Finite Group Databasebeta

← #29607003

Subgroups — #29607003

Order 15616.

1622 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
1C2×#16839673C211yesmaximal, characteristic
2C2×#16839669C211yesmaximal, characteristic, complemented
3C2×#16839609C211yesmaximal, characteristic
4#21444083C211yesmaximal, characteristic, complemented
5#21444080C212yesmaximal, complemented
6#21444065C211yesmaximal, characteristic
7#21444063C212yesmaximal
8#21444054C212yesmaximal, complemented
9#21444044C211yesmaximal, characteristic, complemented
10#21444040C212yesmaximal
11C61×#1446C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15C22×#11758198C2211yescharacteristic
16C22×#11758198C2211yescharacteristic
17C22×#11758198C2212yes—
18C22×#11758198C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20C22×C4×Dic61C2211yescharacteristic
21#16839673C418yescomplemented
22#16839669—24no—
23#16839609—24no—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2212yes—
33C23×Q8×C61C2211yescharacteristic
34C2×#95×C61C2211yescharacteristic
35C2×#95×C61C2211yescharacteristic
36C2×#95×C61C2212yes—
37C2×#95×C61C2212yes—
38C24×D61C2311yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—42no—
47C22×C4×D61—42no—
48C22×C4×D61Q814yescomplemented
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—42no—
51C2×#11758196—24no—
52C2×#11758196—42no—
53C2×#11758196—42no—
54C2×#11758198—22no—
55C2×#11758198—22no—
56C2×#11758198—22no—
57C2×#11758198—22no—
58C2×#11758198—22no—
59C2×#11758198—24no—
60C2×#11758198—42no—
61C2×#11758198—42no—
62C2×#11758198—42no—
63C2×#11758198—42no—
64C2×#11758198C2×C414yes—
65C2×#11758198C2×C418yes—
66C2×C4×Dic61—22no—
67C2×C4×Dic61—42no—
68C2×C4×Dic61—42no—
69C2×C4×Dic61D414yes—
70#14519684—22no—
71#14519684—24no—
72#14519684—24no—
73#14519684—24no—
74#14519684—42no—
75#14519684—42no—
76#14519684—42no—
77#14519684C2×C418yescomplemented
78#14519684C2×C414yescomplemented
79C22×Q8×C61—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61—42no—
82C22×Q8×C61—42no—
83C22×Q8×C61C2×C414yescomplemented
84C23×C4×C61C2312yes—
85C23×C4×C61C2312yes—
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C23×C4×C61C2311yescharacteristic
89#95×C61—22no—
90#95×C61—22no—
91#95×C61—42no—
92#95×C61—42no—
93C24×C61C2411yescharacteristic
94C23×D61—21no—
95C23×D61—21no—
96C23×D61—21no—
97C23×D61—21no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61C22×C412yes—
103C23×D61C2×Q812yes—
104C23×D61#4112yes—
105C2×Q8×C61—28no—
106C2×Q8×C61—48no—
107C2×Q8×C61—44no—
108C2×Q8×C61—44no—
109C2×Q8×C61—48no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
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—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×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61C22×C414yes—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×Q812yes—
137C22×C4×C61#4112yes—
138C22×C4×C61#4112yes—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
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×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61C22×C412yes—
163C22×Dic61C22×C414yes—
164C22×Dic61C2×D412yes—
165C22×Dic61C2×D412yes—
166C22×Dic61C2×Q812yes—
167C22×Dic61#4112yes—
168C2×C4×D61—28no—
169C2×C4×D61—48no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61—48no—
173#11758198—28no—
174#11758198—28no—
175#11758198—48no—
176#11758198—48no—
177C4⋊Dic61—44no—
178#11758196—48no—
179C4×Dic61—44no—
180C4×Dic61—44no—
181C23×C61C23×C411yescharacteristic
182C23×C61C22×D411yescharacteristic
183C23×C61C22×Q811yescharacteristic
184C23×C61C2×#4111yescharacteristic
185C23×C61C2×#4111yescharacteristic
186C23×C61C2×#4111yescharacteristic
187C23×C61C2×#4111yescharacteristic
188C23×C61H(5,𝔽2)12yes—
189C23×C61H(5,𝔽2)12yes—
190C23×C61H(5,𝔽2)12yes—
191C23×C612−1+412yes—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—24no—
