Finite Group Databasebeta

← #29713076

Subgroups — #29713076

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊D61C211yesmaximal, characteristic
2C2×#16840098C211yesmaximal, characteristic, complemented
3C22×#14519701C211yesmaximal, characteristic, complemented
4#21450137C211yesmaximal, characteristic
5#21447064C212yesmaximal, complemented
6#21447021C212yesmaximal, complemented
7#21446240C211yesmaximal, characteristic, complemented
8#21446109C211yesmaximal, characteristic
9#21446075C212yesmaximal, complemented
10#21445752C212yesmaximal, complemented
11C61×#1787C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14Q8×C22⋊D61—24no—
15Q8×C22⋊D61—24no—
16C2×#14519775—21no—
17C2×#14519775C2211yescharacteristic
18C2×#14519756—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519753—21no—
21C2×#14519753C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×C4×C22⋊D61C2211yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic
26#16840098—24no—
27C2×#14519711C2212yes—
28C2×#14519702C2212yes—
29C2×#14519701—24no—
30C2×#14519701C2212yes—
31C2×#14519701C2212yes—
32C2×#14519700C2212yes—
33C2×#14519689C2212yes—
34#16839702—21no—
35#16839699—21no—
36#16839661C2212yes—
37#16839639—21no—
38#16839637—21no—
39#16839619C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C23×Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×#103×C61C2212yes—
45C2×C22⋊C8×C61C2211yescharacteristic
46C61×#380C2211yescharacteristic
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—22no—
51C2×Q8×D61—24no—
52C2×Q8×D61—42no—
53C22×C22⋊D61D411yescharacteristic
54C23×Dic61—21no—
55C22×C4×D61—21no—
56C22×C4×D61C2311yescharacteristic
57C22×#92418—22no—
58C22×#92418—42no—
59C22×#92418C2312yes—
60#14519775—28no—
61#14519775—22no—
62#14519775—22no—
63#14519756—28no—
64#14519756—22no—
65#14519756D414yes—
66#14519753—28no—
67#14519753—22no—
68#14519753—22no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71C4×C22⋊D61—28no—
72C4×C22⋊D61—22no—
73C4×C22⋊D61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61C2312yes—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196D411yescharacteristic
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61D411yescharacteristic
88C2×#11758198—21no—
89C2×#11758198—22no—
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519711—24no—
94#14519702—24no—
95#14519701—24no—
96#14519701—24no—
97#14519700—42no—
98#14519689—42no—
99#14519684—22no—
100#14519684—22no—
101C22×Q8×C61—21no—
102C22×Q8×C61—22no—
103C22×Q8×C61—24no—
104C22×Q8×C61—42no—
105C22×Q8×C61D414yes—
106C22×Q8×C61C2311yescharacteristic
107C22×Q8×C61C2311yescharacteristic
108C23×C4×C61—21no—
109C23×C4×C61C2311yescharacteristic
110C22×C8×C61C2312yes—
111C2×C4⋊C4×C61C2312yes—
112C2×C22⋊C4×C61—21no—
113#103×C61—24no—
114#103×C61—42no—
115C22⋊C8×C61—22no—
116#95×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61C2×D411yescharacteristic
119C2×Q8×C61—24no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—22no—
123C2×Q8×C61—28no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—44no—
127C2×Q8×C61—44no—
128C2×Q8×C61—48no—
129C2×Q8×C61C2×D412yes—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61#4114yes—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61—44no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—22no—
144C2×C22⋊D61—22no—
145C2×C22⋊D61Q1614yes—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—42no—
150C22×Dic61—82no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154Q8×D61—48no—
155Q8×D61—44no—
156Q8×D61—44no—
157C2×C4×D61—24no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—44no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—24no—
166C2×Dic122—21no—
167C2×Dic122—44no—
