Finite Group Databasebeta

← #29609022

Subgroups — #29609022

Order 15616.

1918 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519765C211yesmaximal, characteristic
2C2×#16839666C212yesmaximal, complemented
3#21444063C212yesmaximal, complemented
4#21444063C212yesmaximal, complemented
5#21444059C212yesmaximal
6#21444055C212yesmaximal, complemented
7#21444055C212yesmaximal, complemented
8#21444046C211yesmaximal, characteristic, complemented
9C61×#1434C211yesmaximal, characteristic, complemented
10C23×Dic122C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519765—28no—
13C2×#14519765C418yescomplemented
14C22×#11758196C2212yes—
15C22×#11758198C2212yes—
16C22×#11758198C2212yes—
17C22×#11758198C2212yes—
18C22×#11758198C2212yes—
19C22×#11758198C2212yes—
20C22×#11758198C2212yes—
21C22×C4×Dic61C2212yes—
22#16839666—28no—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2211yescharacteristic
29C22×C42×C61C2211yescharacteristic
30C2×#95×C61C2212yes—
31C2×#95×C61C2212yes—
32C2×#95×C61C2211yescharacteristic
33C2×#95×C61C2211yescharacteristic
34C24×D61C2312yes—
35C22×Dic122—41no—
36C22×Dic122—42no—
37C22×Dic122—41no—
38C22×Dic122C2×C414yes—
39C22×Dic122D418yes—
40C22×D244—24no—
41C22×D244—42no—
42C22×D244—41no—
43C22×D244—41no—
44C22×D244C2×C414yes—
45C23×Dic61C2312yes—
46C23×Dic61C2312yes—
47C23×Dic61C2312yes—
48#14519765—232no—
49C2×#11758196—24no—
50C2×#11758196—42no—
51C2×#11758196—42no—
52C2×#11758198—24no—
53C2×#11758198—24no—
54C2×#11758198—24no—
55C2×#11758198—24no—
56C2×#11758198—24no—
57C2×#11758198—24no—
58C2×#11758198—24no—
59C2×#11758198—24no—
60C2×#11758198—42no—
61C2×#11758198—42no—
62C2×#11758198—42no—
63C2×#11758198—42no—
64C2×#11758198—42no—
65C2×#11758198—42no—
66C2×#11758198—42no—
67C2×#11758198—42no—
68C2×#11758198C2×C418yes—
69C2×#11758198C2×C418yes—
70C2×C4×Dic61—24no—
71C2×C4×Dic61—42no—
72C2×C4×Dic61—42no—
73#14519684—24no—
74#14519684—24no—
75#14519684—24no—
76#14519684—24no—
77#14519684—41no—
78#14519684—41no—
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2312yes—
83C23×C4×C61C2311yescharacteristic
84C2×C42×C61—24no—
85C2×C42×C61C2×C414yes—
86#95×C61—24no—
87#95×C61—24no—
88#95×C61—41no—
89#95×C61—41no—
90C24×C61C2411yescharacteristic
91C23×D61—22no—
92C23×D61—22no—
93C23×D61—22no—
94C23×D61—22no—
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61C22×C414yes—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
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×C61C22×C412yes—
133C22×C4×C61C2×D414yes—
134C22×C4×C61#4114yes—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
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×Dic61C22×C414yes—
153C22×Dic61C2×D414yes—
154C22×Dic61C2×D414yes—
155C22×Dic61#4114yes—
156C2×D244—28no—
157C2×D244—44no—
158C2×D244—44no—
159C2×D244—48no—
160C2×D244—48no—
161C2×Dic122—44no—
162C2×Dic122—44no—
163C2×Dic122—48no—
164C2×Dic122—48no—
165C2×Dic122C22⋊C4116yescomplemented
166C42×C61—28no—
167#11758198—216no—
168#11758198—216no—
169#11758198—48no—
170#11758198—48no—
171#11758198—48no—
172#11758198—48no—
173#11758196—48no—
174C4×Dic61—48no—
175C23×C61C23×C411yescharacteristic
176C23×C61C22×D412yes—
177C23×C61C2×#4112yes—
178C23×C61C2×#4112yes—
179C23×C61H(5,𝔽2)12yes—
180C23×C61H(5,𝔽2)11yescharacteristic
181C23×C61H(5,𝔽2)11yescharacteristic
182C23×C61H(5,𝔽2)11yescharacteristic
183C23×C61H(5,𝔽2)11yescharacteristic
184C23×C612−1+411yescharacteristic
185C23×C612−1+411yescharacteristic
186C22×D61—24no—
187C22×D61—24no—
188C22×D61—24no—
189C22×D61—24no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61C2×C22⋊C414yes—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
