Finite Group Databasebeta

← #29715689

Subgroups — #29715689

Order 15616.

1606 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758201C211yesmaximal, characteristic, complemented
2C2×#16839994C212yesmaximal, complemented
3C2×#16839825C211yesmaximal, characteristic, complemented
4#21449178C211yesmaximal, characteristic, complemented
5#21449176C211yesmaximal, characteristic
6#21447133C212yesmaximal, complemented
7#21446821C212yesmaximal, complemented
8#21446670C212yesmaximal, complemented
9#21445842C211yesmaximal, characteristic, complemented
10#21445751C211yesmaximal, characteristic, complemented
11C61×#1787C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758201—28no—
14C22×#11758201—24no—
15C22×#11758201C2212yes—
16C22×#11758201C2212yes—
17C23×#92418C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic, complemented
19C2×#14519753C2212yescomplemented
20C2×#14519752C2212yes—
21C2×#14519747—21no—
22C2×C4×C22⋊D61C2211yescharacteristic
23#16839994—28no—
24#16839825—24no—
25C2×#14519708C2212yescomplemented
26C2×#14519706C2211yescharacteristic, complemented
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29C2×#14519689C2211yescharacteristic, complemented
30C2×C8×Dic61C2211yescharacteristic, complemented
31#16839681—21no—
32#16839680—21no—
33#16839673—21no—
34#16839672—21no—
35#16839669C2212yescomplemented
36#16839663C2212yes—
37#16839641—21no—
38#16839635—21no—
39#16839622C2211yescharacteristic
40#16839617—21no—
41#16839524C2212yescomplemented
42C23×Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yescomplemented
44C2×#103×C61C2212yescomplemented
45C2×C22⋊C8×C61C2211yescharacteristic
46C61×#380C2211yescharacteristic
47C24×D61—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—22no—
53C22×D244—42no—
54C22×D244C2312yes—
55C22×C22⋊D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C2×#11758201—232no—
58C2×#11758201—24no—
59C2×#11758201—24no—
60C2×#11758201—24no—
61C2×#11758201—24no—
62C2×#11758201—24no—
63C2×#11758201D418yescomplemented
64C22×#92418—24no—
65C22×#92418D414yescomplemented
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C2×C8×D61—24no—
69C2×C8×D61—22no—
70#14519753—24no—
71#14519753—42no—
72#14519752—24no—
73#14519752—42no—
74#14519747—44no—
75C4×C22⋊D61—22no—
76C4×C22⋊D61—22no—
77C2×C4⋊Dic61C2312yes—
78C2×#11758196—22no—
79C2×#11758196D411yescharacteristic
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61D411yescharacteristic
82C2×#11758198—22no—
83C2×#11758198—22no—
84C2×#11758198—22no—
85C2×#11758198—22no—
86C2×#11758198D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519708—24no—
89#14519706—22no—
90#14519699—24no—
91#14519699—42no—
92#14519689—22no—
93C8×Dic61—22no—
94#14519684—21no—
95#14519684—22no—
96#14519684—22no—
97C22×Q8×C61—21no—
98C22×Q8×C61—24no—
99C22×Q8×C61—22no—
100C22×Q8×C61—22no—
101C22×Q8×C61—42no—
102C22×Q8×C61C2312yes—
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—21no—
109#103×C61—24no—
110#103×C61—42no—
111C22⋊C8×C61—22no—
112#95×C61—21no—
113C24×C61C2×D411yescharacteristic
114C23×D61—22no—
115C23×D61—44no—
116C23×D61—42no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—82no—
120C23×D61C2×D411yescharacteristic
121C2×Q8×C61—22no—
122C2×Q8×C61—28no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C2×Q8×C61—48no—
129C2×Q8×C61—44no—
130C2×Q8×C61—42no—
131C2×Q8×C61C2×D414yes—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—44no—
137C22×C4×C61—42no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D412yes—
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—22no—
143C2×C22⋊D61—22no—
144C2×C22⋊D61SD1614yescomplemented
145C22×Dic61—22no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×D244—28no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—48no—
156C2×D244—44no—
157C2×D244—44no—
158C2×D244—42no—
159C2×D244—44no—
160C2×D244—44no—
161C2×D244—82no—
162C2×D244C2×D414yes—
163C2×C4×D61—21no—
164C2×C4×D61—22no—
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C22⋊C4×C61—42no—
174C22⋊Dic61—22no—
175C22⋊Dic61—42no—
176#11758201—216no—
177#11758201—216no—
178#11758198—22no—
179#11758198—44no—
180#11758198—44no—
181#11758198—44no—
182#11758198—82no—
183C4⋊Dic61—22no—
