Finite Group Databasebeta

← #29886861

Subgroups — #29886861

Order 15616.

2422 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840391C211yesmaximal, characteristic, complemented
2C22×#14519758C211yesmaximal, characteristic
3#21454104C212yesmaximal, complemented
4#21452338C214yesmaximal, complemented
5#21452334C214yesmaximal, complemented
6#21452102C214yesmaximal, complemented
7#21452094C214yesmaximal, complemented
8#21451628C212yesmaximal, complemented
9#21451200C211yesmaximal, characteristic, complemented
10#21451194C211yesmaximal, characteristic, complemented
11#21450090C212yesmaximal, complemented
12#21450090C212yesmaximal, complemented
13#21449466C211yesmaximal, characteristic
14#21448419C211yesmaximal, characteristic, complemented
15C61×#2435C211yesmaximal, characteristic
16C24×Dic61C2211yescharacteristic
17#16840391—28no—
18C2×#14519776C2212yescomplemented
19C2×#14519775C2212yescomplemented
20C2×#14519767C2212yescomplemented
21C2×#14519767C2212yescomplemented
22C2×#14519764C2212yes—
23C2×#14519758—28no—
24C2×#14519758C2214yes—
25C2×#14519758C2214yes—
26C2×#14519756C2211yescharacteristic
27C2×#14519754C2212yes—
28C2×#14519754C2214yescomplemented
29C2×#14519733C2212yes—
30C22×#11758196C2211yescharacteristic
31C22×#11758196C2211yescharacteristic
32C22×C22⋊Dic61C2211yescharacteristic
33C22×C22⋊Dic61C2211yescharacteristic
34C22×C22⋊Dic61C2211yescharacteristic
35#16839698C2214yescomplemented
36#16839698C2214yescomplemented
37#16839698C2214yescomplemented
38#16839694C2212yescomplemented
39#16839693C2212yescomplemented
40#16839690C2214yescomplemented
41#16839690C2214yescomplemented
42#16839685C2214yescomplemented
43#16839685C2214yescomplemented
44#16839683C2212yescomplemented
45#16839681C2212yescomplemented
46#16839676C2214yes—
47#16839674C2214yescomplemented
48#16839665C2212yes—
49#16839660C2214yescomplemented
50#16839657C2214yescomplemented
51#16839656C2214yescomplemented
52#16839656C2214yescomplemented
53#16839655C2214yescomplemented
54#16839655C2214yescomplemented
55#16839646C2214yescomplemented
56#16839644C2214yescomplemented
57#16839637C2212yes—
58#16839637C2212yes—
59#16839637C2214yescomplemented
60#16839634C2212yes—
61#16839634C2212yes—
62#16839634C2214yescomplemented
63#16839623C2212yes—
64#16839620C2212yes—
65#16839620C2214yescomplemented
66#16839618C2212yes—
67#16839618C2212yes—
68C4×C22⋊Dic61C2211yescharacteristic
69C23×D4×C61C2211yescharacteristic
70C2×#124×C61C2212yes—
71C2×C22⋊Q8×C61C2211yescharacteristic
72C61×#394C2214yes—
73C61×#390C2212yes—
74C61×#379C2212yes—
75C61×#379C2212yes—
76C4×C22⋊C4×C61C2211yescharacteristic
77C22×Dic122C2311yescharacteristic
78C22×Dic122C2312yes—