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×Dic61—42no—
270C2×Dic61C2×C22⋊C418yes—
271D244—416no—
272Dic122—416no—
273#14492—611nomaximal, Sylow
274C22×C61C22×C22⋊C411yescharacteristic
275C22×C61C4×#4111yescharacteristic
276C22×C61C4×#4111yescharacteristic
277C22×C61#51811yescharacteristic
278C22×C61#51811yescharacteristic
279C22×C61#52011yescharacteristic
280C22×C61#52011yescharacteristic
281C22×C61#53412yes—
282C22×C61#53412yes—
283C22×C61#53812yes—
284C22×C61#53812yes—
285C22×C61#54212yes—
286C22×C61#54212yes—
287C22×C61#54812yes—
288C22×C61#54812yes—
289C22×C61#55012yes—
290C22×C61#55012yes—
291C22×C61#55112yes—
292C22×C61#55512yes—
293C22×C61H(3,𝔽4)11yescharacteristic
294C22×C61#56211yescharacteristic
295C22×C61#56211yescharacteristic
296C22×C61#56311yescharacteristic
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—44no—
301C2×D61—44no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—82no—
305C2×D61—82no—
306C2×D61—82no—
307C2×D61—82no—
308C2×D61—82no—
309C2×D61—82no—
310C2×D61—82no—
311C2×D61—82no—
312C4×C61—24no—
313C4×C61—42no—
314C4×C61—42no—
315C4×C61—44no—
316C4×C61—82no—
317Dic61—28no—
318Dic61—44no—
319Dic61—82no—
320C22×#124—611no—
321C2×#391—612no—
322#1439—612no—
323#1439—612no—
324#1438—612no—
325#1437—612no—
326#1437—612no—
327#1434—611no—
328#1434D6111yescharacteristic, complemented
329C2×C61#230811yescharacteristic
330C2×C61#233711yescharacteristic
331C2×C61#233711yescharacteristic
332C2×C61#234111yescharacteristic
333C2×C61#234111yescharacteristic
334C2×C61#237211yescharacteristic
335C2×C61#237911yescharacteristic
336C2×C61#273012yes—
337C2×C61#273012yes—
338C2×C61#273112yes—
339C2×C61#273312yes—
340D61—84no—
341C23×Q8—611no—
342C23×D4—611no—
343C2×#124—618no—
344C2×#124—1228no—
345C22×C4⋊C4—612no—
346C22×C22⋊C4—612no—
347C22×C22⋊C4—612no—
348C22×C22⋊C4—612no—
349C22×C22⋊C4—612no—
350C22×C22⋊C4—612no—
351C22×C22⋊C4—612no—
352C22×C42—612no—
353C22×C42C2×D6111yescharacteristic
354#391—1228no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—611no—
360C2×#95—611no—
361C2×#95C2×D6112yes—
362C2×#95C2×D6112yes—
363C2×#95C2×D6111yescharacteristic
364C2×#95C2×D6111yescharacteristic
365C61#1449211yescharacteristic, complemented, Sylow
366C25—612no—
367C22×Q8—614no—
368C22×Q8—618no—
369C22×Q8—2441no—
370C22×Q8—2442no—
371C22×Q8—2441no—
372C22×D4—614no—
373C22×D4—1224no—
374C22×D4—2441no—
375C22×D4—2441no—
376C22×D4—2442no—
377C23×C4—612no—
378C23×C4—612no—
379C23×C4—612no—
380C23×C4C22×D6112yes—
381C23×C4C22×D6111yescharacteristic
382C23×C4C22×D6111yescharacteristic
383C23×C4C22×D6112yes—
384C23×C4C22×D6111yescharacteristic
385#124—12232no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—2442no—
388C2×C4⋊C4—2442no—
389C2×C22⋊C4—618no—
390C2×C22⋊C4—618no—
391C2×C22⋊C4—1224no—
392C2×C22⋊C4—1224no—
393C2×C22⋊C4—1224no—
394C2×C22⋊C4—1224no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4—1224no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—2442no—
400C2×C22⋊C4—2442no—
401C2×C22⋊C4—2442no—
402C2×C22⋊C4—2442no—
403C2×C22⋊C4—2442no—
404C2×C22⋊C4—2442no—
405C2×C22⋊C4—2442no—
406C2×C22⋊C4—2442no—
407C2×C42—24no—
408C2×C42—1224no—
409C2×C42—2442no—
410C2×C42—2442no—
411C2×C42C4×D6114yes—
412#95—24no—
413#95—24no—
414#95—41no—
415#95—41no—
416#95—1224no—
417#95—1224no—
418#95—1224no—
419#95—1224no—
420#95—2441no—
421#95—2441no—
422C24—614no—
423C24—1222no—
424C24—1222no—
425C24—1222no—
426C24—1222no—
427C24—1222no—
428C24—1222no—
429C24—1222no—
430C24—1222no—
431C24—1222no—
432C24—1222no—
433C24—1222no—
434C24—1222no—
435C24—1222no—
436C24—1222no—
