Finite Group Databasebeta

← #29602544

Subgroups — #29602544

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519733C211yesmaximal, characteristic
2C2×#16839620C211yesmaximal, characteristic
3C2×C22⋊C4×Dic61C211yesmaximal, characteristic
4#21444068C211yesmaximal, characteristic
5#21444066C211yesmaximal, characteristic
6#21444066C211yesmaximal, characteristic
7#21444047C212yesmaximal, complemented
8#21444047C212yesmaximal, complemented
9#21444045C212yesmaximal, complemented
10#21444044C212yesmaximal, complemented
11C61×#1440C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C2×#14519733—24no—
14C2×#14519733—24no—
15C2×#14519733C418yescomplemented
16C22×#11758196C2212yes—
17C22×#11758196C2211yescharacteristic
18C22×#11758196C2211yescharacteristic
19C22×C22⋊Dic61C2211yescharacteristic
20C22×C22⋊Dic61C2211yescharacteristic
21C22×C22⋊Dic61C2211yescharacteristic
22C22×C4×Dic61C2212yes—
23C22×C4×Dic61C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25#16839620—24no—
26C22⋊C4×Dic61—24no—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C22⋊C4×C61C2211yescharacteristic
36C22×C22⋊C4×C61C2211yescharacteristic
37C22×C22⋊C4×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2212yes—
40C23×Dic61—24no—
41C23×Dic61—24no—
42C23×Dic61C2×C414yes—
43C23×Dic61D414yes—
44C23×Dic61D414yes—
45C23×Dic61C2312yes—
46C23×Dic61C2311yescharacteristic
47C23×Dic61C2311yescharacteristic
48C23×Dic61C2312yes—
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2311yescharacteristic
51#14519733—232no—
52C2×#11758196—22no—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×#11758196—24no—
57C2×#11758196—24no—
58C2×#11758196—24no—
59C2×#11758196C2×C414yes—
60C2×#11758196C2×C414yes—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—24no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—42no—
67C2×C22⋊Dic61C2×C414yes—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61—24no—
73C2×C4×Dic61—24no—
74C2×C4×Dic61—24no—
75C2×C4×Dic61C2×C414yes—
76C2×C4×Dic61C2×C414yes—
77#14519684—24no—
78#14519684—24no—
79#14519684—24no—
80#14519684—24no—
81#14519684—24no—
82#14519684—42no—
83#14519684—42no—
84#14519684—42no—
85C25×C61C2311yescharacteristic
86C23×C4×C61C2312yes—
87C23×C4×C61C2311yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2312yes—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—24no—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61—42no—
96C2×C22⋊C4×C61C2×C414yes—
97#95×C61—24no—
98#95×C61—42no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61C22×C412yes—
106C24×C61C2×D412yes—
107C24×C61C2×D412yes—
108C24×C61C2411yescharacteristic
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
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×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C22×C412yes—
132C22×C4×C61#4114yes—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—28no—
136C22×Dic61—28no—
137C22×Dic61—216no—
138C22×Dic61—216no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
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×Dic61C22⋊C4116yes—
161C22×Dic61C22×C412yes—
162C22×Dic61C22×C412yes—
163C22×Dic61C22×C412yes—
164C22×Dic61C22×C412yes—
165C22×Dic61C2×D412yes—
166C22×Dic61C2×D412yes—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D412yes—
169C22×Dic61#4114yes—
170C22×Dic61#4114yes—
171C22×Dic61#4114yes—
172C22⋊C4×C61—28no—
173C22⋊C4×C61—28no—
174C22⋊C4×C61—44no—
175C22⋊Dic61—28no—
176C22⋊Dic61—28no—
177C22⋊Dic61—44no—
178#11758196—28no—
179#11758196—28no—
180#11758196—48no—
181C4×Dic61—28no—
182C4×Dic61—28no—
