Finite Group Databasebeta

← #29689751

Subgroups — #29689751

Order 15616.

1234 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840297C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic, complemented
3#384×D61C211yesmaximal, characteristic, complemented
4#21446789C211yesmaximal, characteristic, complemented
5#21446635C211yesmaximal, characteristic, complemented
6#21446633C211yesmaximal, characteristic, complemented
7#21446135C212yesmaximal
8#21446125C212yesmaximal, complemented
9#21445823C212yesmaximal, complemented
10#21445818C212yesmaximal, complemented
11C61×#1992C211yesmaximal, characteristic, complemented
12#16840297—24no—
13#16840297C418yescomplemented
14C2×#14519772—21no—
15C2×#14519765C2211yescharacteristic
16C2×#14519762—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519752C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519739—21no—
21C2×#14519739C2211yescharacteristic
22C2×#14519735—21no—
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C2212yes—
26C2×C42×D61C2211yescharacteristic
27#16839817—24no—
28C2×#14519706C2212yes—
29C2×#14519702C2212yes—
30C2×#14519700C2212yes—
31C2×#14519699C2212yes—
32C2×#14519697C2212yes—
33C2×#14519689C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839632C2211yescharacteristic
36#16839631—21no—
37#16839631—21no—
38#16839631C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×#97×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×D244—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61—22no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×#92418C2312yes—
55#14519772—28no—
56#14519765—22no—
57#14519765C2×C414yescomplemented
58#14519762—28no—
59#14519757—22no—
60#14519757C2×C414yes—
61#14519752—22no—
62#14519752C2×C414yes—
63#14519742—22no—
64#14519742C2×C414yescomplemented
65#14519739—28no—
66#14519739—22no—
67#14519739C2×C414yescomplemented
68#14519735—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C4⋊C4×D61—42no—
78C4⋊C4×D61—42no—
79C4⋊C4×D61—44no—
80C4⋊C4×D61—44no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C42×D61C2×C414yes—
91C42×D61D414yes—
92#14519706—24no—
93#14519702—42no—
94#14519700—42no—
95#14519699—42no—
96#14519697—42no—
97#14519689—24no—
98#14519688—22no—
99C22×C8×C61C2312yes—
100#125×C61—22no—
101#125×C61C2×C414yescomplemented
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110#97×C61—22no—
111C23×D61—21no—
112C23×D61C2×D411yescharacteristic
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—22no—
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—24no—
127C2×D244—21no—
128C2×D244—42no—
129C2×D244—42no—
130C2×D244C22×C412yes—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—24no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61—44no—
141C2×C4×D61—44no—
142C2×C4×D61—44no—
143C2×C4×D61C22⋊C414yes—
144C2×C4×D61C22⋊C414yes—
145C2×C4×D61C22×C412yes—
146C2×C4×D61C2×D412yes—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—21no—
150C2×Dic122C22×C412yes—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61C22×C412yes—
163C4⋊C4×C61C22×C412yes—
164C42×C61C22×C412yes—
165C42×C61C2×D412yes—
166#11758198—24no—
167#11758198—24no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171C4⋊Dic61—24no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—42no—
175C4⋊Dic61—42no—
176#11758196—24no—
177#11758196—44no—
178#11758196—44no—
179#11758196—44no—
180C4×Dic61—24no—
181C4×Dic61C22⋊C414yes—
182C4×Dic61C22⋊C414yes—
183C4×Dic61C22×C412yes—
184C4×Dic61C2×D412yes—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61C22×D411yescharacteristic
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61C2×C22⋊C412yes—
201C22×D61AGL(1,ℤ/8)12yes—
202C22×D61#13712yes—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61H(5,𝔽2)12yes—
221C2×C4×C612−1+412yes—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61AGL(1,ℤ/8)12yes—
236C2×Dic61#13712yes—
237D244—24no—
238D244—48no—
239D244—42no—
240C4×D61—28no—
241C4×D61—24no—
242C4×D61—24no—
243C4×D61—28no—
244C4×D61—48no—
245C4×D61—88no—
246Dic122—24no—
247Dic122—42no—
248C8×C61—42no—
249#92418—42no—
250#21867—611nomaximal, Sylow
251C22×C61C22×C22⋊C411yescharacteristic
252C22×C61#53511yescharacteristic
253C22×C61#53611yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61C2×#13711yescharacteristic
257C22×C61C2×#13711yescharacteristic
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61#41714yes—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61—82no—
277C4×C61#52012yes—
278C4×C61#53712yes—
279Dic61—22no—
280Dic61—22no—
281Dic61—42no—
282Dic61—82no—
283Dic61#41714yes—
284C2×#529—611no—
285C2×#432—611no—
286C2×#384—611no—
287#1992—611no—
288#1992D6111yescharacteristic, complemented
289#1959—611no—
290#1959—611no—
291#1874—612no—
292#1872—612no—
293#1803—612no—
294#1802—612no—
295C2×C61#231111yescharacteristic
296C2×C61C2×#41711yescharacteristic
297C2×C61C2×#41711yescharacteristic
298C2×C61#321711yescharacteristic
299C2×C61#321811yescharacteristic
