Finite Group Databasebeta

← #29647716

Subgroups — #29647716

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459581C211yesmaximal, characteristic
2#21454842C211yesmaximal, characteristic, complemented
3#21448730C211yesmaximal, characteristic, complemented
4#21444385C214yesmaximal, complemented
5#21444380C211yesmaximal, characteristic, complemented
6#21444379C211yesmaximal, characteristic, complemented
7#21444313C211yesmaximal, characteristic, complemented
8#327×D61C214yesmaximal, complemented
9C61×#1502C211yesmaximal, characteristic, complemented
10C2×#14519823—21no—
11#16840417C412yescomplemented
12#16840415C412yescomplemented
13#16840408—22no—
14#16840405—22no—
15D4×D244C412yescomplemented
16#16840397C412yescomplemented
17#16840340—21no—
18#16840320—22no—
19#16840312C2211yescharacteristic
20#121×D61C2214yes—
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C8×D244—22no—
24#16839827—22no—
25#16839815C2211yescharacteristic, complemented
26C22⋊C8×D61C2214yescomplemented
27#16839660—21no—
28#16839633—22no—
29#16839628C2211yescharacteristic, complemented
30#16839621C2212yescomplemented
31#95×D61C2212yescomplemented
32#16839497C2214yescomplemented
33#16839496C2214yescomplemented
34#16839495C2212yescomplemented
35#16839495C2212yescomplemented
36C61×#532C2211yescharacteristic
37C61×#431C2211yescharacteristic, complemented
38C4×C4⋊C4×C61C2211yescharacteristic, complemented
39C61×#327C2212yescomplemented
40C61×#327C2212yescomplemented
41#14519823—44no—
42#14519823—44no—
43C2×#11758226—22no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C22×D244—22no—
47C22×D244D411yescharacteristic
48C2×D4×D61—24no—
49C2×D4×D61—22no—
50C2×D4×D61C2×C414yescomplemented
51C22×C4×D61—22no—
52C22×C4×D61C2312yes—
53C2×C8×D61—24no—
54#14519777C2×C411yescharacteristic, complemented
55#14519774C2×C411yescharacteristic, complemented
56#14519765—21no—
57#14519762—24no—
58#14519759—24no—
59#14519756—22no—
60#14519753—22no—
61#14519752C2×C414yescomplemented
62#14519751—21no—
63#14519751—22no—
64#14519751C2312yes—
65#14519750—24no—
66#14519749—24no—
67#14519748—22no—
68#14519748C2×C414yescomplemented
69#14519747—24no—
70#14519745—24no—
71C4×D244—21no—
72C4×D244—44no—
73C4×D244C2×C412yes—
74C4×D244C2×C412yes—
75C4×D244C2×C412yescomplemented
76C4×D244D412yes—
77C4×D244D412yes—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×#11758196—22no—
81C4⋊C4×D61C2×C414yescomplemented
82C2×#11758198—22no—
83C2×#11758198D412yes—
84C22⋊C4×D61—24no—
85C4⋊#92418—21no—
86C22⋊#92418C2312yes—
87#14519689—24no—
88C4×#92418—21no—
89#14519684—21no—
90#14519684—22no—
91#14519684C2312yes—
92C2×#41×C61—21no—
93C4⋊Q8×C61C2×C411yescharacteristic, complemented
94#127×C61C2×C411yescharacteristic, complemented
95#124×C61—21no—
96#121×C61—21no—
97#121×C61C2312yes—
98C4×D4×C61—21no—
99C4×D4×C61C2×C412yescomplemented
100C2×C4⋊C4×C61—21no—
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—21no—
104C22⋊C8×C61C2312yes—
105C4×C8×C61—21no—
106#95×C61C2311yescharacteristic
107#95×C61C2311yescharacteristic
108C23×D61—24no—
109C23×D61C2×D412yes—
110#41×C61—42no—
111#41×C61—42no—
