Finite Group Databasebeta

← #29602708

Subgroups — #29602708

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519733C211yesmaximal, characteristic
2C2×#16839626C211yesmaximal, characteristic, complemented
3C2×#16839608C211yesmaximal, characteristic, complemented
4#21444070C212yesmaximal, complemented
5#21444062C212yesmaximal
6#21444056C211yesmaximal, characteristic
7#21444054C211yesmaximal, characteristic, complemented
8#21444052C211yesmaximal, characteristic, complemented
9#21444045C212yesmaximal, complemented
10#21444041C212yesmaximal
11C61×#1437C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C2×#14519733—24no—
14C2×#14519733—24no—
15C2×#14519733C418yescomplemented
16C22×#11758196C2211yescharacteristic
17C22×#11758196C2211yescharacteristic
18C22×#11758196C2212yes—
19C22×C22⋊Dic61C2212yes—
20C22×C22⋊Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24#16839626—24no—
25#16839608—24no—
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C22⋊C4×C61C2212yes—
36C22×C22⋊C4×C61C2211yescharacteristic
37C22×C42×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2212yes—
40C23×Dic61—24no—
41C23×Dic61—22no—
42C23×Dic61—22no—
43C23×Dic61—24no—
44C23×Dic61C2×C414yes—
45C23×Dic61C2312yes—
46C23×Dic61C2311yescharacteristic
47C23×Dic61C2312yes—
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2311yescharacteristic
51#14519733—232no—
52C2×#11758196—22no—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×#11758196—24no—
57C2×#11758196—42no—
58C2×#11758196—42no—
59C2×#11758196C2×C414yes—
60C2×#11758196C2×C414yes—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—24no—
64C2×C22⋊Dic61—24no—
65C2×C22⋊Dic61—42no—
66C2×C22⋊Dic61C2×C414yes—
67C2×C4×Dic61—22no—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—24no—
71C2×C4×Dic61C2×C414yes—
72C2×C4×Dic61C2×C414yes—
73C2×C4×Dic61D414yes—
74C2×C4×Dic61D414yes—
75#14519684—22no—
76#14519684—24no—
77#14519684—24no—
78#14519684—24no—
79#14519684—24no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83#14519684—42no—
84C25×C61C2311yescharacteristic
85C23×C4×C61C2312yes—
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—24no—
92C2×C22⋊C4×C61—24no—
93C2×C22⋊C4×C61—42no—
94C2×C22⋊C4×C61C2×C414yes—
95C2×C42×C61—22no—
96C2×C42×C61—24no—
97#95×C61—22no—
98#95×C61—42no—
99C24×C61—21no—
100C24×C61—21no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61—22no—
106C24×C61C22×C412yes—
107C24×C61#4114yes—
108C24×C61C2411yescharacteristic
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61#4114yes—
133C22×Dic61—216no—
134C22×Dic61—216no—
135C22×Dic61—28no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—28no—
144C22×Dic61—28no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61C22×C412yes—
163C22×Dic61C22×C412yes—
164C22×Dic61C22×C412yes—
165C22×Dic61C22×C412yes—
166C22×Dic61C2×D412yes—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D412yes—
169C22×Dic61C2×D412yes—
170C22×Dic61#4114yes—
171C22×Dic61#4114yes—
172C22⋊C4×C61—28no—
173C22⋊C4×C61—48no—
174C42×C61—28no—
175C22⋊Dic61—28no—
176C22⋊Dic61—48no—
177#11758196—28no—
178#11758196—28no—
179#11758196—48no—
180C4×Dic61—28no—
181C4×Dic61—28no—
182C4×Dic61C22⋊C4116yescomplemented
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61#11714yes—
204C23×C61C23×C411yescharacteristic
205C23×C61C22×D411yescharacteristic
206C23×C61C22×D411yescharacteristic
207C23×C61C2×#4112yes—
208C23×C61C2×#4112yes—
209C23×C61C2×#4112yes—
210C23×C61C2×#4112yes—
211C23×C61H(5,𝔽2)12yes—
212C23×C612−1+412yes—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—42no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C414yes—
237C2×Dic61—232no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—42no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61C2×C22⋊C414yes—
270C2×Dic61C2×C22⋊C414yes—
271C2×Dic61#11714yes—
272#13957—611nomaximal, Sylow
273C22×C61—28no—
274C22×C61—24no—
275C22×C61—42no—
276C22×C61—44no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61—44no—
281C22×C61—42no—
282C22×C61C22×C22⋊C411yescharacteristic
283C22×C61C2×#11711yescharacteristic
284C22×C61C2×#11711yescharacteristic
285C22×C61C4×#4112yes—
286C22×C61#52012yes—
287C22×C61#52511yescharacteristic
288C22×C61#52511yescharacteristic
289C22×C61#52512yes—
290C22×C61#52512yes—
291C22×C61#53212yes—
292C22×C61#53511yescharacteristic
293C22×C61#53611yescharacteristic
294C22×C61#53712yes—
295C22×C61#55112yes—
296C22×C61#55112yes—
297C22×C61#55212yes—
298C22×C61#55212yes—
299C22×C61#55212yes—
300C22×C61#55212yes—
301C22×C61#55512yes—
302C22×C61#55512yes—
303C4×C61—24no—
304C4×C61—42no—
305C4×C61—44no—
306C4×C61—82no—
307Dic61—24no—
308Dic61—24no—
309Dic61—24no—
310Dic61—44no—
311Dic61—42no—
312Dic61—82no—
313C22×#117—611no—
314C2×#382—611no—
315C2×#376—611no—
316#1441—612no—
317#1439—612no—
318#1437—611no—
319#1437D6111yescharacteristic
