Finite Group Databasebeta

← #29713065

Subgroups — #29713065

Order 15616.

1338 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊D61C211yesmaximal, characteristic
2C2×#16840093C211yesmaximal, characteristic, complemented
3C22×#14519701C211yesmaximal, characteristic, complemented
4#21450128C211yesmaximal, characteristic
5#21447064C212yesmaximal, complemented
6#21447014C212yesmaximal, complemented
7#21446240C211yesmaximal, characteristic, complemented
8#21446109C211yesmaximal, characteristic
9#21446089C212yesmaximal, 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×#14519761—21no—
19C2×#14519756—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519753—21no—
22C2×#14519753C2211yescharacteristic
23C2×Q8×Dic61C2211yescharacteristic
24C2×C4×C22⋊D61—21no—
25C2×C4×C22⋊D61C2211yescharacteristic
26C22×C4⋊Dic61C2211yescharacteristic
27#16840093—24no—
28C2×#14519709C2212yes—
29C2×#14519702C2212yes—
30C2×#14519701—24no—
31C2×#14519701C2212yes—
32C2×#14519701C2212yes—
33C2×#14519700C2212yes—
34C2×#14519689C2212yes—
35#16839694—21no—
36#16839689—21no—
37#16839661C2212yes—
38#16839623—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#14519761—44no—
64#14519756—28no—
65#14519756—22no—
66#14519756D414yes—
67#14519753—28no—
68#14519753—22no—
69#14519753—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72C4×C22⋊D61—28no—
73C4×C22⋊D61—22no—
74C4×C22⋊D61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—22no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—22no—
80C2×C4⋊Dic61C2312yes—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196D411yescharacteristic
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61D411yescharacteristic
89C2×#11758198—21no—
90C2×#11758198—22no—
91C2×#11758198D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519709—24no—
95#14519702—24no—
96#14519701—24no—
97#14519701—24no—
98#14519700—42no—
99#14519689—42no—
100#14519684—22no—
101#14519684—22no—
102C22×Q8×C61—21no—
103C22×Q8×C61—22no—
104C22×Q8×C61—24no—
105C22×Q8×C61—42no—
106C22×Q8×C61D414yes—
107C22×Q8×C61C2311yescharacteristic
108C22×Q8×C61C2311yescharacteristic
109C23×C4×C61—21no—
110C23×C4×C61C2311yescharacteristic
111C22×C8×C61C2312yes—
112C2×C4⋊C4×C61C2312yes—
113C2×C22⋊C4×C61—21no—
114#103×C61—24no—
115#103×C61—42no—
116C22⋊C8×C61—22no—
117#95×C61—21no—
118C24×C61C2×D411yescharacteristic
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—24no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—22no—
124C2×Q8×C61—28no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C2×Q8×C61—44no—
129C2×Q8×C61—48no—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61#4114yes—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—24no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—44no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—22no—
145C2×C22⋊D61—22no—
146C2×C22⋊D61SD1614yes—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—22no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156Q8×D61—48no—
157Q8×D61—44no—
158Q8×D61—44no—
159C2×C4×D61—24no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—44no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—24no—
168C2×Dic122—21no—
169C2×Dic122—44no—
170C2×Dic122C2×D412yes—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C22⋊C4×C61—42no—
177C22⋊Dic61—24no—
178C22⋊Dic61—22no—
179#11758198—24no—
180#11758198—22no—
181#11758198—44no—
182C4⋊Dic61—24no—
183C4⋊Dic61—21no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61—42no—
187C4⋊Dic61—44no—
188C4⋊Dic61C2×D412yes—
189#11758196—24no—
190#11758196—24no—
