Finite Group Databasebeta

← #29679210

Subgroups — #29679210

Order 15616.

3998 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758226C211yesmaximal, characteristic
2C2×#16839757C214yesmaximal, complemented
3C2×C22⋊C8×D61C211yesmaximal, characteristic, complemented
4#21448201C211yesmaximal, characteristic, complemented
5#21445758C214yesmaximal, complemented
6#21445754C211yesmaximal, characteristic, complemented
7#21445751C211yesmaximal, characteristic, complemented
8#21445749C211yesmaximal, characteristic, complemented
9C61×#1787C211yesmaximal, characteristic, complemented
10C22×#11758226—216no—
11C22×#11758226—24no—
12C22×#11758226—21no—
13C22×#11758226—21no—
14C22×#11758226C418yescomplemented
15C22×#11758226C2212yes—
16C22×#11758226C2212yes—
17C23×D244—21no—
18C23×D244C2211yescharacteristic
19C23×C4×D61—21no—
20C23×C4×D61C2211yescharacteristic
21C22×C8×D61C2212yescomplemented
22C2×#14519735C2212yescomplemented
23#16839757—216no—
24C2×#14519701C2214yescomplemented
25C2×#14519699C2214yescomplemented
26C2×#14519697C2214yescomplemented
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519689C2212yescomplemented
29C22⋊C8×D61—24no—
30#16839624—21no—
31#16839622C2211yescharacteristic
32#16839619C2212yescomplemented
33#16839617—21no—
34C22⋊C4×Dic61—21no—
35#16839609—21no—
36C23×Q8×C61C2211yescharacteristic
37C2×#103×C61C2214yescomplemented
38C2×C22⋊C8×C61C2211yescharacteristic
39C61×#380C2211yescharacteristic
40C24×D61—21no—
41C24×D61D411yescharacteristic
42C2×#11758226—216no—
43C2×#11758226—264no—
44C2×#11758226—216no—
45C2×#11758226—28no—
46C2×#11758226—264no—
47C2×#11758226—28no—
48C2×#11758226—28no—
49C2×#11758226—28no—
50C2×#11758226—24no—
51C2×#11758226—24no—
52C2×#11758226—28no—
53C2×#11758226C2×C418yes—
54C2×#11758226C2×C418yes—
55C2×#11758226D4116yes—
56C22×D244—216no—
57C22×D244—28no—
58C22×D244—22no—
59C22×D244—22no—
60C22×D244—21no—
61C22×D244—22no—
62C22×D244—28no—
63C22×D244C2×C414yes—
64C22×D244C2312yes—
65C23×Dic61D411yescharacteristic
66C22×C4×D61—216no—
67C22×C4×D61—28no—
68C22×C4×D61—22no—
69C22×C4×D61—22no—
70C22×C4×D61—28no—
71C22×C4×D61C2×C414yes—
72C22×C4×D61D411yescharacteristic
73C22×C4×D61D411yescharacteristic
74C22×C4×D61D414yes—
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2311yescharacteristic
77C22×#92418C2312yes—
78C2×C8×D61—24no—
79C2×C8×D61—28no—
80#14519735—28no—
81C2×C4⋊Dic61C2312yes—
82C2×C22⋊Dic61—21no—
83C2×#11758198—22no—
84C2×#11758198—22no—
85C2×C4×Dic61—22no—
86#14519701—28no—
87#14519699—28no—
88#14519697—28no—
89C22⋊#92418—22no—
90#14519689—24no—
91#14519684—21no—
92#14519684—22no—
93C22×Q8×C61—28no—
94C22×Q8×C61—22no—
95C22×Q8×C61—21no—
96C22×Q8×C61C2×C414yes—
97C22×Q8×C61C2312yes—
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61C2312yes—
101C2×C4⋊C4×C61C2312yes—
102C2×C22⋊C4×C61—21no—
103#103×C61—28no—
104C22⋊C8×C61—22no—
105#95×C61—21no—
106C24×C61C2×D411yescharacteristic
107C23×D61—28no—
108C23×D61—24no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—24no—
113C23×D61—42no—
114C23×D61—44no—
115C23×D61—42no—
116C23×D61—42no—
117C23×D61C22⋊C414yes—
118C23×D61C2×D411yescharacteristic
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—216no—
121C2×Q8×C61—28no—
122C2×Q8×C61—216no—
123C2×Q8×C61—24no—
124C2×Q8×C61—22no—
125C2×Q8×C61—24no—