195C22×D61—24no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61C4×Q814yes—
215Q8×C61—416no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
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×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×C61C4×D414yes—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—22no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—41no—
258C2×Dic61—41no—
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×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61C22⋊Q814yes—
274C4×D61—416no—
275#14319—611nomaximal, Sylow
276C22×C61C2×C4×D411yescharacteristic
277C22×C61C2×C4×Q811yescharacteristic
278C22×C61C4×#4111yescharacteristic
279C22×C61#51812yes—
280C22×C61#52012yes—
281C22×C61C2×C22⋊Q811yescharacteristic
282C22×C61#52511yescharacteristic
283C22×C61#52911yescharacteristic
284C22×C61#53311yescharacteristic
285C22×C61#53411yescharacteristic
286C22×C61#53511yescharacteristic
287C22×C61#53511yescharacteristic
288C22×C61#53611yescharacteristic
289C22×C61#53812yes—
290C22×C61#53812yes—
291C22×C61#53812yes—
292C22×C61#53812yes—
293C22×C61#53912yes—
294C22×C61#54212yes—
295C22×C61#54212yes—
296C22×C61#54212yes—
297C22×C61#54312yes—
298C22×C61#54412yes—
299C2×D61—28no—
300C2×D61—24no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—44no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—44no—
309C2×D61—44no—
310C4×C61—24no—
311C4×C61—44no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61—82no—
315Dic61—24no—
316Dic61—42no—
317Dic61—44no—
318Dic61—42no—
319Dic61—82no—
320C2×#393—611no—
321C2×#391—611no—
322C2×C4×C22⋊C4—611no—
323#1446D6111yescharacteristic, complemented
324#1444—611no—
325#1443—612no—
326#1439—612no—
327#1439—611no—
328#1436—612no—
329#1434—611no—
330C22⋊#95—612no—
331C2×C61C4×C22⋊Q811yescharacteristic
332C2×C61#232111yescharacteristic
333C2×C61#232611yescharacteristic
334C2×C61#232912yes—
335C2×C61#233712yes—
336C2×C61#262611yescharacteristic
337C2×C61#262811yescharacteristic
338C2×C61#263111yescharacteristic
339C2×C61#263811yescharacteristic
340C2×C61#265312yes—
341C2×C61#266512yes—
342D61—44no—
343C23×Q8C2×D6111yescharacteristic
344C24×C4—611no—
345C22×C4⋊C4—612no—
346C22×C4⋊C4—611no—
347C22×C22⋊C4—611no—
348C22×C22⋊C4—612no—
349C22×C22⋊C4—611no—
350C22×C22⋊C4—612no—
351C22×C42—611no—
352C22×C42—611no—
353#393—618no—
354#391—1224no—
355C4×C22⋊C4—1224no—
356C2×#95—611no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95—611no—
362C2×#95—612no—
363C2×#95—612no—
364C2×#95—612no—
365C2×#95C2×D6111yescharacteristic
366C2×#95C2×D6111yescharacteristic
367C2×#95C2×D6112yes—
368C2×#95C2×D6112yes—
369C61#1431911yescharacteristic, complemented, Sylow
370C25—611no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—42no—
374C22×Q8—42no—
375C22×Q8C4×D6114yescomplemented
376C23×C4—611no—
377C23×C4—612no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—614no—
381C23×C4—612no—
382C23×C4—1222no—
383C23×C4—1222no—
384C23×C4—2442no—
385C23×C4—2442no—
386C23×C4C22×D6112yes—
387C23×C4C22×D6112yes—
388C23×C4C22×D6111yescharacteristic
389C23×C4C22×D6111yescharacteristic
390C23×C4C22×D6111yescharacteristic
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1224no—
393C2×C4⋊C4—2442no—
394C2×C4⋊C4—2442no—
395C2×C4⋊C4—2442no—
396C2×C22⋊C4—614no—
397C2×C22⋊C4—618no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—2442no—
405C2×C22⋊C4—2442no—
406C2×C22⋊C4—2442no—
407C2×C22⋊C4—2442no—
408C2×C42—614no—
409C2×C42—1222no—
410C2×C42—2442no—
411C2×C42—2442no—
412#95—22no—
413#95—22no—
414#95—42no—
415#95—42no—
416#95—618no—
417#95—614no—
418#95—1222no—
419#95—1224no—
420#95—1224no—
421#95—1224no—
422#95—2442no—
423#95—2442no—
424#95—2442no—
425C24—612no—
426C24—612no—
427C24—612no—
428C24—1221no—
429C24—1221no—
430C24—1221no—
431C24—1221no—
