Finite Group Databasebeta

← #29648402

Subgroups — #29648402

Order 15616.

996 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#21459116C211yesmaximal, characteristic
2#432×D61C211yesmaximal, characteristic, complemented
3C2×#16839814C211yesmaximal, characteristic, complemented
4#21444526C212yesmaximal, complemented
5#21444523C212yesmaximal, complemented
6#21444519C211yesmaximal, characteristic, complemented
7#21444515C211yesmaximal, characteristic, complemented
8#21444512C211yesmaximal, characteristic, complemented
9#21444475C212yesmaximal, complemented
10#21444467C212yesmaximal, complemented
11C61×#1552C211yesmaximal, characteristic, complemented
12#16840317C2211yescharacteristic
13#16840317C2211yescharacteristic
14#16840313C2211yescharacteristic
15#16840313C2211yescharacteristic
16#16840312C2211yescharacteristic
17#16840299—21no—
18#16840299—21no—
19#16840298—21no—
20#16840298—21no—
21#16840297C412yes—
22#16840297C412yes—
23#16840275—21no—
24#16840275—21no—
25#16840274C412yes—
26C4×#11758226—21no—
27C4×#11758226—21no—
28#16840230C412yes—
29C2×C42×D61C2211yescharacteristic
30#16839819C2211yescharacteristic
31#16839818C2212yes—
32#16839817—21no—
33#16839816—21no—
34#16839814—21no—
35#16839814—21no—
36#16839814C2212yes—
37#16839814C2212yes—
38C2×C4⋊#92418C2211yescharacteristic
39C4⋊C8×D61C2212yes—
40C2×#14519689—21no—
41C22⋊C8×D61—21no—
42#97×D61—21no—
43C2×C8×Dic61—21no—
44#16839510C2212yescomplemented
45#16839510C2212yescomplemented
46#16839508C2212yescomplemented
47#16839508C2212yescomplemented
48#16839505C2212yescomplemented
49#16839505C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#432C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C61×#332C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×#11758226—21no—
56C2×#11758226—21no—
57C2×#11758222C2×C412yes—
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×#92418—21no—
62C2×C8×D61—22no—
63#14519777C2311yescharacteristic
64#14519774C2311yescharacteristic
65#14519772—22no—
66#14519772—22no—
67#14519768—21no—
68#14519768—21no—
69#14519766—21no—
70#14519766—21no—
71#14519765C2×C412yes—
72#14519762—22no—
73#14519762—22no—
74#14519761—22no—
75#14519757C2×C412yes—
76#14519757C2×C412yes—
77#14519753—21no—
78#14519753—21no—
79#14519752—21no—
80#14519752—21no—
81#14519752C2×C412yes—
82#14519752C2×C412yes—
83#14519751—21no—
84#14519751—21no—
85#14519747—22no—
86Q8×Dic61—21no—
87Q8×Dic61—21no—
88#14519742C2×C412yes—
89#14519742C2×C412yes—
90C4×D244—22no—
91C4×D244—22no—
92C4×D244—21no—
93C4×D244—21no—
94#14519739—22no—
95#14519739—22no—
96#14519739—21no—
97#14519739—21no—
98#14519739C2×C412yes—
99#14519739C2×C412yes—
100#14519737—22no—
101#14519737—22no—
102#14519735—21no—
103#14519735—21no—
104#14519735—22no—
105#14519735—22no—
106#14519734—22no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C42×D61—22no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊#92418—21no—
120C4⋊#92418C2312yes—
121C4⋊#92418C2312yes—
122#14519706—22no—
123C22⋊#92418—21no—
124#14519689—22no—
125#14519689—22no—
126#14519689—22no—
127#14519688—21no—
128#14519687—22no—
129C8×Dic61—22no—
130C8×Dic61—22no—
131C22×C8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133C4⋊Q8×C61C2311yescharacteristic
134#125×C61C2×C412yes—
135C22⋊Q8×C61—21no—
136C22⋊Q8×C61—21no—
137C4×Q8×C61—21no—
138C4×Q8×C61—21no—
139#117×C61C2×C412yes—
140C2×C42×C61C2311yescharacteristic
141C4⋊C8×C61—21no—
142C4⋊C8×C61C2312yes—
143C4⋊C8×C61C2312yes—
144C22⋊C8×C61—21no—
145#97×C61—21no—
146C23×D61C22⋊C411yescharacteristic
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155C22×Dic61C22⋊C411yescharacteristic
156#11758226—42no—
157#11758226—42no—
158#11758226—44no—
159#11758222—44no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244C22×C412yes—
165C2×C4×D61—24no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—41no—
170C2×C4×D61—41no—
171C2×C4×D61—42no—
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22×C411yescharacteristic