300C2×C61#323811yescharacteristic
301C2×C61#323911yescharacteristic
302D61—24no—
303D61—82no—
304#529—618no—
305#529—1224no—
306C2×#126—1221no—
307C2×#125C2×D6111yescharacteristic
308C2×#124—611no—
309C2×#123—1221no—
310C2×C22⋊Q8—611no—
311C2×#121—611no—
312C2×C4×Q8—611no—
313C2×C4×D4—611no—
314C2×C4×D4—1221no—
315C2×#117—1221no—
316C22×C4⋊C4—612no—
317C22×C4⋊C4—1221no—
318C22×C4⋊C4—1221no—
319C22×C42—611no—
320#432—1224no—
321C2×C4⋊C8—612no—
322C2×#103—612no—
323C2×#103—612no—
324C2×#102—612no—
325C2×#102—612no—
326C2×C22⋊C8—612no—
327C2×#97—611no—
328C2×#97C2×D6111yescharacteristic
329#384—612no—
330#384—611no—
331#384—1221no—
332#384—1221no—
333#384C2×D6111yescharacteristic
334#340—612no—
335#340—612no—
336#340C2×D6112yes—
337#340C2×D6112yes—
338C61#2186711yescharacteristic, complemented, Sylow
339C22×Q8—611no—
340C22×D4—611no—
341C22×D4—1221no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—1221no—
346C23×C4—1222no—
347C22×C8—612no—
348C22×C8C22×D6112yes—
349#126—1228no—
350#125—22no—
351#125C4×D6114yescomplemented
352#124—614no—
353#124—1222no—
354#123—1228no—
355C22⋊Q8—614no—
356C22⋊Q8—1222no—
357#121—614no—
358#121—1222no—
359C4×Q8—614no—
360C4×Q8—1222no—
361C4×D4—614no—
362C4×D4—1222no—
363C4×D4—1228no—
364#117—1228no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—2442no—
378C2×C4⋊C4—2442no—
379C2×C4⋊C4—2444no—
380C2×C4⋊C4—2444no—
381C2×C4⋊C4C22×D6112yes—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—614no—
393C2×C42—614no—
394C2×C42—1221no—
395C2×C42C22×D6111yescharacteristic
396C4⋊C8—1224no—
397#103—2442no—
398#103—2442no—
399#102—2442no—
400#102—2442no—
401C22⋊C8—1224no—
402#97—22no—
403#97—1222no—
404C24—611no—
405C24—1221no—
406C2×Q8—612no—
407C2×Q8—1221no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×D4—612no—
411C2×D4—1224no—
412C2×D4—1221no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—2442no—
416C2×D4—2442no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—614no—
426C22×C4—614no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1222no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2444no—
439C22×C4—2444no—
440C22×C4—2444no—
441C22×C4D4×D6111yescharacteristic
442C22×C4D4×D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—1222no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—2442no—
463C4⋊C4—2442no—
464C4⋊C4—2444no—
465C4⋊C4—2444no—
466C4⋊C4—2444no—
467C4⋊C4C2×C4×D6112yes—
468C4⋊C4C2×C4×D6112yes—
469C22⋊C4—1224no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C42—612no—
475C42—612no—
476C42—614no—
477C42—614no—
478C42—1224no—
479C42C2×C4×D6112yes—
480C42D4×D6112yes—
481C23—612no—
482C23—612no—
483C23—612no—
484C23—1221no—
485C23—1221no—
486C23—1221no—
487C23—1221no—
488C23—1222no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23C2×D4×D6111yescharacteristic
496Q8—1224no—
497Q8—2442no—
498D4—1224no—
499D4—2448no—
500D4—2442no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—1228no—
520C2×C4—1228no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1222no—
528C2×C4—4888no—
529C2×C4—2448no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4C22⋊C4×D6112yes—
536C2×C4C22⋊C4×D6112yes—
537C2×C4C22×C4×D6111yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4#1451982312yes—
540C2×C4#1451982512yes—
541C8—42no—
542C8—2442no—
543C22—614no—
544C22—1222no—
545C22—1222no—
546C22—1222no—
547C22—1222no—
548C22—1222no—
549C22—1222no—
550C22—4881no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22C2×C22⋊C4×D6111yescharacteristic
558C22#1684033011yescharacteristic
559C22#1684033711yescharacteristic
560C22AGL(1,ℤ/8)×D6111yescharacteristic
561C22AGL(1,ℤ/8)×D6111yescharacteristic
562C22#137×D6111yescharacteristic
563C22#137×D6111yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—82no—
569C4—614no—
570C4—1222no—
571C4—1222no—
572C4—4882no—
573C4—2442no—
574C4#1684024412yes—
575C4#1684034212yes—
576C2—1224no—
577C2—4882no—
578C2#2144841011yescharacteristic
579C2#417×D6111yescharacteristic
580C2#417×D6111yescharacteristic
581C2#2145683811yescharacteristic
582C2#2145684711yescharacteristic
583C2#2145704111yescharacteristic
584C2#2145705011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1992C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1992D61
hypocenterC61#21867
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
2. C2×#16839817C2semidirect
3. #384×D61C2semidirect
4. #21446789C2semidirect
5. #21446635C2semidirect
6. #21446633C2semidirect
8. #21446125C2semidirect
9. #21445823C2semidirect
10. #21445818C2semidirect
11. C61×#1992C2semidirect
13. #16840297C4semidirect
57. #14519765C2×C4semidirect
64. #14519742C2×C4semidirect
67. #14519739C2×C4semidirect
101. #125×C61C2×C4semidirect
288. #1992D61semidirect
338. C61#21867semidirect
351. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402972
C2×#168398172
#384×D612
#214467892
#214466352
#214466332
#214461352
#214461252
#214458232
#214458182
C61×#19922
#2186761