79C23×Dic61—24no—
80C23×Dic61—44no—
81C23×Dic61C2314yes—
82C23×Dic61C2311yescharacteristic
83C23×Dic61C2311yescharacteristic
84C23×Dic61C2311yescharacteristic
85C23×Dic61C2311yescharacteristic
86#14519776—28no—
87#14519775—28no—
88#14519767—28no—
89#14519767—28no—
90#14519764—44no—
91#14519758—216no—
92#14519758—216no—
93#14519756—24no—
94#14519754—28no—
95#14519754—48no—
96#14519733—28no—
97C2×C4⋊Dic61C2312yes—
98C2×C4⋊Dic61C2311yescharacteristic
99C2×C4⋊Dic61C2314yes—
100C2×C4⋊Dic61C2314yes—
101C2×#11758196—24no—
102C2×#11758196D418yes—
103C2×#11758196C2312yes—
104C2×#11758196C2312yes—
105C2×#11758196C2312yes—
106C2×#11758196C2312yes—
107C2×#11758196C2312yes—
108C2×#11758196C2312yes—
109C2×#11758196C2312yes—
110C2×#11758196C2314yes—
111C2×#11758196C2314yes—
112C2×#11758196C2314yes—
113C2×#11758196C2312yes—
114C2×#11758196C2312yes—
115C2×C22⋊Dic61—24no—
116C2×C22⋊Dic61—24no—
117C2×C22⋊Dic61—24no—
118C2×C22⋊Dic61C2314yes—
119C2×C22⋊Dic61C2314yes—
120C2×C22⋊Dic61C2314yes—
121C2×C22⋊Dic61C2314yes—
122C2×C22⋊Dic61C2312yes—
123C2×C22⋊Dic61C2312yes—
124C2×C22⋊Dic61C2314yes—
125C2×C22⋊Dic61C2314yes—
126C2×C22⋊Dic61C2314yes—
127C2×C4×Dic61C2312yes—
128C2×C4×Dic61C2312yes—
129C2×C4×Dic61C2312yes—
130#14519684C2314yes—
131#14519684C2312yes—
132#14519684C2314yes—
133#14519684C2314yes—
134#14519684C2314yes—
135#14519684C2314yes—
136#14519684C2314yes—
137#14519684C2314yes—
138C25×C61C2311yescharacteristic
139C25×C61C2311yescharacteristic
140C22×Q8×C61C2311yescharacteristic
141C22×D4×C61—24no—
142C22×D4×C61—44no—
143C22×D4×C61C2312yes—
144C22×D4×C61C2312yes—
145C23×C4×C61C2311yescharacteristic
146#124×C61—44no—
147C22⋊Q8×C61—24no—
148C2×C4⋊C4×C61C2311yescharacteristic
149C2×C4⋊C4×C61C2312yes—
150C2×C22⋊C4×C61C2314yes—
151C2×C22⋊C4×C61C2312yes—
152C2×C22⋊C4×C61C2312yes—
153C2×C22⋊C4×C61C2314yes—
154C2×C22⋊C4×C61C2314yes—
155C2×C22⋊C4×C61C2314yes—
156C2×C42×C61C2312yes—
157#95×C61C2312yes—
158C24×C61—22no—
159C24×C61—24no—
160C24×C61—22no—
161C24×C61—24no—
162C24×C61—24no—
163C24×C61—42no—
164C24×C61—42no—
165C24×C61—42no—
166C24×C61C2412yes—
167C24×C61C2412yes—
168C24×C61C2411yescharacteristic
169C2×Q8×C61—22no—
170C2×Q8×C61—42no—
171C2×D4×C61—24no—
172C2×D4×C61—24no—
173C2×D4×C61—24no—
174C2×D4×C61—24no—
175C2×D4×C61—42no—
176C2×D4×C61—42no—
177C2×D4×C61—416no—
178C22×C4×C61—42no—
179C22×C4×C61C2×D414yes—
180C22×C4×C61C2412yes—
181C22×C4×C61C2414yes—
182C22×C4×C61C2412yes—
183C22×C4×C61C2411yescharacteristic
184C22×C4×C61C2411yescharacteristic
185C22×Dic61—24no—
186C22×Dic61—22no—
187C22×Dic61—22no—
188C22×Dic61—28no—
189C22×Dic61—28no—
190C22×Dic61—42no—
191C22×Dic61—44no—
192C22×Dic61—42no—