184C4⋊Dic61—22no—
185#11758196—22no—
186#11758196—44no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C2×#92418—22no—
190C2×#92418—28no—
191C2×#92418C2×D412yes—
192C2×#92418C2×D412yes—
193C2×#92418C2×D414yes—
194C8×D61—44no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61#12112yes—
198C23×C61C2×SD1612yes—
199C23×C61#13512yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—21no—
202C22×D61—22no—
203C22×D61—21no—
204C22×D61—22no—
205C22×D61—24no—
206C22×D61—48no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—88no—
214C22×D61—88no—
215C22×D61—84no—
216C22×D61—84no—
217C22×D61C2×SD1612yescomplemented
218Q8×C61—24no—
219Q8×C61—24no—
220Q8×C61—48no—
221Q8×C61—42no—
222Q8×C61—42no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—48no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61#12112yes—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D412yes—
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61H(5,𝔽2)12yes—
239C22⋊D61—28no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61C2×SD1612yescomplemented
250D244—24no—
251D244—24no—
252D244—48no—
253D244—42no—
254D244—42no—
255D244—88no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—42no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—22no—
262#92418#12014yescomplemented
263#24264—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—22no—
268C22×C61#46514yescomplemented
269C22×C61C2×#12111yescharacteristic
270C22×C61D4212yes—
271C22×C61C22×SD1611yescharacteristic
272C22×C61C2×#13511yescharacteristic
273C22×C61#57712yes—
274C2×D61—24no—
275C2×D61—22no—
276C2×D61—41no—
277C2×D61—42no—
278C2×D61—41no—
279C2×D61—44no—
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—816no—
283C2×D61—82no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—82no—
287C4×C61—22no—
288C4×C61—44no—
289C4×C61—82no—
290C4×C61C2×#12011yescharacteristic
291C4×C61#53511yescharacteristic
292C4×C61#54612yes—
293Dic61—22no—
294Dic61—41no—
295Dic61—41no—
296Dic61—82no—
297C23×SD16—611no—
298C2×#465—612no—
299C2×C8×D4—611no—
300#2409—611no—
301#2408—611no—
302#2055—612no—
303#2000—612no—
304#1968—612no—
305#1807—611no—
306#1787D6111yescharacteristic, complemented
307#1786—611no—
308C2×C61—24no—
309C2×C61#240811yescharacteristic
310C2×C61C2×#46511yescharacteristic
311C2×C61#305711yescharacteristic
312C2×C61D4×SD1612yes—
313C2×C61#328212yes—
314D61—42no—
315D61—84no—
316C23×Q8C2×D6111yescharacteristic
317C23×D4—611no—
318C22×SD16—612no—
319C22×SD16—612no—
320C22×SD16—1224no—
321C22×SD16—1228no—
322C23×C8—611no—
323C23×C8—611no—
324C2×C22⋊Q8—612no—
325C2×#121—612no—
326C2×#120—1221no—
327C2×C4×D4—611no—
328#465—1228no—
329C8×D4—1224no—
330C2×#106—612no—
331C2×C4⋊C8—611no—
332C2×#103C2×D6112yescomplemented
333C2×#103C2×D6112yescomplemented
334C2×#102—612no—
335C2×#102—612no—
336C2×C22⋊C8—611no—
337C2×C22⋊C8C2×D6111yescharacteristic
338C2×C4×C8—611no—
339#395—1221no—
340#394—1221no—
341#393—1221no—
342#392—1221no—
343#391—612no—
344#390—612no—
345#386—1221no—
346#385—1221no—
347#380—611no—
348#380C2×D6111yescharacteristic
349#379—1221no—
350#340—612no—
351C61#2426411yescharacteristic, complemented, Sylow
352C25—1221no—
353C22×Q8—21no—
354C22×Q8—24no—
355C22×Q8—22no—
356C22×Q8—22no—
357C22×Q8—42no—
358C22×Q8C22×D6112yes—
359C22×D4—611no—
360C22×D4—612no—
361C22×D4—1224no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—1221no—
366C22×D4—2442no—
367C23×C4—21no—
368C23×C4—611no—
369C23×C4C22×D6111yescharacteristic
370C2×SD16—618no—
371C2×SD16—1224no—
372C2×SD16—1224no—
373C2×SD16—1224no—
374C2×SD16—1224no—
375C2×SD16—12232no—
376C2×SD16—1224no—
377C22×C8—611no—
378C22×C8—614no—
379C22×C8—611no—
380C22×C8—1222no—
381C22×C8—1224no—
382C22×C8—1224no—
383C22×C8C22×D6111yescharacteristic
384C22×C8C22×D6111yescharacteristic
385C22⋊Q8—1224no—
386C22⋊Q8—2442no—
387#121—1224no—
388#121—2442no—
389#120—2444no—
390C4×D4—1222no—
391C4×D4—1222no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4C22×D6112yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—611no—
398C2×C22⋊C4—611no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1221no—
404C2×C42—611no—
405#106—1224no—
406C4⋊C8—1222no—
407#103—24no—
