441no—
656C4—2441no—
657C4#1683974412yes—
658C2—21no—
659C2—21no—
660C2—42no—
661C2—42no—
662C2—4881no—
663C2—2442no—
664C2#2145426611yescharacteristic
665C2#2145427911yescharacteristic
666C2C2×#1683974411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2974C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840462C2
cosocleC22×C4×C61C24
hypercenter#2974D61
hypocenterC61#17424
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #21454854C2semidirect
3. #21444176C2semidirect
4. #21444174C2semidirect
5. #21444168C2semidirect
6. #21444165C2semidirect
7. #21444159C2semidirect
8. #21444155C2semidirect
9. #21444139C2semidirect
10. #21444135C2semidirect
11. C61×#2974C2semidirect
327. #2974D61semidirect
386. C61#17424semidirect

Maximal subgroups

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

subgroupindex
C2×#168404622
#214548542
#214441762
#214441742
#214441682
#214441652
#214441592
#214441552
#214441392
#214441352
C61×#29742
#1742461