Finite Group Databasebeta

← #29646194

Subgroups — #29646194

Order 15616.

1424 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840408C211yesmaximal, characteristic
2#21454265C212yesmaximal, complemented
3#21444178C212yesmaximal, complemented
4#21444166C214yesmaximal, complemented
5#21444157C211yesmaximal, characteristic, complemented
6#21444144C212yesmaximal, complemented
7#323×D61C212yesmaximal, complemented
8C61×#1459C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×D244C411yescharacteristic
11C2×#14519783—22no—
12#16840408—22no—
13#16840408—24no—
14#16840408—22no—
15C2×#14519765C411yescharacteristic
16C2×#14519762C2212yes—
17C2×#14519756C411yescharacteristic
18C2×#14519751C2211yescharacteristic
19C2×#14519750C412yes—
20C2×#14519745C412yes—
21C2×C4×D244C411yescharacteristic
22C2×C22⋊C4×D61C2212yes—
23#16839728—22no—
24#16839727—24no—
25#16839726C2212yescomplemented
26#16839724C2214yescomplemented
27C22⋊C8×D61C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yes—
30#16839485C2214yescomplemented
31#16839485C2214yescomplemented
32C2×C4×D4×C61C2211yescharacteristic
33C2×C22⋊C8×C61C2212yescomplemented
34C61×#324C2212yescomplemented
35C61×#323C2211yescharacteristic
36C61×#323C2211yescharacteristic
37C24×D61C2×C412yes—
38C2×#11758222—44no—
39C2×#11758222D412yes—
40C2×#11758222D414yes—
41C2×#11758222D412yes—
42C22×Dic122C2×C411yescharacteristic
43C22×D244—21no—
44C22×D244—22no—
45C22×D244—21no—
46C22×D244—44no—
47C22×D244C2×C411yescharacteristic
48C22×D244C2×C412yes—
49C22×D244C2×C411yescharacteristic
50C22×C22⋊D61C2×C412yes—
51C22×C22⋊D61C2×C412yes—
52C22×C4×D61C2×C412yes—
53C22×C4×D61C2312yes—
54#14519783—48no—
55C2×#11758194—22no—
56C2×#11758189—24no—
57C2×C8×D61—24no—
58#14519765—22no—
59#14519765—22no—
60#14519762—24no—
61#14519762—24no—
62#14519756—22no—
63#14519756—22no—
64#14519751—22no—
65#14519751—22no—
66#14519750—24no—
67#14519750—24no—
68#14519745—24no—
69#14519745—24no—
70C4×D244—22no—
71C4×D244—21no—
72C4×D244—21no—
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×#11758196C2×C412yes—
75C2×C22⋊Dic61C2×C412yes—
76C2×#11758198C2×C412yes—
77C2×#11758198C2×C412yes—
78C2×#11758198C2×C412yes—
79C2×#11758198C2312yes—
80C22⋊C4×D61—24no—
81C22⋊C4×D61—24no—
82C22⋊#92418—22no—
83C22⋊#92418C2314yes—
84#14519689—24no—
85#14519689—24no—
86C22×D4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C411yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C22×C8×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61C2×C411yescharacteristic
94C2×C22⋊C4×C61C2311yescharacteristic
95C2×C42×C61C2×C411yescharacteristic
96C22⋊C8×C61—22no—
97C22⋊C8×C61C2312yes—
98C22⋊C8×C61C2312yes—
99C24×C61C22×C411yescharacteristic
100C24×C61C22×C411yescharacteristic
101C23×D61—24no—
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—42no—
105C23×D61—44no—
106C23×D61—82no—
107C23×D61C22×C412yes—
108C23×D61C22×C412yes—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—41no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—24no—
124C2×C22⋊D61—44no—
125C2×C22⋊D61—82no—
126C2×C22⋊D61C22⋊C414yes—
127C2×C22⋊D61C22⋊C414yes—
128C22×Dic61C22×C412yes—
129C22×Dic61C22×C412yes—
130#11758222—28no—
131#11758222—44no—
132#11758222—48no—
133C2×D244—24no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—48no—
139C2×D244—44no—
140C2×D244—44no—
141C2×D244—42no—
142C2×D244—44no—
143C2×D244C22⋊C412yes—
144C2×D244C22⋊C411yescharacteristic
145C2×D244C22⋊C411yescharacteristic
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—44no—
149C2×C4×D61—82no—
150C2×C4×D61C2×D414yes—
151C2×C4×D61C2×D414yes—
152C2×Dic122—42no—
153C2×Dic122C22⋊C412yes—
154C2×Dic122C22⋊C412yes—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—44no—
158C4⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C42×C61—21no—
162C42×C61—21no—
163C22⋊Dic61—24no—
164#11758198—22no—
165#11758198—22no—
166#11758198—24no—
167#11758198—24no—