179C2×C4×D61C22×C411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×Dic122C22×C412yes—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C2×C8×C61—42no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61C22×C412yes—
193C4⋊C4×C61C22×C412yes—
194C22⋊C4×C61—22no—
195C42×C61—22no—
196C42×C61C22×C411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C42×C61C2411yescharacteristic
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207#11758198—22no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—22no—
211C4⋊Dic61—22no—
212#11758196—22no—
213#11758196—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22×C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—42no—
231C8×D61—44no—
232C23×C61C2×C22⋊C411yescharacteristic
233C22×D61—22no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—41no—
237C22×D61C2×C22⋊C411yescharacteristic
238C22×D61C2×C22⋊C411yescharacteristic
239Q8×C61—42no—
240Q8×C61—41no—
241Q8×C61—41no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C22⋊D61—44no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265D244—42no—
266D244—44no—
267D244—44no—
268D244—42no—
269D244—41no—
270D244—41no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—21no—
274C4×D61—21no—
275C4×D61—21no—
276C4×D61—21no—
277C4×D61—22no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—44no—
285C4×D61—44no—
286C4×D61—44no—
287C4×D61#13514yes—
288Dic122—42no—
289C8×C61—42no—
290C8×C61—42no—
291#92418—42no—
292#92418—42no—
293#17713—611nomaximal, Sylow
294C22×C61—21no—
295C22×C61—21no—
296C22×C61#41312yes—
297C22×C61C22×C22⋊C411yescharacteristic
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—21no—
303C2×D61—44no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61#41612yes—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—21no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61C2×#13512yes—
315C4×C61#57511yescharacteristic
316C4×C61#57511yescharacteristic
317Dic61—21no—
318Dic61—21no—
319Dic61—21no—
320Dic61—21no—
321Dic61—21no—
322Dic61—42no—
323Dic61#41612yes—
324#3444—611no—
325C2×#432—611no—
326C2×#431—611no—
327#1554—612no—
328#1553—612no—
329#1552—611no—
330#1552D6111yescharacteristic, complemented
331#1551—611no—
332#1551—611no—
333#1542—612no—
334#1540—612no—
335C2×C61—22no—
336C2×C61C2×#41311yescharacteristic
337C2×C61C2×#41611yescharacteristic
338C2×C61#289611yescharacteristic
339D61—22no—
340D61—41no—
341D61—82no—
342#533—611no—
343#533—611no—
344#533C2×D6111yescharacteristic
345#532—611no—
346#532—611no—
347#532—611no—
348#529—612no—
349#529—612no—
350#529—1221no—
351#529—1221no—
352#529—1221no—
353#529—1221no—
354#525—612no—
355#525—1221no—
356#525—1221no—
357C4×#41—612no—
358C4×#41—1221no—
359C4×#41—1221no—
360C22×C42—611no—
361#432—611no—
362#432—612no—
363#432—1221no—
364#432—1221no—
365#432C2×D6111yescharacteristic
366#431—612no—
367#431—612no—
368#431—1221no—
369#431—1221no—
370C2×C4⋊C8—611no—
371C2×C4⋊C8—612no—
372C2×C4⋊C8C2×D6111yescharacteristic
373C2×C22⋊C8—1221no—
374C2×C22⋊C8—1221no—
375C2×#97—1221no—
376C2×C4×C8—1221no—
377#333—612no—
378#333—612no—
379#332—612no—
380#332—612no—
381#332C2×D6112yescomplemented
382#332C2×D6112yescomplemented
383#331—612no—
384#331—612no—
385C61#1771311yescharacteristic, complemented, Sylow
386C2×#41—612no—
387C2×#41—1221no—
388C2×#41—1221no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C22×C8—21no—
393C22×C8—1222no—
394C22×C8—1221no—
395C4⋊Q8—611no—
396C4⋊Q8C22×D6111yescharacteristic
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#126—1222no—
400#126—1222no—
401#125—1221no—
402#125—1221no—
403#125C4×D6112yes—
404#124—612no—
405#124—1221no—
406#124—1221no—
407#123—1222no—
408#123—1222no—
409#123—1222no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—612no—
413C22⋊Q8—612no—
414C22⋊Q8—1221no—
415C22⋊Q8—1221no—
416#121—612no—
417#121—612no—
418#121—1221no—
419#121—1221no—
420#121—1221no—
421#121—1221no—
422#120—1222no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—612no—
426C4×Q8—612no—
427C4×Q8—1221no—
428C4×Q8—1221no—
429C4×D4—612no—
430C4×D4—612no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1221no—