193C22×Dic61—42no—
194C22×Dic61—416no—
195C22×Dic61C2×D414yes—
196C22×Dic61C2×D414yes—
197C22×Dic61C2412yes—
198C22×Dic61C2412yes—
199C22×Dic61C2412yes—
200C22×Dic61C2414yes—
201C22×Dic61C2412yes—
202C22×Dic61C2412yes—
203C22×Dic61C2412yes—
204C2×Dic122—22no—
205C2×Dic122—24no—
206C2×Dic122—44no—
207C2×Dic122—42no—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—24no—
210C22⋊C4×C61—24no—
211C22⋊C4×C61—28no—
212C22⋊C4×C61—24no—
213C22⋊C4×C61—44no—
214C22⋊C4×C61—44no—
215C22⋊C4×C61—44no—
216C42×C61—42no—
217C22⋊Dic61—24no—
218C22⋊Dic61—28no—
219C22⋊Dic61—24no—
220C22⋊Dic61—28no—
221C22⋊Dic61—28no—
222C22⋊Dic61—28no—
223C22⋊Dic61—44no—
224C22⋊Dic61—44no—
225C22⋊Dic61—44no—
226C4⋊Dic61—22no—
227C4⋊Dic61—24no—
228C4⋊Dic61—44no—
229C4⋊Dic61—44no—
230#11758196—24no—
231#11758196—24no—
232#11758196—24no—
233#11758196—24no—
234#11758196—24no—
235#11758196—24no—
236#11758196—24no—
237#11758196—44no—
238#11758196—44no—
239#11758196—44no—
240#11758196C2×D418yes—
241#11758196C2×D418yes—
242C4×Dic61—24no—
243C4×Dic61—24no—
244C4×Dic61—42no—
245C23×C61—22no—
246C23×C61—22no—
247C23×C61—28no—
248C23×C61—22no—
249C23×C61—22no—
250C23×C61—24no—
251C23×C61—24no—
252C23×C61—22no—
253C23×C61—22no—
254C23×C61—22no—
255C23×C61—24no—
256C23×C61—24no—
257C23×C61—416no—
258C23×C61—44no—
259C23×C61—44no—
260C23×C61—44no—
261C23×C61—44no—
262C23×C61—44no—
263C23×C61—88no—
264C23×C61C22×D412yes—
265C23×C61H(5,𝔽2)14yes—
266C23×C612−1+414yes—
267C23×C61C2511yescharacteristic
268Q8×C61—82no—
269D4×C61—48no—
270D4×C61—48no—
271C2×C4×C61—22no—
272C2×C4×C61—22no—
273C2×C4×C61—22no—
274C2×C4×C61—22no—
275C2×C4×C61—24no—
276C2×C4×C61—24no—
277C2×C4×C61—24no—
278C2×C4×C61—24no—
279C2×C4×C61—24no—
280C2×C4×C61—44no—
281C2×C4×C61C22×D412yes—
282C2×C4×C61C22×D412yes—
283C2×Dic61—28no—
284C2×Dic61—24no—
285C2×Dic61—24no—
286C2×Dic61—22no—
287C2×Dic61—22no—
288C2×Dic61—22no—
289C2×Dic61—22no—
290C2×Dic61—24no—
291C2×Dic61—24no—
292C2×Dic61—24no—
293C2×Dic61—24no—
294C2×Dic61—22no—
295C2×Dic61—24no—
296C2×Dic61—22no—
297C2×Dic61—24no—
298C2×Dic61—44no—
299C2×Dic61—416no—
300C2×Dic61—44no—
301C2×Dic61—44no—
302C2×Dic61C22×D414yes—
303C2×Dic61C22×D414yes—
304Dic122—84no—
305Dic122—82no—
306#36711—611nomaximal, Sylow
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—28no—
310C22×C61—24no—
311C22×C61—28no—
312C22×C61—416no—
313C22×C61—42no—
314C22×C61—42no—
315C22×C61—44no—
316C22×C61—42no—
317C22×C61—42no—
318C22×C61—42no—
319C22×C61—44no—
320C22×C61—44no—
321C22×C61—88no—
322C22×C61#53614yes—
323C22×C61C23×D411yescharacteristic
324C22×C61C2×H(5,𝔽2)12yes—
325C22×C61C2×2−1+412yes—
326C22×C61#58512yes—
327C4×C61—22no—
328C4×C61—22no—
