Finite Group Databasebeta

← #29715680

Subgroups — #29715680

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
1C2×#16839988C212yesmaximal, complemented
2C2×C8×C22⋊D61C211yesmaximal, characteristic, complemented
3#21449178C211yesmaximal, characteristic, complemented
4#21449169C211yesmaximal, characteristic
5#21447132C212yesmaximal, complemented
6#21446819C212yesmaximal, complemented
7#21446675C212yesmaximal, complemented
8#21445844C211yesmaximal, characteristic, complemented
9#21445754C211yesmaximal, characteristic, complemented
10#21445750C211yesmaximal, characteristic, complemented
11C23×SD16×C61C211yesmaximal, characteristic, complemented
12C22×C8×D61C2211yescharacteristic, complemented
13C2×#14519753C2212yescomplemented
14C2×#14519748C2212yes—
15C2×#14519746—21no—
16C2×C4×C22⋊D61C2211yescharacteristic
17#16839988—28no—
18C8×C22⋊D61—24no—
19C2×#14519709C2212yescomplemented
20C2×#14519706C2211yescharacteristic, complemented
21C2×#14519701C2212yescomplemented
22C2×#14519701C2212yescomplemented
23C2×#14519698C2212yes—
24C2×#14519698C2212yes—
25C2×C22⋊#92418C2211yescharacteristic
26C2×#14519689C2211yescharacteristic, complemented
27C2×C8×Dic61C2211yescharacteristic, complemented
28#16839681—21no—
29#16839676—21no—
30#16839673—21no—
31#16839670—21no—
32#16839669C2212yescomplemented
33#16839665C2212yes—
34#16839641—21no—
35#16839635—21no—
36#16839622C2211yescharacteristic
37#16839622C2211yescharacteristic
38#16839618—21no—
39#16839525C2212yescomplemented
40C23×Q8×C61C2211yescharacteristic
41C23×D4×C61C2211yescharacteristic
42C22×SD16×C61—28no—
43C22×SD16×C61—24no—
44C22×SD16×C61C2212yes—
45C22×SD16×C61C2212yes—
46C23×C8×C61C2211yescharacteristic
47C22×C22⋊D61—22no—
48C22×C22⋊D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C22×#92418C2311yescharacteristic
52C2×C8×D61—24no—
53C2×C8×D61—22no—
54#14519753—24no—
55#14519753—42no—
56#14519748—24no—
57#14519748—42no—
58#14519746—44no—
59C4×C22⋊D61—22no—
60C4×C22⋊D61—22no—
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—22no—
64C2×#11758196D411yescharacteristic
65C2×C22⋊Dic61—21no—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61—22no—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61D411yescharacteristic
71C2×#11758198—22no—
72C2×#11758198D411yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74#14519709—24no—
75#14519706—22no—
76#14519701—24no—
77#14519701—42no—
78#14519698—24no—
79#14519698—42no—
80C22⋊#92418—22no—
81#14519689—22no—
82C8×Dic61—22no—
83#14519684—21no—
84#14519684—21no—
85#14519684—22no—
86#14519684—22no—
87C25×C61—21no—
88C22×Q8×C61—21no—
89C22×Q8×C61—24no—
90C22×Q8×C61—22no—
91C22×Q8×C61—22no—
92C22×Q8×C61—42no—
93C22×Q8×C61C2312yes—
94C22×D4×C61—24no—
95C22×D4×C61—22no—
96C22×D4×C61—21no—
97C22×D4×C61—22no—
98C22×D4×C61—42no—
99C22×D4×C61C2312yes—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C2×SD16×C61—232no—
103C2×SD16×C61—24no—
104C2×SD16×C61—24no—
105C2×SD16×C61—24no—
106C2×SD16×C61—24no—
107C2×SD16×C61—24no—
108C2×SD16×C61D418yescomplemented
109C22×C8×C61—24no—
110C22×C8×C61D414yescomplemented
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C24×C61—22no—
114C24×C61—44no—
115C24×C61—42no—
116C24×C61—42no—
117C24×C61—42no—
118C24×C61—82no—
119C24×C61C2×D411yescharacteristic
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—
132C2×D4×C61—28no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—48no—
139C2×D4×C61—44no—
140C2×D4×C61—44no—
141C2×D4×C61—42no—
142C2×D4×C61C2×D414yes—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—44no—
147C22×C4×C61—42no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61#4112yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—44no—
155C2×C22⋊D61—44no—
156C2×C22⋊D61—82no—
157C2×C22⋊D61SD1614yescomplemented
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—22no—
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167SD16×C61—216no—
168SD16×C61—216no—
169C2×C8×C61—22no—
170C2×C8×C61—28no—
171C2×C8×C61C2×D412yes—