408#103—42no—
409#102—1224no—
410#102—2442no—
411C22⋊C8—22no—
412C22⋊C8—1222no—
413C4×C8—1222no—
414#95—21no—
415#95—1222no—
416#95—1222no—
417#95—1221no—
418C24—611no—
419C24—1222no—
420C24—4882no—
421C24—2444no—
422C24—2442no—
423C24—2442no—
424C24—2442no—
425C24D4×D6111yescharacteristic
426C2×Q8—22no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—44no—
433C2×Q8—48no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8D4×D6114yes—
437C2×D4—614no—
438C2×D4—614no—
439C2×D4—1228no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1222no—
447C2×D4—4882no—
448C2×D4—2448no—
449C2×D4—2444no—
450C2×D4—2444no—
451C2×D4—2444no—
452C2×D4—2444no—
453C2×D4—2442no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4#1175822212yes—
469C22×C4D4×D6112yes—
470C22×C4D4×D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472SD16—12216no—
473SD16—12216no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—612no—
479C2×C8—614no—
480C2×C8—612no—
481C2×C8—1228no—
482C2×C8—1222no—
483C2×C8—2444no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—2444no—
490C22⋊C4—42no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—4882no—
494C22⋊C4—2444no—
495C22⋊C4—2444no—
496C22⋊C4—2444no—
497C22⋊C4—2442no—
498C42—1221no—
499C42—1221no—
500C23—22no—
501C23—22no—
502C23—612no—
503C23—1221no—
504C23—1221no—
505C23—1222no—
506C23—1222no—
507C23—1224no—
508C23—4884no—
509C23—4888no—
510C23—4888no—
511C23—4884no—
512C23—2448no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23#1451975012yes—
520C23SD16×D6112yes—
521C23#1451980012yes—
522C23C2×D4×D6111yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—48no—
526Q8—42no—
527Q8—42no—
528D4—1228no—
529D4—1224no—
530D4—1224no—
531D4—4888no—
532D4—2448no—
533D4—2442no—
534D4—2442no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—612no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4#1451975012yes—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×D4×D6112yes—
562C2×C4C2×#1175822211yescharacteristic
563C2×C4#1451982212yes—
564C8—41no—
565C8—41no—
566C8—614no—
567C8—1222no—
568C22—24no—
569C22—22no—
570C22—24no—
571C22—22no—
572C22—1222no—
573C22—1224no—
574C22—48816no—
575C22—4882no—
576C22—4882no—
577C22—4882no—
578C22—4882no—
579C22—2444no—
580C22—2444no—
581C22—2444no—
582C22—2441no—
583C22—2442no—
584C22—2441no—
585C22#1683999414yescomplemented
586C22C2×#1451975011yescharacteristic
587C22#1684039412yes—
588C22C2×SD16×D6111yescharacteristic
589C22C2×#1451980011yescharacteristic
590C22#1684057112yes—
591C4—22no—
592C4—44no—
593C4—82no—
594C4—1222no—
595C4—4882no—
596C4—2441no—
597C4—2441no—
598C4#120×D6111yescharacteristic
599C4#1684032711yescharacteristic
600C4#1684040212yes—
601C2—24no—
602C2—4884no—
603C2—2442no—
604C2#2144917411yescharacteristic
605C2C2×#1683999411yescharacteristic
606C2#2145546111yescharacteristic
607C2#2145750812yes—
608C2#2145751912yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radical#21449176C2
hypercenter#1787D61
hypocenterC61#24264
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C23×#11758201C2semidirect
2. C2×#16839994C2semidirect
3. C2×#16839825C2semidirect
4. #21449178C2semidirect
6. #21447133C2semidirect
7. #21446821C2semidirect
8. #21446670C2semidirect
9. #21445842C2semidirect
10. #21445751C2semidirect
11. C61×#1787C2semidirect
18. C22×C8×D61C22semidirect
19. C2×#14519753C22semidirect
25. C2×#14519708C22semidirect
26. C2×#14519706C22semidirect
29. C2×#14519689C22semidirect
30. C2×C8×Dic61C22semidirect
35. #16839669C22semidirect
41. #16839524C22semidirect
43. C2×#103×C61C22semidirect
44. C2×#103×C61C22semidirect
63. C2×#11758201D4semidirect
65. C22×#92418D4semidirect
144. C2×C22⋊D61SD16semidirect
217. C22×D61C2×SD16semidirect
249. C2×Dic61C2×SD16semidirect
262. #92418#120semidirect
268. C22×C61#465semidirect
306. #1787D61semidirect
332. C2×#103C2×D61semidirect
333. C2×#103C2×D61semidirect
351. C61#24264semidirect
585. C22#16839994semidirect

Maximal subgroups

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

subgroupindex
C23×#117582012
C2×#168399942
C2×#168398252
#214491782
#214491762
#214471332
#214468212
#214466702
#214458422
#214457512
C61×#17872
#2426461