126C2×Q8×C61C22×C414yes—
127C2×Q8×C61C2×D418yes—
128C22×C4×C61—28no—
129C22×C4×C61—24no—
130C22×C4×C61—22no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—24no—
138C22×Dic61—22no—
139C22×Dic61C22⋊C414yes—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142#11758226—264no—
143#11758226—232no—
144#11758226—264no—
145#11758226—216no—
146#11758226—216no—
147#11758226—216no—
148#11758226—232no—
149#11758226C22⋊C4132yescomplemented
150C2×D244—216no—
151C2×D244—28no—
152C2×D244—264no—
153C2×D244—216no—
154C2×D244—28no—
155C2×D244—28no—
156C2×D244—216no—
157C2×D244—24no—
158C2×D244—24no—
159C2×D244—22no—
160C2×D244—216no—
161C2×D244C22×C414yes—
162C2×D244C2×D418yes—
163C2×C4×D61—216no—
164C2×C4×D61—264no—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—28no—
168C2×C4×D61—28no—
169C2×C4×D61—216no—
170C2×C4×D61—24no—
171C2×C4×D61—216no—
172C2×C4×D61—216no—
173C2×C4×D61C22⋊C414yes—
174C2×C4×D61C22⋊C414yes—
175C2×C4×D61C22×C412yes—
176C2×C4×D61C22×C412yes—
177C2×C4×D61C2×D412yes—
178C2×C4×D61C2×D418yes—
179C2×C4×D61C2×D412yes—
180C2×C8×C61—22no—
181C2×C8×C61—24no—
182C4⋊C4×C61—24no—
183C22⋊C4×C61—42no—
184C22⋊Dic61—42no—
185#11758198—44no—
186#11758198—44no—
187C4⋊Dic61—24no—
188C4×Dic61—44no—
189C2×#92418—22no—
190C2×#92418—24no—
191C8×D61—48no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61C2×C22⋊C412yes—
195C23×C61#13512yes—
196C23×C61#13512yes—
197C23×C61C22×D411yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—22no—
201C22×D61—216no—
202C22×D61—24no—
203C22×D61—22no—
204C22×D61—28no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—22no—
208C22×D61—28no—
209C22×D61—28no—
210C22×D61—44no—
211C22×D61—48no—
212C22×D61—48no—
213C22×D61—48no—
214C22×D61—416no—
215C22×D61—416no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—44no—
219C22×D61—48no—
220C22×D61C2×C22⋊C412yes—
221C22×D61C2×C22⋊C412yes—
222C22×D61#12014yes—
223Q8×C61—216no—
224Q8×C61—24no—
225Q8×C61—28no—
226Q8×C61C2×C22⋊C418yes—
227C2×C4×C61—216no—
228C2×C4×C61—24no—
229C2×C4×C61—28no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D414yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×Dic61—22no—
241C2×Dic61—28no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61#12014yes—
248D244—232no—
249D244—216no—
250D244—232no—
251D244—24no—
252D244—28no—
253D244C2×C22⋊C418yes—
254C4×D61—232no—
255C4×D61—28no—
256C4×D61—24no—
257C4×D61—24no—
258C4×D61—232no—
259C4×D61—28no—
260C4×D61C2×C22⋊C418yes—
261C4×D61#12018yes—
262C8×C61—42no—
263#92418—42no—
264#20597—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61—22no—
269C22×C61#41614yescomplemented
270C22×C61C22×C22⋊C411yescharacteristic
271C22×C61C2×#12012yes—
272C22×C61C2×#13511yescharacteristic
273C22×C61C2×#13511yescharacteristic
274C22×C61#57512yes—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—24no—
278C2×D61—24no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—28no—
283C2×D61—416no—
284C2×D61—432no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—48no—
290C2×D61—48no—
291C2×D61#37914yes—
292C4×C61—28no—
293C4×C61—22no—
294C4×C61—82no—
295C4×C61C22×C22⋊C412yes—
296C4×C61C2×#12012yes—
297Dic61—22no—
298Dic61—82no—
299Dic61#37914yes—
300C23×#41—611no—
301C2×#416—614no—
302C22×C22⋊C8—611no—
303#2280—611no—
304#1788—614no—
305#1787—611no—