329C4×C61—42no—
330C4×C61—42no—
331C4×C61—44no—
332Dic61—24no—
333Dic61—24no—
334Dic61—42no—
335Dic61—42no—
336Dic61—42no—
337Dic61—42no—
338Dic61—44no—
339C2×#544—611no—
340C22×#123—611no—
341#2848—612no—
342#2694—614no—
343#2694—614no—
344#2674—614no—
345#2674—614no—
346#2626—612no—
347#2583—611no—
348#2583—611no—
349#2482—612no—
350#2482—612no—
351#2435—611no—
352#2435D6111yescharacteristic
353#2313—611no—
354C2×C61—24no—
355C2×C61—44no—
356C2×C61—82no—
357C2×C61C2×#53611yescharacteristic
358C2×C61#348411yescharacteristic
359C2×C61#348711yescharacteristic
360C2×C61#353514yes—
361C23×D4C2×D6111yescharacteristic
362C24×C4—611no—
363#544—1228no—
364C2×C4⋊Q8—612no—
365C2×C4⋊Q8—612no—
366C2×#125—612no—
367C2×#125—612no—
368C2×#124—612no—
369C2×#124C2×D6112yes—
370C2×#123—614no—
371C2×#123—614no—
372C2×#123—1228no—
373C2×C22⋊Q8—611no—
374C2×C22⋊Q8—612no—
375C2×C22⋊Q8—614no—
376C2×C22⋊Q8C2×D6111yescharacteristic
377C2×#117—612no—
378C22×C4⋊C4—611no—
379C22×C4⋊C4—611no—
380C22×C22⋊C4—611no—
381C22×C22⋊C4—611no—
382C22×C22⋊C4—611no—
383#399—614no—
384#399—614no—
385#399—614no—
386#398—612no—
387#398—612no—
388#397—614no—
389#397—614no—
390#396—614no—
391#396—614no—
392#395—612no—
393#395—612no—
394#394—614no—
395#394C2×D6114yes—
396#393—614no—
397#390—612no—
398#390C2×D6112yes—
399#389—614no—
400#389—614no—
401#388—614no—
402#388—614no—
403#388—614no—
404#388—614no—
405#387—614no—
406#387—614no—
407#385—612no—
408#385—612no—
409#385—612no—
410#385—612no—
411#385—614no—
412#385—614no—
413#381—612no—
414#380—612no—
415#380—614no—
416#379—612no—
417#379—612no—
418#379C2×D6112yes—
419#379C2×D6112yes—
420C4×C22⋊C4—611no—
421C4×C22⋊C4C2×D6111yescharacteristic
422C61#3671111yescharacteristic, complemented, Sylow
423C25C22×D6111yescharacteristic
424C25C22×D6111yescharacteristic
425C22×Q8—612no—
426C22×Q8—611no—
427C22×Q8C22×D6111yescharacteristic
428C22×D4—24no—
429C22×D4—44no—
430C22×D4C22×D6112yes—
431C22×D4C22×D6112yes—
432C23×C4—611no—
433C23×C4—611no—
434C23×C4—614no—
435C23×C4—611no—
436C23×C4—611no—
437C23×C4—1224no—
438C23×C4—2444no—
439C23×C4C22×D6111yescharacteristic
440C4⋊Q8—1228no—
441C4⋊Q8—1228no—
442#125—1228no—
443#125—1228no—
444#124—44no—
445#124—2444no—
446#123—12216no—
447#123—12216no—
448C22⋊Q8—24no—
449C22⋊Q8—1228no—
450C22⋊Q8—1224no—
451C22⋊Q8—2448no—
452#117—1228no—
453C2×C4⋊C4—618no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—612no—
456C2×C4⋊C4—612no—
457C2×C4⋊C4—612no—
458C2×C4⋊C4—612no—
459C2×C4⋊C4—612no—
460C2×C4⋊C4—612no—
461C2×C4⋊C4—611no—
462C2×C4⋊C4—612no—
463C2×C4⋊C4—612no—
464C2×C4⋊C4—612no—
465C2×C4⋊C4—614no—
466C2×C4⋊C4—614no—
467C2×C4⋊C4—614no—
468C2×C4⋊C4—614no—
