Finite Group Databasebeta

← #29607283

Subgroups — #29607283

Order 15616.

1566 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
1C2×#16839624C211yesmaximal, characteristic
2C2×#16839619C211yesmaximal, characteristic
3C2×#16839609C211yesmaximal, characteristic
4#21444084C211yesmaximal, characteristic, complemented
5#21444078C211yesmaximal, characteristic, complemented
6#21444075C211yesmaximal, characteristic, complemented
7#21444065C212yesmaximal
8#21444061C212yesmaximal
9#21444048C212yesmaximal, complemented
10#21444044C212yesmaximal, complemented
11C61×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×C4⋊Dic61C2212yes—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×#11758198C2212yes—
16C22×#11758198C2211yescharacteristic
17C22×#11758198C2211yescharacteristic
18C22×#11758198C2212yes—
19C22×C4×Dic61C2212yes—
20#16839624—24no—
21#16839619—24no—
22#16839609C418yescomplemented
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2211yescharacteristic
33C22×C4⋊C4×C61C2212yes—
34C22×C4⋊C4×C61C2211yescharacteristic
35C2×#95×C61C2211yescharacteristic
36C2×#95×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C24×D61C2311yescharacteristic
39C23×Dic61C2311yescharacteristic
40C23×Dic61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2312yes—
43C23×Dic61C2311yescharacteristic
44C22×C4×D61—42no—
45C22×C4×D61—42no—
46C22×C4×D61C2×C414yes—
47C22×C4×D61D414yes—
48C22×C4×D61Q814yes—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—24no—
51C2×C4⋊Dic61—42no—
52C2×C4⋊Dic61—42no—
53C2×C4⋊Dic61—42no—
54C2×#11758198—24no—
55C2×#11758198—24no—
56C2×#11758198—24no—
57C2×#11758198—22no—
58C2×#11758198—22no—
59C2×#11758198—42no—
60C2×#11758198—42no—
61C2×#11758198—42no—
62C2×#11758198—42no—
63C2×#11758198C2×C418yes—
64C2×C4×Dic61—42no—
65C2×C4×Dic61—42no—
66C2×C4×Dic61C2×C418yes—
67#14519684—22no—
68#14519684—24no—
69#14519684—24no—
70#14519684—22no—
71#14519684—22no—
72#14519684—42no—
73#14519684—42no—
74#14519684—42no—
75#14519684—42no—
76#14519684C2×C414yes—
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2312yes—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—24no—
84C2×C4⋊C4×C61—42no—
85C2×C4⋊C4×C61—42no—
86C2×C4⋊C4×C61—42no—
87#95×C61—22no—
88#95×C61—24no—
89#95×C61C2×C414yes—
90C24×C61C2411yescharacteristic
91C23×D61—22no—
92C23×D61—22no—
93C23×D61—22no—
94C23×D61—22no—
95C23×D61—21no—
96C23×D61—21no—
97C23×D61—21no—
98C23×D61—21no—
99C23×D61C22×C412yes—
100C23×D61C2×D412yes—
101C23×D61C2×Q812yes—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C414yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×Q812yes—
129C22×C4×C61#4114yes—
130C22×C4×C61#4114yes—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C22×C414yes—
157C22×Dic61C22×C412yes—
158C22×Dic61C2×D412yes—
159C22×Dic61C2×Q812yes—
160C2×C4×D61—48no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×C4×D61—48no—
164C2×C4×D61C4⋊C4116yescomplemented
165C4⋊C4×C61—44no—
166C4⋊C4×C61—48no—
167#11758198—216no—
168#11758198—44no—
169#11758198—44no—
170#11758198—48no—
171C4⋊Dic61—44no—
172C4⋊Dic61—48no—
173C4×Dic61—48no—
174C23×C61C23×C411yescharacteristic
175C23×C61C22×D411yescharacteristic
176C23×C61C22×Q811yescharacteristic
177C23×C61C2×#4112yes—
178C23×C61C2×#4112yes—
179C23×C61H(5,𝔽2)12yes—
180C23×C61H(5,𝔽2)12yes—
181C23×C61H(5,𝔽2)11yescharacteristic
182C23×C61H(5,𝔽2)11yescharacteristic
183C23×C612−1+411yescharacteristic
184C23×C612−1+411yescharacteristic
185C22×D61—24no—
186C22×D61—24no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—24no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—24no—
197C22×D61—22no—
198C22×D61—22no—