306#1787D6111yescharacteristic, complemented
307#1786—611no—
308#1786—611no—
309C2×C61—24no—
310C2×C61C2×#37911yescharacteristic
311C2×C61C2×#41611yescharacteristic
312C2×C61#289611yescharacteristic
313C2×C61#300212yes—
314C2×C61#300212yes—
315D61—24no—
316D61—42no—
317D61—48no—
318C22×#41—618no—
319C22×#41—612no—
320C22×#41—612no—
321C22×#41—1221no—
322C22×#41—1221no—
323C22×#41—1224no—
324C22×#41—12216no—
325C23×Q8C2×D6111yescharacteristic
326C23×D4—611no—
327C23×D4—1221no—
328C24×C4—611no—
329C24×C4—1221no—
330C23×C8—612no—
331C2×#117—612no—
332#416—12216no—
333C2×#103—614no—
334C2×#103C2×D6114yescomplemented
335C2×#102—614no—
336C2×#102—614no—
337C2×C22⋊C8—611no—
338C2×C22⋊C8—612no—
339C2×C22⋊C8—1224no—
340C2×C22⋊C8C2×D6111yescharacteristic
341#381—1221no—
342#380—611no—
343#380—612no—
344#380C2×D6111yescharacteristic
345#379—1221no—
346C4×C22⋊C4—1221no—
347C4×C22⋊C4—1221no—
348C61#2059711yescharacteristic, complemented, Sylow
349C25—611no—
350C25—1221no—
351C2×#41—618no—
352C2×#41—618no—
353C2×#41—6116no—
354C2×#41—1228no—
355C2×#41—1224no—
356C2×#41—1224no—
357C2×#41—1228no—
358C2×#41—1228no—
359C2×#41—1228no—
360C2×#41—12264no—
361C2×#41—12216no—
362C2×#41—12264no—
363C2×#41—12216no—
364C2×#41—1228no—
365C22×Q8—28no—
366C22×Q8—21no—
367C22×Q8—22no—
368C22×Q8C4×D6114yes—
369C22×Q8C22×D6112yes—
370C22×D4—614no—
371C22×D4—612no—
372C22×D4—1228no—
373C22×D4—1222no—
374C22×D4—1222no—
375C22×D4—1228no—
376C22×D4—12216no—
377C22×D4—1222no—
378C22×D4—1221no—
379C23×C4—21no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—614no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—614no—
386C23×C4—611no—
387C23×C4—1228no—
388C23×C4—1222no—
389C23×C4—1222no—
390C23×C4—1228no—
391C23×C4—12216no—
392C23×C4C22×D6111yescharacteristic
393C22×C8—612no—
394C22×C8—1224no—
395C22×C8—1228no—
396C22×C8C22×D6112yes—
397#117—1228no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4C22×D6112yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1221no—
404C2×C42—1222no—
405#103—28no—
406#103—1228no—
407#102—1228no—
408#102—1228no—
409C22⋊C8—22no—
410C22⋊C8—1224no—
411C22⋊C8—1222no—
412#95—21no—
413#95—1222no—
414#95—1221no—
415C24—611no—
416C24—611no—
417C24—614no—
418C24—1224no—
419C24—1228no—
420C24—1222no—
421C24—1224no—
422C24—1222no—
423C24—1222no—
424C24—2442no—
425C24—2444no—
426C24—2442no—
427C24—2442no—
428C24D4×D6111yescharacteristic
429#41—6132no—
430#41—12216no—
431#41—12232no—
432#41—12216no—
433#41—12264no—
434#41—12264no—
435#41—12232no—
436#41—12216no—
437C2×Q8—216no—
438C2×Q8—28no—
439C2×Q8—216no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8C2×C4×D6114yes—
444C2×Q8D4×D6118yes—
445C2×D4—614no—
446C2×D4—618no—
447C2×D4—12216no—
448C2×D4—1228no—
449C2×D4—1228no—
450C2×D4—12216no—
451C2×D4—12264no—
452C2×D4—12216no—
453C2×D4—1228no—
454C2×D4—12216no—
455C2×D4—1224no—
456C2×D4—1222no—
457C2×D4—1224no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—611no—
464C22×C4—612no—
465C22×C4—618no—
466C22×C4—612no—
467C22×C4—611no—
468C22×C4—614no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—1224no—
474C22×C4—12216no—
475C22×C4—12216no—
476C22×C4—1228no—
477C22×C4—1228no—
478C22×C4—12216no—
479C22×C4—1224no—
480C22×C4—12264no—
481C22×C4—12216no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—1222no—