112C2×Q8×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61C22×C412yes—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—22no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C22×Dic61—22no—
127C22×Dic61C2×D412yes—
128#11758226—48no—
129D4×D61—48no—
130D4×D61—48no—
131D4×D61—48no—
132D4×D61—84no—
133#11758222—42no—
134#11758222—42no—
135#11758222—48no—
136C2×D244—24no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—48no—
140C2×D244C22⋊C412yes—
141C2×D244C22⋊C412yes—
142C2×D244D814yes—
143C2×D244SD1614yes—
144C2×D244C22×C412yes—
145C2×C4×D61—24no—
146C2×C4×D61—22no—
147C2×C4×D61—44no—
148C2×C4×D61—44no—
149C2×C4×D61—82no—
150C2×C4×D61—82no—
151C2×C4×D61C22×C414yes—
152C2×C4×D61C2×D414yes—
153C2×Dic122—21no—
154C2×C8×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61C22×C412yes—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C42×C61—42no—
161C42×C61C22×C411yescharacteristic
162C42×C61C22×C411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C22⋊Dic61—24no—
166#11758198—22no—
167#11758198—22no—
168#11758198—24no—
169#11758198—44no—
170#11758198C22⋊C414yes—
171#11758198C22⋊C414yes—
172C4⋊Dic61—21no—
173C4⋊Dic61—42no—
174C4⋊Dic61C22⋊C412yes—
175C4⋊Dic61C22⋊C412yes—
176C4⋊Dic61C22×C412yes—
177#11758196—24no—
178#11758196—82no—
179C2×#92418—22no—
180C8×D61—84no—
181C23×C61—22no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—48no—
187C22×D61—84no—
188C22×D61—84no—
189C22×D61C2×C22⋊C414yes—
190C22×D61C2×D814yes—
191C22×D61C2×SD1614yes—
192Q8×C61—42no—
193D4×C61—42no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—81no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—44no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×C4×C61C2×D812yes—
213C2×C4×C61C2×SD1612yes—
214C2×C4×C61C23×C411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C22⋊D61—42no—
217C22⋊D61—42no—
218C22⋊D61—48no—
219C22⋊D61—84no—
220C2×Dic61—22no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61C2×C22⋊C414yes—
226D244—42no—
227D244—48no—
228D244—42no—
229D244—81no—
230D244#10218yes—
231D244#10218yes—
232C4×D61—44no—
233C4×D61—44no—
234C4×D61—48no—
235C4×D61—44no—
236C4×D61—88no—
237Dic122—42no—
238C8×C61—42no—
239#92418—42no—
240#17613—611nomaximal, Sylow
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61C22×C22⋊C411yescharacteristic
247C22×C61C22×D811yescharacteristic
248C22×C61C22×SD1611yescharacteristic
249C2×D61—24no—
250C2×D61—24no—
251C2×D61—44no—
252C2×D61—44no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—168no—
257C2×D61C2×#10218yes—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—81no—
265C4×C61C2×#10212yes—
266C4×C61C2×#10212yes—
267Dic61—24no—
268Dic61—42no—
269Dic61—82no—
270#3496—611no—
271#2972—611no—
272#2349—611no—
273#1519—614no—
274#1517—611no—
275#1517—611no—
276#1502—611no—
277#1502D6111yescharacteristic, complemented
278C2×#327—614no—
279C2×C61—21no—
280C2×C61—21no—
281C2×C61—82no—
282C2×C61#288311yescharacteristic
283C2×C61C22×#10211yescharacteristic
284C2×C61#290111yescharacteristic
285D61—28no—
286D61—82no—
287C2×H(5,𝔽2)—1221no—
288#547—612no—
289#547—612no—
290#546—1222no—
291#546—1222no—
292D42—612no—
293D42—612no—
294#537—1221no—