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×D61C2×C4⋊C414yes—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
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×C61C2×C4⋊C414yes—
236C2×C4×C61#11718yes—
237C2×Dic61—24no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—42no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—41no—
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×Dic61C2×C4⋊C414yes—
266C4×D61—416no—
267#14355—611nomaximal, Sylow
268C22×C61C22×C4⋊C411yescharacteristic
269C22×C61C2×#11712yes—
270C22×C61#51812yes—
271C22×C61#52011yescharacteristic
272C22×C61#52011yescharacteristic
273C22×C61#53211yescharacteristic
274C22×C61#53211yescharacteristic
275C22×C61#53312yes—
276C22×C61#53411yescharacteristic
277C22×C61#53411yescharacteristic
278C22×C61#53511yescharacteristic
279C22×C61#53611yescharacteristic
280C22×C61#53812yes—
281C22×C61#53812yes—
282C22×C61#53912yes—
283C22×C61#53912yes—
284C22×C61#54012yes—
285C22×C61#54012yes—
286C22×C61#54212yes—
287C22×C61#54212yes—
288C22×C61#54312yes—
289C22×C61#54411yescharacteristic
290C22×C61#54411yescharacteristic
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—24no—
294C2×D61—44no—
295C2×D61—44no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—44no—
301C2×D61—44no—
302C4×C61—24no—
303C4×C61—28no—
304C4×C61—81no—
305C4×C61—82no—
306C4×C61—81no—
307Dic61—24no—
308Dic61—44no—
309Dic61—82no—
310Dic61—81no—
311Dic61—81no—
312C2×#381—611no—
313C2×#380—611no—
314C2×C4×C22⋊C4—611no—
315#1444—611no—
316#1444D6111yescharacteristic, complemented
317#1442—611no—
318#1442—611no—
319#1439—612no—
320#1439—612no—
321C4⋊#95—612no—
322#1434—612no—
323C2×C61#228511yescharacteristic
324C2×C61#231511yescharacteristic
325C2×C61#231711yescharacteristic
326C2×C61#231812yes—
327C2×C61#231912yes—
328C2×C61#266111yescharacteristic
329C2×C61#266111yescharacteristic
330C2×C61#266311yescharacteristic
331C2×C61#266311yescharacteristic
332C2×C61#266512yes—
333C2×C61#266712yes—
334D61—44no—
335C24×C4—611no—
336C22×C4⋊C4—612no—
337C22×C4⋊C4—611no—
338C22×C4⋊C4C2×D6112yes—
339C22×C4⋊C4C2×D6111yescharacteristic
340C22×C22⋊C4—612no—
341C22×C22⋊C4—612no—
342C22×C22⋊C4—611no—
343C22×C22⋊C4—611no—
344C22×C42—612no—
345#381—1224no—
346#380—1224no—
347C4×C22⋊C4—618no—
348C2×#95—611no—
349C2×#95—611no—
350C2×#95—612no—
351C2×#95—612no—
352C2×#95—611no—
353C2×#95—611no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95C2×D6112yes—
359C2×#95C2×D6111yescharacteristic
360C2×#95C2×D6111yescharacteristic
361C61#1435511yescharacteristic, complemented, Sylow
362C25—611no—
363C23×C4—611no—
364C23×C4—614no—
365C23×C4—614no—
366C23×C4—614no—
367C23×C4—612no—
368C23×C4—611no—
369C23×C4—612no—
370C23×C4—611no—
371C23×C4—2442no—
372C23×C4—2442no—
373C23×C4C22×D6111yescharacteristic
374C23×C4C22×D6112yes—
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C23×C4C22×D6112yes—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—42no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1224no—
385C2×C4⋊C4—2442no—
386C2×C4⋊C4—2442no—
387C2×C4⋊C4—2442no—
388C2×C22⋊C4—618no—
389C2×C22⋊C4—1224no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4—1224no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—2442no—
395C2×C22⋊C4—2442no—
396C2×C22⋊C4—2442no—
397C2×C22⋊C4—2442no—
398C2×C42—618no—
399C2×C42—2442no—
400C2×C42—2442no—
401#95—22no—
402#95—24no—
403#95—614no—
404#95—1224no—
405#95—1222no—
406#95—1222no—
407#95—1222no—
408#95—1224no—
409#95—2442no—
410#95—2442no—
411#95—2442no—
412#95—2442no—
413#95C4×D6114yes—
414C24—612no—
415C24—612no—
416C24—612no—
417C24—1222no—
418C24—1222no—