168C2×Dic122C2×D412yes—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C22⋊C4×C61—42no—
175C22⋊Dic61—24no—
176C22⋊Dic61—22no—
177#11758198—24no—
178#11758198—22no—
179#11758198—82no—
180C4⋊Dic61—24no—
181C4⋊Dic61—21no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—42no—
185C4⋊Dic61—82no—
186C4⋊Dic61C2×D412yes—
187#11758196—24no—
188#11758196—24no—
189#11758196—24no—
190#11758196—24no—
191#11758196—22no—
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#12012yes—
202C23×C61C2×Q1612yes—
203C23×C61#13712yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—21no—
206C22×D61—22no—
207C22×D61—21no—
208C22×D61—22no—
209C22×D61C2×Q1612yes—
210Q8×C61—22no—
211Q8×C61—24no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—48no—
215Q8×C61—42no—
216Q8×C61#12314yes—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—48no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61#12012yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4112yes—
233C2×C4×C61H(5,𝔽2)12yes—
234C22⋊D61—28no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—84no—
249C2×Dic61C2×Q1612yes—
250C4×D61—48no—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—42no—
254Dic122—24no—
255Dic122—48no—
256Dic122—41no—
257Dic122—41no—
258C8×C61—42no—
259#92418—42no—
260#24042—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—24no—
265C22×C61#45114yes—
266C22×C61C2×#12011yescharacteristic
267C22×C61#54612yes—
268C22×C61C22×Q1611yescharacteristic
269C22×C61C2×#13711yescharacteristic
270C22×C61#57712yes—
271C2×D61—22no—
272C2×D61—24no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—42no—
276C4×C61—24no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—82no—
280C4×C61C2×#12311yescharacteristic
281C4×C61#53411yescharacteristic
282C4×C61#54612yes—
283Dic61—22no—
284Dic61—42no—
285Dic61—41no—
286Dic61—41no—
287Dic61—42no—
288Dic61—82no—
289C2×D4×Q8—611no—
290C2×#484—611no—
291C22×#103—611no—
292#2487—611no—
293#2043—612no—
294#2036—612no—
295#1892—611no—
296#1869—611no—
297#1863—612no—
298#1787—612no—
299#1787D6111yescharacteristic, complemented
300C2×C61—24no—
301C2×C61#240511yescharacteristic
302C2×C61C2×#45111yescharacteristic
303C2×C61#301011yescharacteristic
304C2×C61#329912yes—
305C2×C61#332012yes—
306D61—42no—
307C23×Q8—611no—
308C23×Q8C2×D6111yescharacteristic
309C23×C8—611no—
310D4×Q8—1224no—
311D4×Q8—1224no—
312C2×C4⋊Q8—611no—
313C2×C4⋊Q8—1221no—
314C2×C22⋊Q8—611no—
315C2×C22⋊Q8—611no—
316C2×C22⋊Q8—1221no—
317C2×C22⋊Q8—1221no—
318C2×C4×Q8—611no—
319C2×C4×D4—611no—
320C2×C4×D4—1221no—
321C22×C4⋊C4—611no—
322#484—1224no—
323C2×#107—612no—
324C2×#103—612no—
325C2×#103—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103—1224no—
329C2×#103C2×D6112yes—
330C2×#103C2×D6112yes—
331C2×C22⋊C8—612no—
332C2×C22⋊C8C2×D6111yescharacteristic
333#400—1221no—
334#399—1221no—
335#389—612no—
336#386—1221no—
337#385—1221no—
338#380—612no—
339#380C2×D6111yescharacteristic
340#340—612no—
341#336—612no—
342C61#2404211yescharacteristic, complemented, Sylow
343C22×Q8—21no—
344C22×Q8—22no—
345C22×Q8—24no—
346C22×Q8—42no—
347C22×Q8—611no—
348C22×Q8—1221no—
349C22×Q8—1222no—
350C22×Q8—1222no—
351C22×Q8—1224no—
352C22×Q8—2442no—
353C22×Q8D24414yes—
354C22×Q8C22×D6111yescharacteristic
355C22×Q8C22×D6111yescharacteristic
356C22×D4—611no—
357C23×C4—21no—
358C23×C4—611no—
359C23×C4—1221no—
360C23×C4—1221no—
361C23×C4C22×D6111yescharacteristic
362C22×C8—612no—
363C22×C8—1222no—
364C22×C8—2442no—
365C22×C8C22×D6112yes—
366C4⋊Q8—1222no—
367C4⋊Q8—1228no—
368C4⋊Q8—1222no—
369C22⋊Q8—614no—
370C22⋊Q8—1222no—
371C22⋊Q8—1228no—
372C22⋊Q8—1228no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375C4×Q8—1222no—
376C4×Q8—1222no—
377C4×D4—1228no—
378C4×D4—1222no—