437C24C23×D6111yescharacteristic
438C2×Q8—6116no—
439C2×Q8—2444no—
440C2×Q8—2444no—
441C2×Q8—2448no—
442C2×Q8—2448no—
443C2×D4—1228no—
444C2×D4—2444no—
445C2×D4—2444no—
446C2×D4—2448no—
447C2×D4—2448no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—614no—
473C22×C4—614no—
474C22×C4—614no—
475C22×C4—614no—
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×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4C2×C4×D6112yes—
494C22×C4C2×C4×D6112yes—
495C22×C4C2×C4×D6112yes—
496C22×C4#1175822214yes—
497C22×C4D4×D6114yes—
498C4⋊C4—2448no—
499C22⋊C4—12216no—
500C22⋊C4—12216no—
501C22⋊C4—2448no—
502C22⋊C4—2448no—
503C22⋊C4—2448no—
504C22⋊C4—2448no—
505C42—28no—
506C42—2448no—
507C23—1224no—
508C23—1224no—
509C23—1224no—
510C23—1224no—
511C23—1224no—
512C23—1224no—
513C23—1224no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—2442no—
525C23—2442no—
526C23—2442no—
527C23—2442no—
528C23—2442no—
529C23—2442no—
530C23—2442no—
531C23—2442no—
532C23—2442no—
533C23—2442no—
534C23—2442no—
535C23—2442no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23—2442no—
540C23—2442no—
541C23—2442no—
542C23C22×C4×D6111yescharacteristic
543C23C2×D4×D6112yes—
544C23#41×D6112yes—
545C23C2×#1175822212yes—
546C23#1451982211yescharacteristic
547C23#1451982311yescharacteristic
548C23#1451982311yescharacteristic
549C23#1451982312yes—
550C23#1451982311yescharacteristic
551C23#1451982411yescharacteristic
552C23#1451982511yescharacteristic
553Q8—24416no—
554D4—24416no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—618no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4C22⋊C4×D6114yes—
605C22—4882no—
606C22—4882no—
607C22—4882no—
608C22—4882no—
609C22—4882no—
610C22—4882no—
611C22—4882no—
612C22—4882no—
613C22—2444no—
614C22—2444no—
615C22—2444no—
616C22—2444no—
617C22—2444no—
618C22—2444no—
619C22—2444no—
620C22C2×C22⋊C4×D6111yescharacteristic
621C22#1684023011yescharacteristic
622C22C4×#1175822211yescharacteristic
623C22#1684023611yescharacteristic
624C22#1684023611yescharacteristic
625C22#1684024211yescharacteristic
626C22#1684024311yescharacteristic
627C22#1684032312yes—
628C22#1684032312yes—
629C22#1684034512yes—
630C22#1684035112yes—
631C22#1684037812yes—
632C22#1684038412yes—
633C22#1684042112yes—
634C22#1684042212yes—
635C22#1684043012yes—
636C22#1684043312yes—
637C22#1684043612yes—
638C22#1684047512yes—
639C22#1684050411yescharacteristic
640C22#1684050611yescharacteristic
641C22#1684051011yescharacteristic
642C22#1684051611yescharacteristic
643C4—24no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—82no—
648C4—1228no—
649C4—4882no—
650C4—2444no—
651C2—4884no—
652C2#2144839611yescharacteristic
653C2#2144862911yescharacteristic
654C2#2144863211yescharacteristic
655C2#2144865611yescharacteristic
656C2#2144865911yescharacteristic
657C2#2144892911yescharacteristic
658C2#2144899011yescharacteristic
659C2#2145281612yes—
660C2#2145282012yes—
661C2#2145282912yes—
662C2#2145284612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1434C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenter#1434D61
hypocenterC61#14492
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#16839666C2semidirect
3. #21444063C2semidirect
4. #21444063C2semidirect
6. #21444055C2semidirect
7. #21444055C2semidirect
8. #21444046C2semidirect
9. C61×#1434C2semidirect
13. C2×#14519765C4semidirect
165. C2×Dic122C22⋊C4semidirect
328. #1434D61semidirect
365. C61#14492semidirect

Maximal subgroups

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

subgroupindex
C22×#145197652
C2×#168396662
#214440632
#214440592
#214440552
#214440462
C61×#14342
#1449261