183C4×Dic61—48no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—24no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—24no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61C2×C22⋊C414yes—
205C23×C61C23×C411yescharacteristic
206C23×C61C22×D411yescharacteristic
207C23×C61C22×D411yescharacteristic
208C23×C61C2×#4112yes—
209C23×C61C2×#4112yes—
210C23×C61C2×#4112yes—
211C23×C61C2×#4112yes—
212C23×C61H(5,𝔽2)12yes—
213C23×C612−1+412yes—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
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×C61—42no—
237C2×C4×C61—42no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—232no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—42no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—42no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—42no—
257C2×Dic61—42no—
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×Dic61C2×C22⋊C414yes—
268C2×Dic61C2×C22⋊C414yes—
269C2×Dic61#11714yes—
270C2×Dic61#11714yes—
271C2×Dic61#12318yes—
272C2×Dic61#12318yes—
273#13939—611nomaximal, Sylow
274C22×C61—24no—
275C22×C61—24no—
276C22×C61—24no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61—44no—
281C22×C61—44no—
282C22×C61—44no—
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61C2×#11711yescharacteristic
285C22×C61C2×#11711yescharacteristic
286C22×C61C4×#4112yes—
287C22×C61#52012yes—
288C22×C61C2×#12312yes—
289C22×C61C2×#12312yes—
290C22×C61#52511yescharacteristic
291C22×C61#52511yescharacteristic
292C22×C61#53212yes—
293C22×C61#53511yescharacteristic
294C22×C61#53611yescharacteristic
295C22×C61#53712yes—
296C22×C61#55212yes—
297C22×C61#55212yes—
298C22×C61#55312yes—
299C22×C61#55312yes—
300C22×C61#55512yes—
301C22×C61#55512yes—
302C22×C61#55612yes—
303C22×C61#55612yes—
304C4×C61—42no—
305C4×C61—44no—
306C4×C61—42no—
307C4×C61—82no—
308Dic61—24no—
309Dic61—42no—
310Dic61—44no—
311Dic61—42no—
312Dic61—82no—
313Dic61#38118yes—
314C22×#117—611no—
315C2×#380—611no—
316C2×C4×C22⋊C4—611no—
317#1440—611no—
318#1440—611no—
319#1440—611no—
320#1440D6111yescharacteristic
321C4⋊#95—612no—
322C4⋊#95—612no—
323#1434—612no—
324#1434—612no—
325C2×C61—44no—
326C2×C61C2×#38111yescharacteristic
327C2×C61#229711yescharacteristic
328C2×C61#231111yescharacteristic
329C2×C61#234612yes—
330C2×C61#234712yes—
331C2×C61#248512yes—
332C2×C61#248512yes—
333C2×C61#249912yes—
334C2×C61#250012yes—
335C24×C4—611no—
336C2×#117—618no—
337C2×#117—1224no—
338C2×#117—1224no—
339C22×C4⋊C4—611no—
340C22×C4⋊C4—611no—
341C22×C4⋊C4—612no—
342C22×C22⋊C4—611no—
343C22×C22⋊C4—611no—
344C22×C22⋊C4—611no—
345C22×C22⋊C4C2×D6111yescharacteristic
346C22×C22⋊C4C2×D6111yescharacteristic
347C22×C22⋊C4C2×D6111yescharacteristic
348C22×C42—611no—
349C22×C42—611no—
350C22×C42—612no—
351#380—1224no—
352C4×C22⋊C4—1224no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95C2×D6112yes—
362C2×#95C2×D6112yes—
363C61#1393911yescharacteristic, complemented, Sylow
364C25C22×D6111yescharacteristic
365C23×C4—611no—
366C23×C4—614no—
367C23×C4—614no—
368C23×C4—614no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—612no—
373C23×C4—612no—
374C23×C4—1224no—
375C23×C4—1224no—
376C23×C4C22×D6112yes—
377C23×C4C22×D6111yescharacteristic
378C23×C4C22×D6111yescharacteristic
379C23×C4C22×D6112yes—
380#117—12232no—
381C2×C4⋊C4—614no—
382C2×C4⋊C4—614no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1224no—
389C2×C4⋊C4—1224no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—614no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—2442no—
403C2×C22⋊C4C4×D6114yes—
404C2×C42—614no—
405C2×C42—614no—