172C2×C8×C61C2×D414yes—
173C2×C8×C61C2×D412yes—
174C22⋊Dic61—22no—
175C22⋊Dic61—44no—
176C22⋊Dic61—42no—
177C22⋊Dic61—42no—
178C22⋊Dic61—44no—
179C22⋊Dic61—82no—
180#11758198—22no—
181#11758198—44no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186#11758196—22no—
187#11758196—44no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—22no—
193C2×#92418—22no—
194C8×D61—44no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—24no—
198C23×C61—48no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—88no—
206C23×C61—88no—
207C23×C61—84no—
208C23×C61—84no—
209C23×C61#12112yes—
210C23×C61C2×SD1612yes—
211C23×C61#13512yes—
212C23×C61C22×D411yescharacteristic
213C22×D61—21no—
214C22×D61—22no—
215C22×D61—21no—
216C22×D61—22no—
217C22×D61C2×SD1612yescomplemented
218Q8×C61—24no—
219Q8×C61—24no—
220Q8×C61—48no—
221Q8×C61—42no—
222Q8×C61—42no—
223D4×C61—24no—
224D4×C61—24no—
225D4×C61—48no—
226D4×C61—42no—
227D4×C61—42no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—48no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61#12112yes—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D412yes—
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61H(5,𝔽2)12yes—
241C22⋊D61—28no—
242C22⋊D61—88no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61C2×SD1612yescomplemented
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—42no—
259C8×C61—22no—
260C8×C61#12014yescomplemented
261#92418—41no—
262#92418—41no—
263#24264—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—22no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61—816no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61—82no—
275C22×C61—82no—
276C22×C61#46514yescomplemented
277C22×C61C2×#12111yescharacteristic
278C22×C61D4212yes—
279C22×C61C22×SD1611yescharacteristic
280C22×C61C2×#13511yescharacteristic
281C22×C61#57712yes—
282C2×D61—24no—
283C2×D61—22no—
284C2×D61—41no—
285C2×D61—42no—
286C2×D61—41no—
287C4×C61—22no—
288C4×C61—44no—
289C4×C61C2×#12011yescharacteristic
290C4×C61#53511yescharacteristic
291C4×C61#54612yes—
292Dic61—22no—
293Dic61—41no—
294Dic61—41no—
295Dic61—82no—
296Dic61—82no—
297C23×SD16D6111yescharacteristic, complemented
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#1787—611no—
307#1786—611no—
308C2×C61—24no—
309C2×C61—84no—
310C2×C61#240811yescharacteristic
311C2×C61C2×#46511yescharacteristic
312C2×C61#305711yescharacteristic
313C2×C61D4×SD1612yes—
314C2×C61#328212yes—
315D61—42no—
316C23×Q8C2×D6111yescharacteristic
317C23×D4C2×D6111yescharacteristic
318C22×SD16—28no—
319C22×SD16—24no—
320C22×SD16C2×D6112yes—
321C22×SD16C2×D6112yes—
322C23×C8—611no—
323C23×C8C2×D6111yescharacteristic
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×#103—612no—
333C2×#103—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×C22⋊C8—611no—
337C2×C22⋊C8—611no—
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#380—611no—
349#379—1221no—
350#340—612no—
351C61#2426411yescharacteristic, complemented, Sylow
352C25—21no—
353C22×Q8—21no—
354C22×Q8—24no—
355C22×Q8—22no—
356C22×Q8—22no—
357C22×Q8—42no—
358C22×Q8C22×D6112yes—
359C22×D4—24no—
360C22×D4—21no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4—42no—
364C22×D4—611no—
365C22×D4—1222no—
366C22×D4C22×D6112yes—
367C23×C4—21no—
368C23×C4—611no—
369C23×C4C22×D6111yescharacteristic
370C2×SD16—232no—
371C2×SD16—24no—
372C2×SD16—24no—
373C2×SD16—24no—
374C2×SD16—24no—
375C2×SD16—24no—
376C2×SD16C22⋊D6118yescomplemented
377C22×C8—24no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—1222no—
381C22×C8—1224no—
382C22×C8C22⋊D6114yescomplemented
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—612no—
395C2×C4⋊C4—1222no—
396C2×C22⋊C4—611no—
397C2×C22⋊C4—611no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C42—611no—
405#106—1224no—
406C4⋊C8—1222no—
407#103—1224no—
408#103—2442no—
409#102—1224no—
410#102—2442no—
411C22⋊C8—1222no—
412C22⋊C8—1222no—
413C4×C8—1222no—
414#95—1222no—
415#95—1222no—
416#95—1221no—
417#95—1221no—
418C24—22no—