485C22×C4C2×C4×D6112yes—
486C22×C4D4×D6112yes—
487C22×C4D4×D6111yescharacteristic
488C22×C4C23×D6111yescharacteristic
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—1222no—
492C2×C8—1224no—
493C2×C8—2448no—
494C4⋊C4—24no—
495C4⋊C4—1224no—
496C22⋊C4—42no—
497C22⋊C4—2444no—
498C22⋊C4—2444no—
499C22⋊C4—2442no—
500C42—2444no—
501C23—22no—
502C23—22no—
503C23—612no—
504C23—614no—
505C23—612no—
506C23—1228no—
507C23—12216no—
508C23—1221no—
509C23—1221no—
510C23—1222no—
511C23—1224no—
512C23—1228no—
513C23—1222no—
514C23—1228no—
515C23—1221no—
516C23—1221no—
517C23—1222no—
518C23—2444no—
519C23—2448no—
520C23—2448no—
521C23—2448no—
522C23—24416no—
523C23—24416no—
524C23—2448no—
525C23—2442no—
526C23—2442no—
527C23—2444no—
528C23C22⋊C4×D6112yes—
529C23#1451979812yes—
530C23#1451980012yes—
531C23C2×D4×D6111yescharacteristic
532Q8—216no—
533Q8—24no—
534Q8—28no—
535Q8C22⋊C4×D6118yes—
536D4—618no—
537D4—12232no—
538D4—12232no—
539D4—12216no—
540D4—1224no—
541D4—1228no—
542C2×C4—216no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—612no—
552C2×C4—614no—
553C2×C4—618no—
554C2×C4—618no—
555C2×C4—612no—
556C2×C4—1228no—
557C2×C4—12232no—
558C2×C4—1228no—
559C2×C4—12232no—
560C2×C4—1228no—
561C2×C4—1222no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4C22⋊C4×D6112yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4C2×D4×D6114yes—
570C2×C4C2×D4×D6111yescharacteristic
571C8—42no—
572C8—2442no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—22no—
577C22—614no—
578C22—1222no—
579C22—1224no—
580C22—1224no—
581C22—1222no—
582C22—1222no—
583C22—1222no—
584C22—1228no—
585C22—1222no—
586C22—24416no—
587C22—24432no—
588C22—2444no—
589C22—2442no—
590C22—2448no—
591C22—2442no—
592C22—2448no—
593C22—2444no—
594C22#1683975714yescomplemented
595C22C2×C22⋊C4×D6111yescharacteristic
596C22#120×D6112yes—
597C22C2×#1451979811yescharacteristic
598C22C2×#1451980011yescharacteristic
599C22#1684056112yes—
600C4—28no—
601C4—22no—
602C4—82no—
603C4—614no—
604C4—1222no—
605C4—4882no—
606C4C2×C22⋊C4×D6112yes—
607C4#120×D6112yes—
608C2—24no—
609C2—1224no—
610C2—2442no—
611C2—2448no—
612C2#379×D6111yescharacteristic
613C2C2×#1683975711yescharacteristic
614C2#2145433711yescharacteristic
615C2#2145503612yes—
616C2#2145504212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC23×#11758226C2
cosocleC22×C4×C61C24
hypercenter#1787D61
hypocenterC61#20597
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. C2×#16839757C2semidirect
3. C2×C22⋊C8×D61C2semidirect
4. #21448201C2semidirect
5. #21445758C2semidirect
6. #21445754C2semidirect
7. #21445751C2semidirect
8. #21445749C2semidirect
9. C61×#1787C2semidirect
14. C22×#11758226C4semidirect
21. C22×C8×D61C22semidirect
22. C2×#14519735C22semidirect
24. C2×#14519701C22semidirect
25. C2×#14519699C22semidirect
26. C2×#14519697C22semidirect
28. C2×#14519689C22semidirect
32. #16839619C22semidirect
37. C2×#103×C61C22semidirect
149. #11758226C22⋊C4semidirect
269. C22×C61#416semidirect
306. #1787D61semidirect
334. C2×#103C2×D61semidirect
348. C61#20597semidirect
594. C22#16839757semidirect

Maximal subgroups

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

subgroupindex
C23×#117582262
C2×#168397572
C2×C22⋊C8×D612
#214482012
#214457582
#214457542
#214457512
#214457492
C61×#17872
#2059761