168#11758198—24no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171#11758196—24no—
172#11758194—44no—
173C2×#92418—24no—
174#11758189—48no—
175C8×D61—48no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61C2×C22⋊C412yes—
185C23×C61#13112yescomplemented
186C23×C61C23×C411yescharacteristic
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—24no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—42no—
194C22×D61—44no—
195C22×D61—44no—
196C22×D61—44no—
197C22×D61—44no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—82no—
201C22×D61—82no—
202C22×D61—88no—
203C22×D61—88no—
204C22×D61C2×C22⋊C412yes—
205C22×D61C2×C22⋊C412yes—
206D4×C61—44no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C22×D412yes—
223C22⋊D61—28no—
224C22⋊D61—44no—
225C22⋊D61—84no—
226C2×Dic61—22no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61C2×C22⋊C414yes—
230D244—24no—
231D244—22no—
232D244—42no—
233D244—44no—
234D244—44no—
235D244—42no—
236D244#9914yes—
237C4×D61—28no—
238C4×D61—44no—
239C4×D61—82no—
240C4×D61—82no—
241Dic122—24no—
242Dic122—42no—
243C8×C61—42no—
244C8×C61—42no—
245#92418—44no—
246#17411—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61C22×C22⋊C411yescharacteristic
260C22×C61C2×#13111yescharacteristic
261C22×C61#56811yescharacteristic
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—44no—
268C2×D61—44no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—88no—
273C2×D61—88no—
274C2×D61—82no—
275C2×D61—82no—
276C2×D61C2×#9914yes—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61C2×#9912yes—
283Dic61—24no—
284Dic61—42no—
285C2×#546—611no—
286#2880—612no—
287#1471—612no—
288#1468—614no—
289#1465—611no—
290#1462—612no—
291#1459D6111yescharacteristic, complemented
292C2×#323—612no—
293C2×C61—21no—
294C2×C61—21no—
295C2×C61—42no—
296C2×C61—42no—
297C2×C61#288011yescharacteristic
298C2×C61C22×#9911yescharacteristic
299C2×C61#288811yescharacteristic
300D61—24no—
301D61—42no—
302D61—82no—
303C22×#41—611no—
304C23×D4—611no—
305C2×#131—1222no—
306#546—1222no—
307#546—1224no—
308#546—1222no—
309C2×#124—611no—
310C2×#123—612no—
311C2×C22⋊Q8—611no—
312C2×#121—611no—
313C2×#121—612no—
314C2×#120—612no—
315C2×C4×D4—611no—
316C2×C4×D4C2×D6111yescharacteristic
317C22×C22⋊C4—612no—
318#408—1224no—
319#408—1222no—
320#407—612no—
321#407—614no—
322C2×C22⋊C8—614no—
323C2×C22⋊C8C2×D6112yescomplemented
324#324—612no—
325#324C2×D6112yescomplemented
326#323—612no—
327#323—614no—
328#323—614no—
329#323C2×D6111yescharacteristic
330#323C2×D6111yescharacteristic
331C61#1741111yescharacteristic, complemented, Sylow
332C25—612no—
333C2×#41—612no—
334C2×#41—614no—
335C2×#41—612no—
336C2×#41—2444no—
337C22×Q8—611no—
338C22×D4—612no—
339C22×D4—612no—
340C22×D4—611no—
341C22×D4—612no—
342C22×D4—611no—
343C22×D4—1221no—
344C22×D4—1222no—
345C22×D4—1221no—
346C22×D4—2444no—
347C22×D4C4×D6111yescharacteristic
348C23×C4—612no—
349C23×C4—612no—
350C23×C4C4×D6111yescharacteristic
351C23×C4C22×D6111yescharacteristic
352#131—2448no—
353C2×M16—1224no—
354C2×M16—1222no—
355C22×C8—22no—
356C22×C8—1224no—
357#124—1222no—
358#124—1222no—
359#123—1224no—
360#123—1224no—
361C22⋊Q8—1222no—
362C22⋊Q8—1222no—
363#121—1224no—
364#121—1224no—
365#121—1222no—
366#121—1222no—
367#120—1224no—
368#120—1224no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—1221no—
372C4×D4—1222no—
373C4×D4—1221no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—612no—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—1224no—
383C2×C22⋊C4—1224no—
384C2×C22⋊C4C4×D6111yescharacteristic
385C2×C22⋊C4C22×D6111yescharacteristic
386C2×C42C4×D6111yescharacteristic
387C22⋊C8—22no—
388C22⋊C8—614no—
389C22⋊C8—1224no—
390C22⋊C8—1224no—
391C22⋊C8—1222no—
392C22⋊C8C22×D6112yes—
393C22⋊C8C22×D6112yes—
394C24—612no—
395C24—612no—
396C24—1222no—
397C24—1222no—
398C24—1224no—
399C24—4882no—
400C24—2442no—
401C24—2444no—