379C4×D4—1222no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6112yes—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—611no—
395C2×C22⋊C4—611no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1222no—
399C2×C42—611no—
400C2×C42—1221no—
401#107—1224no—
402#103—24no—
403#103—42no—
404#103—1224no—
405#103—1224no—
406#103—1224no—
407#103—2442no—
408C22⋊C8—22no—
409C22⋊C8—2442no—
410#95—21no—
411#95—1222no—
412#95—1222no—
413C24—611no—
414C24D4×D6111yescharacteristic
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—28no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—48no—
425C2×Q8—612no—
426C2×Q8—1224no—
427C2×Q8—1221no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—2444no—
431C2×Q8—2448no—
432C2×Q8—2444no—
433C2×Q8—2444no—
434C2×Q8C2×D24412yes—
435C2×Q8C2×D24412yes—
436C2×Q8#1175822414yes—
437C2×D4—614no—
438C2×D4—1222no—
439C2×D4—1222no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—44no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—4882no—
459C22×C4—2444no—
460C22×C4—2442no—
461C22×C4C2×D24412yes—
462C22×C4D4×D6112yes—
463C22×C4D4×D6111yescharacteristic
464C22×C4C23×D6111yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—2444no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—612no—
475C4⋊C4—1224no—
476C4⋊C4—1224no—
477C4⋊C4—1224no—
478C4⋊C4—1224no—
479C4⋊C4—1221no—
480C4⋊C4—1222no—
481C4⋊C4—1224no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—4882no—
485C4⋊C4—2442no—
486C22⋊C4—42no—
487C22⋊C4—1224no—
488C22⋊C4—1222no—
489C22⋊C4—1224no—
490C22⋊C4—1222no—
491C22⋊C4—4882no—
492C42—1224no—
493C42—1221no—
494C42—1221no—
495C23—22no—
496C23—22no—
497C23—612no—
498C23—1221no—
499C23—1221no—
500C23—1222no—
501C23—1222no—
502C23#1451974712yes—
503C23Q16×D6112yes—
504C23#1451980612yes—
505C23C2×D4×D6111yescharacteristic
506Q8—22no—
507Q8—24no—
508Q8—22no—
509Q8—22no—
510Q8—48no—
511Q8—42no—
512Q8—1224no—
513Q8—2448no—
514Q8—2441no—
515Q8—2441no—
516Q8#1451976114yes—
517D4—1228no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—612no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1222no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—4884no—
540C2×C4—2448no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4#1451974712yes—
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C22×D24411yescharacteristic
552C2×C4C2×#1175822412yes—
553C2×C4#1451982212yes—
554C8—42no—
555C8—2442no—
556C22—22no—
557C22—24no—
558C22—24no—
559C22—22no—
560C22—1224no—
561C22—1222no—
562C22—2441no—
563C22—2442no—
564C22—2441no—
565C22#1683991914yes—
566C22C2×#1451974711yescharacteristic
567C22#1684039912yes—
568C22C2×Q16×D6111yescharacteristic
569C22C2×#1451980611yescharacteristic
570C22#1684057112yes—
571C4—22no—
572C4—24no—
573C4—42no—
574C4—82no—
575C4—1222no—
576C4—4882no—
577C4—2441no—
578C4—2442no—
579C4—2441no—
580C4—2442no—
581C4C2×#1451976111yescharacteristic
582C4#1684032411yescharacteristic
583C4#1684039912yes—
584C2—24no—
585C2—2442no—
586C2#2144914711yescharacteristic
587C2C2×#1683991911yescharacteristic
588C2#2145510611yescharacteristic
589C2#2145768812yes—
590C2#2145799412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenter#1787D61
hypocenterC61#24042
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
2. C2×#16840098C2semidirect
3. C22×#14519701C2semidirect
5. #21447064C2semidirect
6. #21447021C2semidirect
7. #21446240C2semidirect
9. #21446075C2semidirect
10. #21445752C2semidirect
11. C61×#1787C2semidirect
299. #1787D61semidirect
342. C61#24042semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊D612
C2×#168400982
C22×#145197012
#214501372
#214470642
#214470212
#214462402
#214461092
#214460752
#214457522
C61×#17872
#2404261