320#1436—611no—
321#1436—611no—
322#1434—612no—
323C22⋊#95—612no—
324C2×C61—44no—
325C2×C61#228011yescharacteristic
326C2×C61#229711yescharacteristic
327C2×C61#231111yescharacteristic
328C2×C61#234312yes—
329C2×C61#234612yes—
330C2×C61#260912yes—
331C2×C61#260912yes—
332C2×C61#261012yes—
333C2×C61#261112yes—
334C24×C4—611no—
335C2×#117—618no—
336C2×#117—1224no—
337C2×#117—1224no—
338C22×C4⋊C4—611no—
339C22×C4⋊C4—611no—
340C22×C4⋊C4—612no—
341C22×C22⋊C4—611no—
342C22×C22⋊C4—612no—
343C22×C22⋊C4C2×D6112yes—
344C22×C22⋊C4C2×D6111yescharacteristic
345C22×C42—611no—
346C22×C42—611no—
347C22×C42—611no—
348C22×C42C2×D6111yescharacteristic
349#382—1224no—
350#376—1224no—
351C2×#95—612no—
352C2×#95—612no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—611no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95C2×D6111yescharacteristic
361C2×#95C2×D6112yes—
362C61#1395711yescharacteristic, complemented, Sylow
363C25C22×D6111yescharacteristic
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—614no—
368C23×C4—611no—
369C23×C4—612no—
370C23×C4—612no—
371C23×C4—1222no—
372C23×C4—1222no—
373C23×C4—1224no—
374C23×C4—1224no—
375C23×C4C22×D6112yes—
376C23×C4C22×D6111yescharacteristic
377C23×C4C22×D6112yes—
378C23×C4C22×D6111yescharacteristic
379#117—12232no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—614no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—2442no—
388C2×C4⋊C4—2442no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—614no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—2442no—
400C2×C22⋊C4C4×D6114yes—
401C2×C42—22no—
402C2×C42—24no—
403C2×C42—614no—
404C2×C42—614no—
405C2×C42—614no—
406C2×C42—614no—
407C2×C42—1222no—
408C2×C42—1222no—
409C2×C42—1224no—
410C2×C42—1222no—
411#95—22no—
412#95—42no—
413#95—1224no—
414#95—1224no—
415#95—1224no—
416#95—1224no—
417#95—1224no—
418#95—1224no—
419#95—1222no—
420#95—1224no—
421#95—2442no—
422C24—21no—
423C24—21no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×C4×D6112yes—
430C24#1175822214yes—
431C24C23×D6111yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—614no—
459C22×C4—612no—
460C22×C4—614no—
461C22×C4—1228no—
462C22×C4—1228no—
463C22×C4—12216no—
464C22×C4—12216no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1228no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4C2×C4×D6112yes—
491C22×C4C2×C4×D6112yes—
492C22×C4D4×D6112yes—
493C22×C4D4×D6112yes—
494C22×C4#1175822414yes—
495C4⋊C4—1228no—
496C4⋊C4—1228no—
497C4⋊C4—2448no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—1228no—
501C22⋊C4—2448no—
502C42—28no—
503C42—6116no—
504C42—1228no—
505C42—1228no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23#1451973414yes—
527C23C22×C4×D6111yescharacteristic
528C23C2×D4×D6111yescharacteristic
529C23C2×D4×D6111yescharacteristic
530C23#41×D6112yes—
531C23#41×D6112yes—
532C23C2×#1175822412yes—
533C23C2×#1175822212yes—
534C23#1451982212yes—
535C23#1451982412yes—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—614no—
560C2×C4—614no—
561C2×C4—614no—
562C2×C4—12232no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1222no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4C22⋊C4×D6114yes—
595C22—24no—
596C22—28no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22C2×C22⋊C4×D6111yescharacteristic
605C22#117×D6111yescharacteristic
606C22C2×#1451973411yescharacteristic
607C22C4×#1175822412yes—
608C22#1684024112yes—
609C22#1684026811yescharacteristic
610C22#1684027011yescharacteristic
611C22#1684027112yes—
612C22#1684027112yes—
613C22#1684030712yes—
614C22#1684032511yescharacteristic
615C22#1684033211yescharacteristic
616C22#1684034112yes—
617C22#1684043812yes—
618C22#1684044112yes—
619C22#1684044512yes—
620C22#1684044812yes—
621C22#1684044912yes—
622C22#1684045212yes—
623C22#1684047412yes—
624C22#1684047612yes—
625C4—24no—
626C4—42no—
627C4—44no—
628C4—82no—
629C4—1224no—
630C4—1224no—
631C4—1224no—
632C4—4882no—
633C4—2442no—
634C4—2444no—
635C2—44no—
636C2#2144820011yescharacteristic
637C2#2144830411yescharacteristic
638C2#2144840911yescharacteristic
639C2#2144867312yes—
640C2#2144870112yes—
641C2#2145149312yes—
642C2#2145149412yes—
643C2#2145150312yes—
644C2#2145151412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1437C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1437D61
hypocenterC61#13957
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#16839626C2semidirect
3. C2×#16839608C2semidirect
4. #21444070C2semidirect
7. #21444054C2semidirect
8. #21444052C2semidirect
9. #21444045C2semidirect
15. C2×#14519733C4semidirect
182. C4×Dic61C22⋊C4semidirect
362. C61#13957semidirect

Maximal subgroups

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

subgroupindex
C22×#145197332
C2×#168396262
C2×#168396082
#214440702
#214440622
#214440562
#214440542
#214440522
#214440452
#214440412
C61×#14372
#1395761