191#11758196—24no—
192#11758196—24no—
193#11758196—22no—
194C4×Dic61—24no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—44no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61#12012yes—
204C23×C61C2×SD1612yes—
205C23×C61#13712yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—21no—
208C22×D61—22no—
209C22×D61—21no—
210C22×D61—22no—
211C22×D61C2×SD1612yes—
212Q8×C61—22no—
213Q8×C61—24no—
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61—48no—
217Q8×C61—42no—
218Q8×C61#12314yescomplemented
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—48no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61#12012yes—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4112yes—
235C2×C4×C61H(5,𝔽2)12yes—
236C22⋊D61—28no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—84no—
251C2×Dic61C2×SD1612yes—
252C4×D61—48no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—42no—
256Dic122—24no—
257Dic122—48no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—42no—
261#92418—42no—
262#24041—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61#44814yes—
268C22×C61C2×#12011yescharacteristic
269C22×C61#54612yes—
270C22×C61C22×SD1611yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C22×C61#57812yes—
273C2×D61—22no—
274C2×D61—24no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—42no—
278C4×C61—24no—
279C4×C61—22no—
280C4×C61—42no—
281C4×C61—82no—
282C4×C61C2×#12311yescharacteristic
283C4×C61#53411yescharacteristic
284C4×C61#54612yes—
285Dic61—22no—
286Dic61—42no—
287Dic61—41no—
288Dic61—41no—
289Dic61—42no—
290Dic61—82no—
291C2×D4×Q8—611no—
292C2×#483—611no—
293C22×#103—611no—
294#2486—611no—
295#2043—612no—
296#2035—612no—
297#1892—611no—
298#1869—611no—
299#1865—612no—
300#1787—612no—
301#1787D6111yescharacteristic, complemented
302C2×C61—24no—
303C2×C61#240511yescharacteristic
304C2×C61C2×#44811yescharacteristic
305C2×C61#301111yescharacteristic
306C2×C61#329812yes—
307C2×C61#331212yes—
308D61—42no—
309C23×Q8—611no—
310C23×Q8C2×D6111yescharacteristic
311C23×C8—611no—
312D4×Q8—1224no—
313D4×Q8—1224no—
314C2×C4⋊Q8—611no—
315C2×C4⋊Q8—1221no—
316C2×#123—1221no—
317C2×C22⋊Q8—611no—
318C2×C22⋊Q8—611no—
319C2×C22⋊Q8—1221no—
320C2×C22⋊Q8—1221no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C2×C4×D4—1221no—
324C22×C4⋊C4—611no—
325#483—1224no—
326C2×#106—612no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#103—612no—
331C2×#103—1224no—
332C2×#103C2×D6112yes—
333C2×#103C2×D6112yes—
334C2×C22⋊C8—612no—
335C2×C22⋊C8C2×D6111yescharacteristic
336#398—1221no—
337#397—1221no—
338#389—612no—
339#381—1221no—
340#380—612no—
341#380C2×D6111yescharacteristic
342#340—612no—
343#336—612no—
344C61#2404111yescharacteristic, complemented, Sylow
345C22×Q8—21no—
346C22×Q8—22no—
347C22×Q8—24no—
348C22×Q8—42no—
349C22×Q8—611no—
350C22×Q8—1221no—
351C22×Q8—1222no—
352C22×Q8—1222no—
353C22×Q8—1224no—
354C22×Q8—2442no—
355C22×Q8D24414yes—
356C22×Q8C22×D6111yescharacteristic
357C22×Q8C22×D6111yescharacteristic
358C22×D4—611no—
359C23×C4—21no—
360C23×C4—611no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4C22×D6111yescharacteristic
364C22×C8—612no—
365C22×C8—1222no—
366C22×C8—2442no—
367C22×C8C22×D6112yes—
368C4⋊Q8—1222no—
369C4⋊Q8—1228no—
370C4⋊Q8—1222no—
371#123—2444no—
372C22⋊Q8—614no—
373C22⋊Q8—1222no—
374C22⋊Q8—1228no—
375C22⋊Q8—1228no—
376C22⋊Q8—1222no—
377C22⋊Q8—1222no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×D4—1228no—
381C4×D4—1222no—
382C4×D4—1222no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4C22×D6112yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—611no—