419C24—1221no—
420C24—1222no—
421C24—1221no—
422C24—1222no—
423C24—1221no—
424C24—1221no—
425C24C23×D6111yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—6116no—
453C22×C4—614no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—2448no—
480C22×C4—2444no—
481C22×C4—2444no—
482C22×C4—2448no—
483C22×C4C2×C4×D6112yes—
484C22×C4C2×C4×D6114yes—
485C22×C4D4×D6112yes—
486C22×C4#1175822414yes—
487C22×C4Q8×D6112yes—
488C22×C4#1175822614yes—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—2444no—
492C4⋊C4—2448no—
493C22⋊C4—12216no—
494C22⋊C4—2444no—
495C22⋊C4—2444no—
496C22⋊C4—2448no—
497C42—2448no—
498C23—614no—
499C23—1224no—
500C23—1224no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—1222no—
505C23—1224no—
506C23—1222no—
507C23—1222no—
508C23—1222no—
509C23—1222no—
510C23—1224no—
511C23—1222no—
512C23—1222no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23—2442no—
520C23—2442no—
521C23C22×C4×D6111yescharacteristic
522C23C2×D4×D6111yescharacteristic
523C23C2×Q8×D6111yescharacteristic
524C23C2×#1175822412yes—
525C23C2×#1175822612yes—
526C23#1451982212yes—
527C23#1451982211yescharacteristic
528C23#1451982312yes—
529C23#1451982311yescharacteristic
530C23#1451982411yescharacteristic
531C23#1451982511yescharacteristic
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
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—1224no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—1224no—
567C2×C4—24416no—
568C2×C4—2441no—
569C2×C4—2442no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
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×C4C4⋊C4×D6114yes—
590C2×C4#1451973518yes—
591C22—1224no—
592C22—1224no—
593C22—1224no—
594C22—2444no—
595C22—2442no—
596C22—2442no—
597C22—2442no—
598C22—2442no—
599C22—2444no—
600C22—2444no—
601C22—2444no—
602C22C2×C4⋊C4×D6111yescharacteristic
603C22C2×#1451973512yes—
604C22#1684023412yes—
605C22#1684024111yescharacteristic
606C22#1684024411yescharacteristic
607C22#1684031011yescharacteristic
608C22#1684031311yescharacteristic
609C22#1684031712yes—
610C22#1684032211yescharacteristic
611C22#1684032311yescharacteristic
612C22#1684032811yescharacteristic
613C22#1684033511yescharacteristic
614C22#1684034712yes—
615C22#1684035212yes—
616C22#1684035612yes—
617C22#1684036112yes—
618C22#1684036512yes—
619C22#1684036812yes—
620C22#1684038512yes—
621C22#1684038612yes—
622C22#1684038712yes—
623C22#1684038911yescharacteristic
624C22#1684039211yescharacteristic
625C4—24no—
626C4—28no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4—1224no—
631C4—4881no—
632C4—4882no—
633C4—4881no—
634C4—2444no—
635C2—2444no—
636C2#2144822411yescharacteristic
637C2#2144843011yescharacteristic
638C2#2144844111yescharacteristic
639C2#2144844712yes—
640C2#2144845812yes—
641C2#2145195011yescharacteristic
642C2#2145195211yescharacteristic
643C2#2145197211yescharacteristic
644C2#2145197411yescharacteristic
645C2#2145199412yes—
646C2#2145201812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1444C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1444D61
hypocenterC61#14355
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
4. #21444084C2semidirect
5. #21444078C2semidirect
6. #21444075C2semidirect
9. #21444048C2semidirect
10. #21444044C2semidirect
11. C61×#1444C2semidirect
22. #16839609C4semidirect
164. C2×C4×D61C4⋊C4semidirect
316. #1444D61semidirect
361. C61#14355semidirect

Maximal subgroups

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

subgroupindex
C2×#168396242
C2×#168396192
C2×#168396092
#214440842
#214440782
#214440752
#214440652
#214440612
#214440482
#214440442
C61×#14442
#1435561