469C2×C4⋊C4—614no—
470C2×C4⋊C4—1224no—
471C2×C4⋊C4C22×D6111yescharacteristic
472C2×C4⋊C4C22×D6112yes—
473C2×C22⋊C4—614no—
474C2×C22⋊C4—614no—
475C2×C22⋊C4—614no—
476C2×C22⋊C4—612no—
477C2×C22⋊C4—614no—
478C2×C22⋊C4—612no—
479C2×C22⋊C4—614no—
480C2×C22⋊C4—614no—
481C2×C22⋊C4—614no—
482C2×C22⋊C4—1224no—
483C2×C22⋊C4—1224no—
484C2×C22⋊C4—1224no—
485C2×C22⋊C4C22×D6112yes—
486C2×C22⋊C4C22×D6114yes—
487C2×C22⋊C4C22×D6112yes—
488C2×C22⋊C4C22×D6114yes—
489C2×C22⋊C4C22×D6114yes—
490C2×C22⋊C4C22×D6114yes—
491C2×C42—612no—
492C2×C42—612no—
493C2×C42—612no—
494C2×C42C22×D6112yes—
495#95—612no—
496#95—614no—
497#95—614no—
498#95—614no—
499#95—614no—
500#95—614no—
501#95—614no—
502#95—614no—
503#95C22×D6112yes—
504C24—22no—
505C24—24no—
506C24—22no—
507C24—24no—
508C24—24no—
509C24—42no—
510C24—42no—
511C24—42no—
512C24C23×D6112yes—
513C24C23×D6112yes—
514C24C23×D6111yescharacteristic
515C2×Q8—22no—
516C2×Q8—42no—
517C2×Q8—1224no—
518C2×Q8—1222no—
519C2×Q8—2444no—
520C2×Q8—2442no—
521C2×D4—24no—
522C2×D4—24no—
523C2×D4—24no—
524C2×D4—24no—
525C2×D4—42no—
526C2×D4—42no—
527C2×D4—416no—
528C22×C4—42no—
529C22×C4—614no—
530C22×C4—612no—
531C22×C4—612no—
532C22×C4—614no—
533C22×C4—614no—
534C22×C4—612no—
535C22×C4—612no—
536C22×C4—612no—
537C22×C4—612no—
538C22×C4—1228no—
539C22×C4—1224no—
540C22×C4—1222no—
541C22×C4—1222no—
542C22×C4—1228no—
543C22×C4—2442no—
544C22×C4—2444no—
545C22×C4—2442no—
546C22×C4—2442no—
547C22×C4—24416no—
548C22×C4D4×D6114yes—
549C22×C4C23×D6114yes—
550C22×C4C23×D6112yes—
551C22×C4C23×D6112yes—
552C22×C4C23×D6111yescharacteristic
553C22×C4C23×D6111yescharacteristic
554C4⋊C4—22no—
555C4⋊C4—24no—
556C4⋊C4—618no—
557C4⋊C4—618no—
558C4⋊C4—1224no—
559C4⋊C4—1224no—
560C4⋊C4—1224no—
561C4⋊C4—1224no—
562C4⋊C4—1224no—
563C4⋊C4—1224no—
564C4⋊C4—1224no—
565C4⋊C4—1224no—
566C4⋊C4—1222no—
567C4⋊C4—2444no—
568C4⋊C4—2444no—
569C4⋊C4—2444no—
570C4⋊C4—2444no—
571C4⋊C4—2444no—
572C22⋊C4—24no—
573C22⋊C4—28no—
574C22⋊C4—24no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—1228no—
579C22⋊C4—1228no—
580C22⋊C4—1228no—
581C22⋊C4—1224no—
582C22⋊C4—1228no—
583C22⋊C4—1224no—
584C22⋊C4—2444no—
585C22⋊C4—2444no—
586C22⋊C4—2444no—
587C42—42no—
588C42—1224no—
589C42—1224no—
590C42—2442no—
591C23—22no—
592C23—22no—
593C23—28no—
594C23—22no—
595C23—22no—
596C23—24no—
597C23—24no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—24no—
602C23—24no—
603C23—416no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—88no—
610C23C2×D4×D6112yes—
611C23#1451982214yes—
612C23#1451982414yes—
613C23C24×D6111yescharacteristic
614Q8—82no—
615Q8—4882no—
616Q8—4884no—
617D4—48no—
618D4—48no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—44no—