419C24—44no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—82no—
424C24—611no—
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×Q8C2×C22⋊D6114yes—
437C2×D4—28no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—48no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—614no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—4882no—
451C2×D4—2444no—
452C2×D4—2444no—
453C2×D4C2×C22⋊D6114yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4D4×D6111yescharacteristic
469C22×C4#1175822412yes—
470C22×C4C2×C22⋊D6112yes—
471C22×C4C23×D6111yescharacteristic
472SD16—216no—
473SD16—216no—
474C2×C8—22no—
475C2×C8—28no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—2444no—
481C2×C8C2×C22⋊D6112yes—
482C2×C8C2×C22⋊D6112yes—
483C2×C8C2×C22⋊D6114yes—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—2444no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—4882no—
493C22⋊C4—2444no—
494C22⋊C4—2444no—
495C22⋊C4—2444no—
496C22⋊C4—2442no—
497C22⋊C4—2442no—
498C42—1221no—
499C42—1221no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—48no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—88no—
511C23—88no—
512C23—84no—
513C23—84no—
514C23—612no—
515C23—1221no—
516C23—1221no—
517C23—1222no—
518C23—1222no—
519C23#1451974812yes—
520C23SD16×D6112yes—
521C23#1451979812yes—
522C23C2×D4×D6111yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—48no—
526Q8—42no—
527Q8—42no—
528D4—24no—
529D4—24no—
530D4—48no—
531D4—42no—
532D4—42no—
533D4—1228no—
534D4—4888no—
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—612no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4#1451974812yes—
560C2×C4C22×C22⋊D6111yescharacteristic
561C2×C4C22×C22⋊D6112yes—
562C2×C4C2×#1175822411yescharacteristic
563C2×C4#1451982212yes—
564C8—22no—
565C8—2441no—
566C8—2441no—
567C8#1451974614yescomplemented
568C22—22no—
569C22—24no—
570C22—24no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—816no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—1222no—
581C22—1224no—
582C22—2441no—
583C22—2442no—
584C22—2441no—
585C22#1683998814yescomplemented
586C22C2×#1451974811yescharacteristic
587C22D4×C22⋊D6112yes—
588C22C2×SD16×D6111yescharacteristic
589C22C2×#1451979811yescharacteristic
590C22#1684056812yes—
591C4—22no—
592C4—44no—
593C4—1222no—
594C4—4882no—
595C4—4882no—
596C4—2441no—
597C4—2441no—
598C4C2×#1451974611yescharacteristic
599C4#1684032911yescharacteristic
600C4#1684040612yes—
601C2—24no—
602C2—84no—
603C2—2442no—
604C2#2144916911yescharacteristic
605C2C2×#1683998811yescharacteristic
606C2#2145545411yescharacteristic
607C2SD16×C22⋊D6112yes—
608C2#2145752612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC23×SD16×C61C2
socleC23×C61C22×D4
prime radical#21449169C2
hypercenterC23×SD16D61
hypocenterC61#24264
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16839988C2semidirect
2. C2×C8×C22⋊D61C2semidirect
3. #21449178C2semidirect
5. #21447132C2semidirect
6. #21446819C2semidirect
7. #21446675C2semidirect
8. #21445844C2semidirect
9. #21445754C2semidirect
10. #21445750C2semidirect
11. C23×SD16×C61C2semidirect
12. C22×C8×D61C22semidirect
13. C2×#14519753C22semidirect
19. C2×#14519709C22semidirect
20. C2×#14519706C22semidirect
21. C2×#14519701C22semidirect
22. C2×#14519701C22semidirect
26. C2×#14519689C22semidirect
27. C2×C8×Dic61C22semidirect
32. #16839669C22semidirect
39. #16839525C22semidirect
108. C2×SD16×C61D4semidirect
110. C22×C8×C61D4semidirect
157. C2×C22⋊D61SD16semidirect
217. C22×D61C2×SD16semidirect
255. C2×Dic61C2×SD16semidirect
260. C8×C61#120semidirect
276. C22×C61#465semidirect
297. C23×SD16D61semidirect
351. C61#24264semidirect
376. C2×SD16C22⋊D61semidirect
382. C22×C8C22⋊D61semidirect
567. C8#14519746semidirect
585. C22#16839988semidirect

Maximal subgroups

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

subgroupindex
C2×#168399882
C2×C8×C22⋊D612
#214491782
#214491692
#214471322
#214468192
#214466752
#214458442
#214457542
#214457502
C23×SD16×C612
#2426461