295#534—1222no—
296#532—611no—
297#532C2×D6111yescharacteristic
298C2×#121—614no—
299C2×C4×D4—611no—
300C2×C4×D4—1221no—
301C8×D4—1222no—
302C8×D4—1222no—
303#431—611no—
304#431C2×D6111yescharacteristic, complemented
305C2×C22⋊C8—614no—
306#389—1221no—
307#385—1222no—
308#382—611no—
309#380—612no—
310C4×C4⋊C4C2×D6111yescharacteristic, complemented
311C2×#95—612no—
312#327—614no—
313#327—614no—
314#327—612no—
315#327—612no—
316#327C2×D6112yescomplemented
317#327C2×D6112yescomplemented
318C61#1761311yescharacteristic, complemented, Sylow
319H(5,𝔽2)—2444no—
320H(5,𝔽2)—2444no—
321C2×#41—21no—
322C2×#41—1222no—
323C2×#41—1222no—
324C2×#41—1221no—
325C22×D4—614no—
326C22×D4—611no—
327C22×D4—1222no—
328C22×D4—1222no—
329C22×D4—1224no—
330C23×C4—612no—
331C23×C4—1222no—
332C22×C8—1224no—
333C4⋊Q8—611no—
334C4⋊Q8C4×D6111yescharacteristic, complemented
335#127—611no—
336#127C4×D6111yescharacteristic, complemented
337#124—21no—
338#124—1221no—
339#123—1224no—
340#123—1224no—
341C22⋊Q8—1222no—
342C22⋊Q8—1222no—
343#121—21no—
344#121—614no—
345#121—612no—
346#121—614no—
347#121—1224no—
348#121—1222no—
349#121—1222no—
350#121—1224no—
351#121—1221no—
352#121C22×D6112yes—
353#120—1224no—
354#120—1224no—
355C4×D4—21no—
356C4×D4—612no—
357C4×D4—612no—
358C4×D4—612no—
359C4×D4—612no—
360C4×D4—612no—
361C4×D4—1221no—
362C4×D4—2444no—
363C4×D4C4×D6112yescomplemented
364C2×C4⋊C4—21no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—614no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C22⋊C4—612no—
370C2×C22⋊C4—1224no—
371C2×C22⋊C4—1222no—
372C2×C42—21no—
373C2×C42C22×D6111yescharacteristic
374C4⋊C8—21no—
375C4⋊C8—1221no—
376C22⋊C8—612no—
377C22⋊C8—1224no—
378C22⋊C8C22×D6112yes—
379C4×C8—21no—
380C4×C8—1221no—
381#95—612no—
382#95—1222no—
383#95—1221no—
384#95C22×D6111yescharacteristic
385#95C22×D6111yescharacteristic
386C24—612no—
387C24—1224no—
388#41—42no—
389#41—42no—
390#41—2448no—
391#41—2448no—
392#41—2442no—
393#41—2442no—
394C2×Q8—21no—
395C2×Q8—1221no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—612no—
400C2×D4—612no—
401C2×D4—612no—
402C2×D4—614no—
403C2×D4—614no—
404C2×D4—1224no—
405C2×D4—1224no—
406C2×D4—1224no—
407C2×D4—1222no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—4884no—
411C2×D4—2448no—
412C2×D4—2448no—
413C2×D4—2448no—
414C2×D4—2448no—
415C2×D4C2×C4×D6112yes—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—612no—
420C22×C4—614no—
421C22×C4—614no—
422C22×C4—1224no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—4882no—
426C22×C4—4882no—
427C22×C4—2444no—
428C22×C4—2444no—
429C22×C4D4×D6111yescharacteristic
430C22×C4D4×D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432C2×C8—22no—
433C2×C8—1222no—
434C2×C8—4884no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—612no—
438C4⋊C4—612no—
439C4⋊C4—612no—
440C4⋊C4—1224no—
441C4⋊C4—1221no—
442C4⋊C4—4882no—
443C4⋊C4—2442no—
444C4⋊C4C2×C4×D6112yes—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—614no—
448C22⋊C4—614no—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C22⋊C4—1222no—
452C22⋊C4—1222no—
453C22⋊C4—2444no—
454C42—42no—
455C42C2×C4×D6111yescharacteristic