432C24—1222no—
433C24—1222no—
434C24—1222no—
435C24—1222no—
436C24C23×D6111yescharacteristic
437C2×Q8—28no—
438C2×Q8—48no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—48no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—614no—
470C22×C4—612no—
471C22×C4—1228no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
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×C4—1222no—
494C22×C4—1222no—
495C22×C4—2448no—
496C22×C4—2444no—
497C22×C4—2444no—
498C22×C4—2448no—
499C22×C4C2×C4×D6114yes—
500C22×C4C2×C4×D6112yes—
501C22×C4D4×D6112yes—
502C22×C4#1175822412yes—
503C22×C4Q8×D6112yes—
504C22×C4#1175822612yes—
505C4⋊C4—2448no—
506C4⋊C4—2444no—
507C22⋊C4—1228no—
508C22⋊C4—1228no—
509C22⋊C4—2448no—
510C22⋊C4—2448no—
511C42—2444no—
512C42—2444no—
513C23—614no—
514C23—1222no—
515C23—1222no—
516C23—1224no—
517C23—1224no—
518C23—1222no—
519C23—1222no—
520C23—1222no—
521C23—1222no—
522C23—1224no—
523C23—1224no—
524C23—1224no—
525C23—1224no—
526C23—2441no—
527C23—2441no—
528C23—2441no—
529C23—2441no—
530C23—2442no—
531C23—2442no—
532C23—2442no—
533C23—2442no—
534C23—2442no—
535C23—2442no—
536C23C22×C4×D6111yescharacteristic
537C23C2×D4×D6111yescharacteristic
538C23C2×Q8×D6111yescharacteristic
539C23#41×D6111yescharacteristic
540C23#41×D6111yescharacteristic
541C23C2×#1175822411yescharacteristic
542C23C2×#1175822611yescharacteristic
543C23#1451982212yes—
544C23#1451982212yes—
545C23#1451982312yes—
546C23#1451982512yes—
547Q8—416no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
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—614no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1222no—
582C2×C4—1224no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—24416no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4#1451973714yes—
607C22—1228no—
608C22—1224no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22—2444no—
613C22—2442no—
614C22—2442no—
615C22—2444no—
616C22—2444no—
617C22—2442no—
618C22C2×#1451973711yescharacteristic
619C22C4×Q8×D6111yescharacteristic
620C22C4×#1175822611yescharacteristic
621C22#1684023512yes—
622C22#1684024412yes—
623C22C22⋊Q8×D6111yescharacteristic
624C22#1684027511yescharacteristic
625C22#1684029111yescharacteristic
626C22#1684031711yescharacteristic
627C22#1684032211yescharacteristic
628C22#1684032711yescharacteristic
629C22#1684033011yescharacteristic
630C22#1684033711yescharacteristic
631C22#1684034612yes—
632C22#1684034612yes—
633C22#1684034712yes—
634C22#1684035212yes—
635C22#1684035812yes—
636C22#1684038212yes—
637C22#1684038512yes—
638C22#1684038612yes—
639C22#1684038712yes—
640C22#1684038912yes—
641C4—24no—
642C4—44no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4—1224no—
647C4—4882no—
648C4—2442no—
649C4—2444no—
650C4—2442no—
651C2—2444no—
652C2#2144835711yescharacteristic
653C2#2144847411yescharacteristic
654C2#2144853311yescharacteristic
655C2#2144855112yes—
656C2#2144863112yes—
657C2#2145162311yescharacteristic
658C2#2145164211yescharacteristic
659C2#2145167711yescharacteristic
660C2#2145174511yescharacteristic
661C2#2145187912yes—
662C2#2145199412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1446C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1446D61
hypocenterC61#14319
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
2. C2×#16839669C2semidirect
4. #21444083C2semidirect
5. #21444080C2semidirect
8. #21444054C2semidirect
9. #21444044C2semidirect
11. C61×#1446C2semidirect
21. #16839673C4semidirect
48. C22×C4×D61Q8semidirect
77. #14519684C2×C4semidirect
78. #14519684C2×C4semidirect
83. C22×Q8×C61C2×C4semidirect
323. #1446D61semidirect
369. C61#14319semidirect
375. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396732
C2×#168396692
C2×#168396092
#214440832
#214440802
#214440652
#214440632
#214440542
#214440442
#214440402
C61×#14462
#1431961