Finite Group Databasebeta

← #29712209

Subgroups — #29712209

Order 15616.

1304 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840464C211yesmaximal, characteristic
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21449178C211yesmaximal, characteristic, complemented
4#21446968C212yesmaximal, complemented
5#21446819C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446155C212yesmaximal, complemented
8#21446128C212yesmaximal, complemented
9#21446109C211yesmaximal, characteristic, complemented
10#21445910C211yesmaximal, characteristic, complemented
11C2×C61×#448C211yesmaximal, characteristic, complemented
12C23×#92418C2211yescharacteristic
13#16840464—24no—
14#16840464—24no—
15C2×#14519770—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519761—21no—
18C2×#14519753C2212yes—
19C2×#14519749—21no—
20C2×#14519748C2211yescharacteristic
21C2×D4×Dic61—21no—
22C2×D4×Dic61C2211yescharacteristic
23C2×C4×C22⋊D61C2211yescharacteristic
24C2×#14519735C2211yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic
26C2×#14519701—24no—
27C2×#14519701C2212yes—
28C2×#14519701C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31#16839681—21no—
32#16839673—21no—
33#16839669C2212yes—
34#16839641—21no—
35#16839635—21no—
36#16839622C2211yescharacteristic
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C23×Q8×C61C2211yescharacteristic
42C22×SD16×C61C2212yes—
43C2×#121×C61C2211yescharacteristic
44C61×#448—24no—
45C2×#103×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×C22⋊D61—21no—
48C22×C22⋊D61D411yescharacteristic
49C23×Dic61—21no—
50C23×Dic61—21no—
51C22×C4×D61C2311yescharacteristic
52C22×#92418—22no—
53C22×#92418—42no—
54C22×#92418C2312yes—
55#14519770—28no—
56#14519768—22no—
57#14519768—22no—
58#14519761—28no—
59#14519753—24no—
60#14519753—42no—
61#14519749—28no—
62#14519748—22no—
63#14519748—22no—
64D4×Dic61—28no—
65D4×Dic61—22no—
66D4×Dic61—22no—
67C4×C22⋊D61—22no—
68C4×C22⋊D61—22no—
69#14519735—22no—
70#14519735—22no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D414yes—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—22no—
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61D411yescharacteristic
85C2×#11758198—21no—
86C2×#11758198—22no—
87C2×#11758198D411yescharacteristic
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519701—24no—
91#14519701—24no—
92#14519698—42no—
93#14519689—42no—
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—
103C22×D4×C61—21no—
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C23×C4×C61C2311yescharacteristic
107C2×SD16×C61—24no—
108C2×SD16×C61—24no—
109C2×SD16×C61—42no—
110C22×C8×C61C2312yes—
111#121×C61—22no—
112#121×C61D414yes—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115#103×C61—24no—
116C22⋊C8×C61—22no—
117C24×C61—21no—
118C24×C61C2×D411yescharacteristic
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—22no—
121C2×Q8×C61—28no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—44no—
127C2×Q8×C61—48no—
128C2×Q8×C61—44no—
129C2×Q8×C61—42no—
130C2×Q8×C61#4114yes—
131C2×D4×C61—24no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61C2×D412yes—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61—44no—
143C22×C4×C61—42no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—22no—
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—24no—
152C2×C22⋊D61—82no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—24no—
156C22×Dic61—24no—
157C22×Dic61—22no—
158C22×Dic61—82no—
159C22×Dic61—82no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167SD16×C61—48no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C2×D412yes—
173C22⋊C4×C61—24no—
174C22⋊Dic61—24no—
175C22⋊Dic61—24no—
176C22⋊Dic61—22no—
177C22⋊Dic61—42no—
178#11758198—24no—
179#11758198—22no—
180#11758198—44no—
181C4⋊Dic61—24no—
182C4⋊Dic61—24no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61C2×D412yes—
188C4⋊Dic61C2×D412yes—
189#11758196—24no—
190#11758196—22no—
191#11758196—44no—
192C4×Dic61—24no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—44no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—42no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—42no—
206C23×C61#12012yes—
207C23×C61#13512yes—
208C23×C61#13712yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—21no—
211C22×D61—22no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—22no—
215Q8×C61—24no—
216Q8×C61—24no—
217Q8×C61—48no—
218Q8×C61—42no—
219Q8×C61—42no—
220D4×C61—24no—
221D4×C61—48no—
222D4×C61—41no—
223D4×C61—41no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—48no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61#12012yes—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4112yes—
239C2×C4×C61H(5,𝔽2)12yes—
240C22⋊D61—44no—
241C22⋊D61—84no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—84no—
257C2×Dic61—84no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—42no—
261C8×C61—42no—
262#92418—42no—
263#23964—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61#45214yes—
274C22×C61C2×#12011yescharacteristic
275C22×C61#54612yes—
276C22×C61C2×#13511yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57712yes—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—42no—
283C2×D61—42no—
284C4×C61—22no—
285C4×C61—44no—
286C4×C61—42no—
287C4×C61#52511yescharacteristic
288C4×C61#53511yescharacteristic
289C4×C61#54612yes—
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61—42no—