406C2×C42—1222no—
407C2×C42—1222no—
408C2×C42—1222no—
409C2×C42—1222no—
410C2×C42—1224no—
411C2×C42—1224no—
412C2×C42—1224no—
413#95—24no—
414#95—42no—
415#95—1224no—
416#95—1224no—
417#95—1224no—
418#95—1224no—
419#95—1224no—
420#95—2442no—
421#95—2442no—
422#95—2442no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×C4×D6112yes—
430C24D4×D6112yes—
431C24D4×D6112yes—
432C24C23×D6111yescharacteristic
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
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—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—614no—
458C22×C4—614no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—6116no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—614no—
466C22×C4—1228no—
467C22×C4—1228no—
468C22×C4—12216no—
469C22×C4—12216no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
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×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4C2×C4×D6112yes—
494C22×C4C2×C4×D6112yes—
495C22×C4#1175822414yes—
496C4⋊C4—1228no—
497C4⋊C4—1228no—
498C4⋊C4—2448no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—44no—
502C22⋊C4—1228no—
503C22⋊C4—1228no—
504C22⋊C4—2444no—
505C42—1228no—
506C42—1228no—
507C42—2448no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C22⋊C4×D6114yes—
529C23C22×C4×D6111yescharacteristic
530C23C2×D4×D6111yescharacteristic
531C23C2×D4×D6111yescharacteristic
532C23#41×D6112yes—
533C23#41×D6112yes—
534C23#41×D6112yes—
535C23C2×#1175822412yes—
536C23#1451982212yes—
537C23#1451982412yes—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
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—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—614no—
563C2×C4—614no—
564C2×C4—614no—
565C2×C4—618no—
566C2×C4—618no—
567C2×C4—614no—
568C2×C4—12232no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—1222no—
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—2442no—
589C2×C4—2442no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2442no—
596C2×C4—2442no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22C2×C22⋊C4×D6111yescharacteristic
607C22#117×D6111yescharacteristic
608C22#117×D6111yescharacteristic
609C22C4×#1175822412yes—
610C22#1684024112yes—
611C22#123×D6112yes—
612C22#123×D6112yes—
613C22#1684026811yescharacteristic
614C22#1684027011yescharacteristic
615C22#1684030712yes—
616C22#1684032711yescharacteristic
617C22#1684033411yescharacteristic
618C22#1684033912yes—
619C22#1684044812yes—
620C22#1684044912yes—
621C22#1684045512yes—
622C22#1684045712yes—
623C22#1684047412yes—
624C22#1684047612yes—
625C22#1684047912yes—
626C22#1684048112yes—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—82no—
631C4—618no—
632C4—1224no—
633C4—4882no—
634C4—2442no—
635C4—2444no—
636C4—2442no—
637C2—44no—
638C2#381×D6111yescharacteristic
639C2#2144830411yescharacteristic
640C2#2144840611yescharacteristic
641C2#2144870112yes—
642C2#2144870912yes—
643C2#2145010912yes—
644C2#2145011112yes—
645C2#2145026712yes—
646C2#2145027612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1440C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1440D61
hypocenterC61#13939
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
7. #21444047C2semidirect
8. #21444047C2semidirect
9. #21444045C2semidirect
10. #21444044C2semidirect
15. C2×#14519733C4semidirect
363. C61#13939semidirect

Maximal subgroups

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

subgroupindex
C22×#145197332
C2×#168396202
C2×C22⋊C4×Dic612
#214440682
#214440662
#214440472
#214440452
#214440442
C61×#14402
#1393961