456C42C2×C4×D6111yescharacteristic
457C42D4×D6111yescharacteristic
458C42D4×D6111yescharacteristic
459C23—22no—
460C23—614no—
461C23—614no—
462C23—614no—
463C23—1222no—
464C23—4884no—
465C23—4884no—
466C23—2442no—
467C23—2442no—
468C23—2448no—
469C23C2×D4×D6111yescharacteristic
470Q8—42no—
471Q8—2442no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—81no—
477D4—618no—
478D4—618no—
479D4—4881no—
480D4—4884no—
481D4—2442no—
482D4—2448no—
483D4—2448no—
484D4—2442no—
485D4—2442no—
486D4—2442no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—614no—
498C2×C4—1222no—
499C2×C4—4888no—
500C2×C4—2444no—
501C2×C4—2444no—
502C2×C4—2448no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2444no—
508C2×C4C22⋊C4×D6111yescharacteristic
509C2×C4C22⋊C4×D6111yescharacteristic
510C2×C4C22⋊C4×D6111yescharacteristic
511C2×C4C22⋊C4×D6111yescharacteristic
512C2×C4D8×D6112yes—
513C2×C4SD16×D6112yes—
514C2×C4C22×C4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C8—42no—
517C8—2442no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—618no—
524C22—1224no—
525C22—1224no—
526C22—9768no—
527C22—2444no—
528C22—2444no—
529C22—2442no—
530C22—2442no—
531C22—2442no—
532C22C2×C22⋊C4×D6111yescharacteristic
533C22C2×D8×D6111yescharacteristic
534C22C2×SD16×D6111yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—42no—
540C4—42no—
541C4—81no—
542C4—1224no—
543C4—4882no—
544C4—2442no—
545C4#102×D6112yes—
546C4#102×D6112yes—
547C2—21no—
548C2—21no—
549C2—82no—
550C2—1228no—
551C2—4882no—
552C2#2145427811yescharacteristic
553C2C2×#102×D6111yescharacteristic
554C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1502C2
socleC22×C61C22×C22⋊C4
prime radical#21459581C2
cosocleC22×C4×C61C24
hypercenter#1502D61
hypocenterC61#17613
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21454842C2semidirect
3. #21448730C2semidirect
4. #21444385C2semidirect
5. #21444380C2semidirect
6. #21444379C2semidirect
7. #21444313C2semidirect
8. #327×D61C2semidirect
9. C61×#1502C2semidirect
11. #16840417C4semidirect
12. #16840415C4semidirect
15. D4×D244C4semidirect
16. #16840397C4semidirect
25. #16839815C22semidirect
26. C22⋊C8×D61C22semidirect
29. #16839628C22semidirect
30. #16839621C22semidirect
31. #95×D61C22semidirect
32. #16839497C22semidirect
33. #16839496C22semidirect
34. #16839495C22semidirect
35. #16839495C22semidirect
37. C61×#431C22semidirect
38. C4×C4⋊C4×C61C22semidirect
39. C61×#327C22semidirect
40. C61×#327C22semidirect
50. C2×D4×D61C2×C4semidirect
54. #14519777C2×C4semidirect
55. #14519774C2×C4semidirect
61. #14519752C2×C4semidirect
68. #14519748C2×C4semidirect
75. C4×D244C2×C4semidirect
81. C4⋊C4×D61C2×C4semidirect
93. C4⋊Q8×C61C2×C4semidirect
94. #127×C61C2×C4semidirect
99. C4×D4×C61C2×C4semidirect
277. #1502D61semidirect
304. #431C2×D61semidirect
310. C4×C4⋊C4C2×D61semidirect
316. #327C2×D61semidirect
317. #327C2×D61semidirect
318. C61#17613semidirect
334. C4⋊Q8C4×D61semidirect
336. #127C4×D61semidirect
363. C4×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214595812
#214548422
#214487302
#214443852
#214443802
#214443792
#214443132
#327×D612
C61×#15022
#1761361