434C4×D4—1221no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4—1221no—
440C4×D4—1221no—
441#117—1222no—
442#117—1222no—
443#117—1222no—
444#117—1221no—
445#117—1221no—
446#117C4×D6112yes—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—1222no—
459C2×C42C22×D6111yescharacteristic
460C4⋊C8—21no—
461C4⋊C8—612no—
462C4⋊C8—612no—
463C4⋊C8—1222no—
464C4⋊C8—1221no—
465C4⋊C8C22×D6112yes—
466C4⋊C8C22×D6112yes—
467C22⋊C8—21no—
468C22⋊C8—1222no—
469C22⋊C8—1222no—
470C22⋊C8—1222no—
471C22⋊C8—1221no—
472#97—21no—
473#97—1222no—
474#97—1221no—
475C4×C8—1222no—
476C4×C8—1222no—
477C24—611no—
478#41—2444no—
479#41—2444no—
480#41—2442no—
481#41—2442no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—612no—
485C2×D4—612no—
486C2×D4—1222no—
487C2×D4—1222no—
488C2×D4—1222no—
489C2×D4—1221no—
490C2×D4—1221no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—1224no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—2441no—
511C22×C4—2441no—
512C22×C4—2442no—
513C22×C4C2×C4×D6111yescharacteristic
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—42no—
519C2×C8—1222no—
520C2×C8—1222no—
521C2×C8—2444no—
522C2×C8—2442no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4C2×C4×D6112yes—
534C4⋊C4C2×C4×D6112yes—
535C22⋊C4—22no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C42—22no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42—1222no—
558C42—1222no—
559C42C2×C4×D6111yescharacteristic
560C42D4×D6111yescharacteristic
561C42D4×D6111yescharacteristic
562C42C23×D6111yescharacteristic
563C23—611no—
564C23—611no—
565C23—1222no—
566C23—1222no—
567C23—1222no—
568C23—2441no—
569C23C22⋊C4×D6111yescharacteristic
570Q8—42no—
571Q8—41no—
572Q8—41no—
573Q8—2442no—
574D4—2442no—
575D4—2444no—
576D4—2444no—
577D4—2444no—
578D4—2442no—
579D4—2441no—
580D4—2441no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—611no—
588C2×C4—614no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—611no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1222no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2444no—
611C2×C4—2444no—
612C2×C4—2444no—
613C2×C4—2441no—
614C2×C4C22⋊C4×D6111yescharacteristic
615C2×C4C22⋊C4×D6111yescharacteristic
616C2×C4C22⋊C4×D6111yescharacteristic
617C2×C4C22×C4×D6111yescharacteristic
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C8—42no—
621C8—42no—
622C8—2442no—
623C8—2442no—
624C22—21no—
625C22—21no—
626C22—612no—
627C22—1221no—
628C22—1221no—
629C22—1221no—
630C22—1221no—
631C22—1221no—
632C22—2444no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22#1683974512yes—
637C22C2×C22⋊C4×D6111yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—42no—
644C4—612no—
645C4—1221no—
646C4—1221no—
647C4—1221no—
648C4—1221no—
649C4—1221no—
650C4—2442no—
651C4#135×D6112yes—
652C4#1684055811yescharacteristic
653C4#1684056111yescharacteristic
654C2—22no—
655C2—1222no—
656C2—4882no—
657C2—2441no—
658C2C2×#1683974511yescharacteristic
659C2#416×D6111yescharacteristic
660C2#2145433511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1552C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC42×C61C24
hypercenter#1552D61
hypocenterC61#17713
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
2. #432×D61C2semidirect
3. C2×#16839814C2semidirect
4. #21444526C2semidirect
5. #21444523C2semidirect
6. #21444519C2semidirect
7. #21444515C2semidirect
8. #21444512C2semidirect
9. #21444475C2semidirect
10. #21444467C2semidirect
11. C61×#1552C2semidirect
44. #16839510C22semidirect
45. #16839510C22semidirect
46. #16839508C22semidirect
47. #16839508C22semidirect
48. #16839505C22semidirect
49. #16839505C22semidirect
53. C61×#332C22semidirect
54. C61×#332C22semidirect
330. #1552D61semidirect
381. #332C2×D61semidirect
382. #332C2×D61semidirect
385. C61#17713semidirect

Maximal subgroups

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

subgroupindex
#214591162
#432×D612
C2×#168398142
#214445262
#214445232
#214445192
#214445152
#214445122
#214444752
#214444672
C61×#15522
#1771361