629C2×C4—614no—
630C2×C4—614no—
631C2×C4—1228no—
632C2×C4—1224no—
633C2×C4—1224no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—1224no—
639C2×C4—1224no—
640C2×C4—1224no—
641C2×C4—1224no—
642C2×C4—1224no—
643C2×C4—1224no—
644C2×C4—1222no—
645C2×C4—1222no—
646C2×C4—2444no—
647C2×C4—24416no—
648C2×C4—2444no—
649C2×C4—2444no—
650C2×C4C2×D4×D6112yes—
651C2×C4C2×D4×D6112yes—
652C22—22no—
653C22—28no—
654C22—22no—
655C22—28no—
656C22—24no—
657C22—416no—
658C22—42no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—88no—
667C22#1684033214yes—
668C22C22×D4×D6111yescharacteristic
669C22C2×#1451982212yes—
670C22C2×#1451982412yes—
671C22#1684059712yes—
672C4—22no—
673C4—22no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—1224no—
678C4—1224no—
679C4—2442no—
680C4—2442no—
681C4—2444no—
682C4—2442no—
683C4—2442no—
684C2—24no—
685C2—44no—
686C2—82no—
687C2C2×#1684033211yescharacteristic
688C2#2145944811yescharacteristic
689C2#2145947511yescharacteristic
690C2#2146001614yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2435C2
prime radicalC23×D4×C61C22
hypercenter#2435D61
hypocenterC61#36711
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2435

Complements

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

NH ≅ G/Ntype
1. C2×#16840391C2semidirect
3. #21454104C2semidirect
4. #21452338C2semidirect
5. #21452334C2semidirect
6. #21452102C2semidirect
7. #21452094C2semidirect
8. #21451628C2semidirect
9. #21451200C2semidirect
10. #21451194C2semidirect
11. #21450090C2semidirect
12. #21450090C2semidirect
14. #21448419C2semidirect
18. C2×#14519776C22semidirect
19. C2×#14519775C22semidirect
20. C2×#14519767C22semidirect
21. C2×#14519767C22semidirect
28. C2×#14519754C22semidirect
35. #16839698C22semidirect
36. #16839698C22semidirect
37. #16839698C22semidirect
38. #16839694C22semidirect
39. #16839693C22semidirect
40. #16839690C22semidirect
41. #16839690C22semidirect
42. #16839685C22semidirect
43. #16839685C22semidirect
44. #16839683C22semidirect
45. #16839681C22semidirect
47. #16839674C22semidirect
49. #16839660C22semidirect
50. #16839657C22semidirect
51. #16839656C22semidirect
52. #16839656C22semidirect
53. #16839655C22semidirect
54. #16839655C22semidirect
55. #16839646C22semidirect
56. #16839644C22semidirect
59. #16839637C22semidirect
62. #16839634C22semidirect
65. #16839620C22semidirect
422. C61#36711semidirect

Maximal subgroups

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

subgroupindex
C2×#168403912
C22×#145197582
#214541042
#214523382
#214523342
#214521022
#214520942
#214516282
#214512002
#214511942
#214500902
#214494662
#214484192
C61×#24352
#3671161