294Dic61—42no—
295Dic61—82no—
296C2×#553—611no—
297C2×#448D6111yescharacteristic, complemented
298C22×#103—611no—
299#2409—611no—
300#2026—612no—
301#2000—612no—
302#1893—611no—
303#1877—612no—
304#1873—612no—
305#1869—611no—
306#1824—611no—
307C2×C61—24no—
308C2×C61—82no—
309C2×C61#240411yescharacteristic
310C2×C61C2×#45211yescharacteristic
311C2×C61#300911yescharacteristic
312C2×C61#330112yes—
313C2×C61#332012yes—
314D61—81no—
315C23×Q8C2×D6111yescharacteristic
316C22×SD16C2×D6112yes—
317C23×C8—611no—
318#553—1224no—
319#553—1224no—
320C2×#126—1221no—
321C2×#125—611no—
322C2×#123—1221no—
323C2×C22⋊Q8—612no—
324C2×#121—611no—
325C2×#121—1221no—
326C2×#121C2×D6111yescharacteristic
327C2×C4×D4—611no—
328C2×C4×D4—611no—
329C2×C4×D4—1221no—
330C2×#117—611no—
331C22×C4⋊C4—611no—
332#448—24no—
333C2×#103—612no—
334C2×#103—612no—
335C2×#103—1224no—
336C2×#103C2×D6112yes—
337C2×#102—612no—
338C2×C22⋊C8—612no—
339C2×C22⋊C8C2×D6111yescharacteristic
340#395—1221no—
341#393—1221no—
342#391—612no—
343#386—1221no—
344#385—1221no—
345#380—611no—
346#340—612no—
347#340—612no—
348#340—612no—
349#336—612no—
350C61#2396411yescharacteristic, complemented, Sylow
351C22×Q8—21no—
352C22×Q8—24no—
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—42no—
356C22×Q8C22×D6112yes—
357C22×D4—21no—
358C22×D4—611no—
359C22×D4—1221no—
360C22×D4C22×D6111yescharacteristic
361C23×C4—21no—
362C23×C4—611no—
363C23×C4—1221no—
364C23×C4—1221no—
365C23×C4C22×D6111yescharacteristic
366C2×SD16—24no—
367C2×SD16—24no—
368C2×SD16—42no—
369C22×C8—612no—
370C22×C8—1222no—
371C22×C8—2442no—
372C22×C8C22×D6112yes—
373#126—1228no—
374#125—1222no—
375#125—1222no—
376#123—1228no—
377C22⋊Q8—1224no—
378C22⋊Q8—2442no—
379#121—22no—
380#121—1222no—
381#121—1228no—
382#121—1222no—
383#121C22⋊D6114yes—
384C4×D4—1222no—
385C4×D4—1228no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1222no—
389#117—1222no—
390#117—1222no—
391C2×C4⋊C4—614no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—611no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1222no—
409C2×C22⋊C4—1221no—
410C2×C42—611no—
411C2×C42—1221no—
412#103—24no—
413#103—1224no—
414#103—1224no—
415#102—2442no—
416C22⋊C8—22no—
417C22⋊C8—2442no—
418#95—1222no—
419#95—1222no—
420#95—1221no—
421C24—21no—
422C24—611no—
423C24D4×D6111yescharacteristic
424C2×Q8—22no—
425C2×Q8—28no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8#1175822414yes—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—1224no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—4882no—
446C2×D4C2×C22⋊D6112yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—1224no—
457C22×C4—1224no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—4882no—
466C22×C4—4882no—
467C22×C4D4×D6112yes—
468C22×C4D4×D6111yescharacteristic
469C22×C4C2×C22⋊D6112yes—
470C22×C4C23×D6111yescharacteristic
471SD16—48no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—2444no—
479C4⋊C4—21no—
480C4⋊C4—612no—
481C4⋊C4—612no—
482C4⋊C4—1224no—
483C4⋊C4—1224no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1224no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—2444no—
491C4⋊C4C2×C22⋊D6112yes—
492C22⋊C4—24no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—2444no—
499C22⋊C4—2442no—
500C42—1224no—
501C42—1221no—
502C42—1221no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1222no—
514C23—1222no—
515C23#1451974512yes—
516C23#1451979812yes—
517C23#1451980612yes—
518C23C2×D4×D6111yescharacteristic
519Q8—24no—
520Q8—24no—
521Q8—48no—
522Q8—42no—
523Q8—42no—
524D4—24no—
525D4—48no—
526D4—41no—
527D4—41no—
528D4—4884no—
529D4—2444no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1222no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—4884no—
551C2×C4—4884no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4#1451974512yes—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4C2×#1175822412yes—
564C2×C4#1451982212yes—
565C8—42no—
566C8—2442no—
567C22—24no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—82no—
575C22—82no—
576C22—2442no—
577C22—2442no—
578C22—2441no—
579C22—2441no—
580C22—2441no—
581C22#1683992114yes—
582C22C2×#1451974511yescharacteristic
583C22#1684039912yes—
584C22C2×#1451979811yescharacteristic
585C22C2×#1451980611yescharacteristic
586C22#1684057112yes—
587C4—22no—
588C4—44no—
589C4—42no—
590C4—4882no—
591C4—2441no—
592C4—2441no—
593C4—2441no—
594C4—2442no—
595C4—2442no—
596C4#1684026811yescharacteristic
597C4#1684032711yescharacteristic
598C4#1684040612yes—
599C2—24no—
600C2—82no—
601C2—4881no—
602C2#2144913511yescharacteristic
603C2C2×#1683992111yescharacteristic
604C2#2145509511yescharacteristic
605C2#2145772512yes—
606C2#2145799412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#448C2
socleC23×C61C22×D4
prime radicalC2×#16840464C2
hypercenterC2×#448D61
hypocenterC61#23964
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. C22×#14519701C2semidirect
3. #21449178C2semidirect
4. #21446968C2semidirect
5. #21446819C2semidirect
6. #21446248C2semidirect
7. #21446155C2semidirect
8. #21446128C2semidirect
9. #21446109C2semidirect
10. #21445910C2semidirect
11. C2×C61×#448C2semidirect
297. C2×#448D61semidirect
350. C61#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#168404642
C22×#145197012
#214491782
#214469682
#214468192
#214462482
#214461552
#214461282
#214461092
#214459102
C2×C61×#4482
#2396461