402C24C2×C4×D6111yescharacteristic
403C24C2×C4×D6111yescharacteristic
404#41—1228no—
405#41—2444no—
406#41—2448no—
407C2×Q8—612no—
408C2×Q8—612no—
409C2×Q8—2442no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—41no—
413C2×D4—614no—
414C2×D4—614no—
415C2×D4—611no—
416C2×D4—612no—
417C2×D4—611no—
418C2×D4—1224no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1224no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—4882no—
425C2×D4—2442no—
426C2×D4—2444no—
427C2×D4—2448no—
428C2×D4—2444no—
429C2×D4—2444no—
430C2×D4—2444no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—612no—
436C22×C4—614no—
437C22×C4—614no—
438C22×C4—612no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—4882no—
442C22×C4—2444no—
443C22×C4C2×C4×D6111yescharacteristic
444C22×C4C2×C4×D6111yescharacteristic
445C22×C4C2×C4×D6111yescharacteristic
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4D4×D6112yes—
448C22×C4D4×D6112yes—
449C22×C4C23×D6111yescharacteristic
450M16—2448no—
451M16—2444no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—44no—
455C2×C8—1224no—
456C2×C8—2448no—
457C4⋊C4—22no—
458C4⋊C4—1224no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—1224no—
464C22⋊C4—1224no—
465C22⋊C4—1222no—
466C22⋊C4—1222no—
467C22⋊C4—1224no—
468C22⋊C4—1224no—
469C42—21no—
470C42—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—612no—
480C23—612no—
481C23—1224no—
482C23—1222no—
483C23—1222no—
484C23—1222no—
485C23—1222no—
486C23—1222no—
487C23—4882no—
488C23—4882no—
489C23—4888no—
490C23—4888no—
491C23—2442no—
492C23—2444no—
493C23—2444no—
494C23—2442no—
495C23—2442no—
496C23—2444no—
497C23—2444no—
498C23C22⋊C4×D6112yes—
499C23#1451978312yescomplemented
500C23C22×C4×D6111yescharacteristic
501Q8—1224no—
502Q8—2442no—
503D4—44no—
504D4—614no—
505D4—1224no—
506D4—1228no—
507D4—1222no—
508D4—4884no—
509D4—2442no—
510D4—2444no—
511D4—2444no—
512D4—2444no—
513D4—2442no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—614no—
528C2×C4—1228no—
529C2×C4—1222no—
530C2×C4—4882no—
531C2×C4—4882no—
532C2×C4—2444no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4C22⋊C4×D6111yescharacteristic
536C2×C4C22⋊C4×D6111yescharacteristic
537C2×C4C2×D4×D6112yes—
538C8—42no—
539C8—42no—
540C8—2444no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—614no—
554C22—1222no—
555C22—1222no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22—4882no—
560C22—4882no—
561C22—4888no—
562C22—4888no—
563C22—2444no—
564C22—2444no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22C2×C22⋊C4×D6111yescharacteristic
569C22C2×#1451978311yescharacteristic
570C22#1684052811yescharacteristic
571C4—22no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—1224no—
577C4—2442no—
578C4#99×D6112yes—
579C2—21no—
580C2—21no—
581C2—42no—
582C2—42no—
583C2—1224no—
584C2—4882no—
585C2—2442no—
586C2#2145426511yescharacteristic
587C2C2×#99×D6111yescharacteristic
588C2#2145429611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1459C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840408C2
cosocleC22×C4×C61C24
hypercenter#1459D61
hypocenterC61#17411
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #21454265C2semidirect
3. #21444178C2semidirect
4. #21444166C2semidirect
5. #21444157C2semidirect
6. #21444144C2semidirect
7. #323×D61C2semidirect
8. C61×#1459C2semidirect
25. #16839726C22semidirect
26. #16839724C22semidirect
27. C22⋊C8×D61C22semidirect
28. #16839488C22semidirect
30. #16839485C22semidirect
31. #16839485C22semidirect
33. C2×C22⋊C8×C61C22semidirect
34. C61×#324C22semidirect
185. C23×C61#131semidirect
291. #1459D61semidirect
323. C2×C22⋊C8C2×D61semidirect
325. #324C2×D61semidirect
331. C61#17411semidirect
499. C23#14519783semidirect

Maximal subgroups

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

subgroupindex
C2×#168404082
#214542652
#214441782
#214441662
#214441572
#214441442
#323×D612
C61×#14592
#1741161