398C2×C22⋊C4—611no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1222no—
402C2×C42—611no—
403C2×C42—1221no—
404#106—1224no—
405#103—24no—
406#103—42no—
407#103—1224no—
408#103—1224no—
409#103—1224no—
410#103—2442no—
411C22⋊C8—22no—
412C22⋊C8—2442no—
413#95—21no—
414#95—1222no—
415#95—1222no—
416C24—611no—
417C24D4×D6111yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—28no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—48no—
428C2×Q8—612no—
429C2×Q8—1224no—
430C2×Q8—1221no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—2444no—
434C2×Q8—2448no—
435C2×Q8—2444no—
436C2×Q8—2444no—
437C2×Q8C2×D24412yes—
438C2×Q8C2×D24412yes—
439C2×Q8#1175822414yes—
440C2×D4—614no—
441C2×D4—1222no—
442C2×D4—1222no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1224no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—2444no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4C2×D24412yes—
466C22×C4D4×D6112yes—
467C22×C4D4×D6111yescharacteristic
468C22×C4C23×D6111yescharacteristic
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—2444no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—612no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C4⋊C4—1224no—
483C4⋊C4—1221no—
484C4⋊C4—1222no—
485C4⋊C4—1224no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—2444no—
489C4⋊C4—2442no—
490C22⋊C4—42no—
491C22⋊C4—1224no—
492C22⋊C4—1222no—
493C22⋊C4—1224no—
494C22⋊C4—1222no—
495C22⋊C4—2444no—
496C42—1224no—
497C42—1221no—
498C42—1221no—
499C23—22no—
500C23—22no—
501C23—612no—
502C23—1221no—
503C23—1221no—
504C23—1222no—
505C23—1222no—
506C23#1451974712yes—
507C23SD16×D6112yes—
508C23#1451980912yes—
509C23C2×D4×D6111yescharacteristic
510Q8—22no—
511Q8—24no—
512Q8—22no—
513Q8—22no—
514Q8—48no—
515Q8—42no—
516Q8—1224no—
517Q8—2448no—
518Q8—2441no—
519Q8—2441no—
520Q8#1451976114yescomplemented
521D4—1228no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—612no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—4884no—
544C2×C4—2448no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4#1451974712yes—
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4C22×D24411yescharacteristic
556C2×C4C2×#1175822412yes—
557C2×C4#1451982212yes—
558C8—42no—
559C8—2442no—
560C22—22no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—1224no—
565C22—1222no—
566C22—2441no—
567C22—2441no—
568C22—2442no—
569C22#1683990214yes—
570C22C2×#1451974711yescharacteristic
571C22#1684039912yes—
572C22C2×SD16×D6111yescharacteristic
573C22C2×#1451980911yescharacteristic
574C22#1684057812yes—
575C4—22no—
576C4—24no—
577C4—42no—
578C4—82no—
579C4—1222no—
580C4—4882no—
581C4—2441no—
582C4—2442no—
583C4—2441no—
584C4—2442no—
585C4C2×#1451976111yescharacteristic
586C4#1684032411yescharacteristic
587C4#1684039912yes—
588C2—24no—
589C2—2442no—
590C2#2144914711yescharacteristic
591C2C2×#1683990211yescharacteristic
592C2#2145511511yescharacteristic
593C2#2145767712yes—
594C2#2145787412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenter#1787D61
hypocenterC61#24041
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×#16840093C2semidirect
3. C22×#14519701C2semidirect
5. #21447064C2semidirect
6. #21447014C2semidirect
7. #21446240C2semidirect
9. #21446089C2semidirect
10. #21445752C2semidirect
11. C61×#1787C2semidirect
218. Q8×C61#123semidirect
301. #1787D61semidirect
344. C61#24041semidirect
520. Q8#14519761semidirect

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×#168400932
C22×#145197012
#214501282
#214470642
#214470142
#214462402
#214461092
#214460892
#214457522
C61×#17872
#2404161