Finite Group Databasebeta

← #30617745

Subgroups — #30617745

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462419C212yesmaximal, complemented
2#21461868C214yesmaximal, complemented
3#21461829C214yesmaximal, complemented
4#21461731C212yesmaximal, complemented
5#21461648C212yesmaximal, complemented
6#21459461C211yesmaximal, characteristic
7#16840829C2214yes—
8#16840743—22no—
9#16840741—22no—
10C2×#14519836—22no—
11C2×#14519836C2214yes—
12D61⋊#99—22no—
13D61⋊#99C2214yes—
14C2×#14519835C2214yescomplemented
15#16840705C2214yescomplemented
16#16840639C2218yescomplemented
17C2×#14519823C2211yescharacteristic
18C22×#11758222C2211yescharacteristic
19#16840417—22no—
20#16840414—22no—
21#16840409—22no—
22#16840408—22no—
23#16840338C411yescharacteristic, complemented
24#16840333C411yescharacteristic, complemented
25#16840322C412yescomplemented
26#16840322C2212yes—
27#16840273C411yescharacteristic, complemented
28#123×D61C412yescomplemented
29#123×D61C2212yes—
30#16840236C2211yescharacteristic
31C61×#525C411yescharacteristic, complemented
32D4×#6923—48no—
33#14519856—24no—
34C2×C22⋊#6923—24no—
35C2×C22⋊#6923C2318yes—
36#14519836—24no—
37#14519836—24no—
38#14519836—24no—
39#14519835—24no—
40#14519835—44no—
41(C2×#6923)⋊C4—44no—
42#14519823—44no—
43C2×#11758226D412yes—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—44no—
47C2×#11758222C2×C411yescharacteristic
48C2×#11758222D412yes—
49C22×Dic122C2×C411yescharacteristic, complemented
50C22×D244D412yes—
51C22×D244C2311yescharacteristic
52C22×C22⋊D61C2×C411yescharacteristic, complemented
53C22×C22⋊D61C2311yescharacteristic
54C2×D4×D61C2×C412yes—
55C2×D4×D61D412yes—
56C2×D4×D61C2312yes—
57C22×C4×D61C2×C411yescharacteristic, complemented
58C22×C4×D61C2311yescharacteristic
59#14519777—21no—
60#14519775—21no—
61#14519765—21no—
62#14519764—21no—
63#14519762—22no—
64#14519762—22no—
65#14519761C2×C412yescomplemented
66#14519761C2×C412yescomplemented
67#14519760C2×C412yescomplemented
68#14519759—22no—
69#14519759—22no—
70#14519759—22no—
71#14519759—22no—
72#14519758C2×C412yescomplemented
73#14519758C2×C412yescomplemented
74#14519757—22no—
75#14519756—22no—
76#14519756—21no—
77#14519751—21no—
78#14519750—22no—
79#14519750—22no—
80#14519748—22no—
81#14519748—22no—
82#14519746C2×C412yescomplemented
83#14519745—22no—
84#14519745—22no—
85C4×D244—22no—
86#14519739—22no—
87C4×C22⋊D61—22no—
88#14519736C2×C411yescharacteristic, complemented
89C4⋊C4×D61—22no—
90C4⋊C4×D61—22no—
91C22⋊C4×D61—22no—
92C22⋊C4×D61—22no—
93C22⋊C4×D61C2×C412yescomplemented
94C22⋊C4×D61C2312yes—
95C2×#41×C61C2×C411yescharacteristic
96C23×C4×C61C2×C411yescharacteristic, complemented
97#123×C61C2×C412yescomplemented
98C22⋊Q8×C61—21no—
99#121×C61—21no—
100#120×C61C2×C412yescomplemented
101C4×D4×C61—22no—
102#117×C61C2×C411yescharacteristic, complemented
103C24×C61C22×C411yescharacteristic
104C23×D61C22⋊C412yes—
105C23×D61C22×C411yescharacteristic
106C23×D61C2×D412yes—
107C23×D61C2411yescharacteristic
108C22×#6923—28no—
109C22×#6923—44no—
110#41×C61—42no—
111C2×Q8×C61C22⋊C411yescharacteristic
112C2×D4×C61—22no—
113C2×D4×C61C22⋊C411yescharacteristic
114C2×D4×C61C22×C412yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61C22⋊C412yes—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61C22⋊C412yes—
127C2×C22⋊D61C22×C412yes—
128C2×C22⋊D61C2×D412yes—
129C22×Dic61C22×C411yescharacteristic
130C22×Dic61C22×C411yescharacteristic
131#11758226—44no—
132D4×D61—28no—
133D4×D61—44no—
134D4×D61—44no—
135#11758222—44no—
136#11758222—48no—
137#11758222—48no—
138#11758222—42no—
139C2×D244—24no—
140C2×D244—44no—
141C2×D244C22⋊C412yes—
142C2×D244C22⋊C411yescharacteristic
143C2×D244C2×D412yes—
144C2×D244C2×D412yes—
145C2×D244C2×D412yes—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—24no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—42no—
152C2×C4×D61C22⋊C412yes—
153C2×C4×D61C22⋊C412yes—
154C2×C4×D61C2×D412yes—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—42no—
158C2×Dic122C22⋊C411yescharacteristic
159C22⋊#6923—28no—
160C22⋊#6923—28no—
161C22⋊#6923—44no—
162C4⋊#6923—44no—
163C4×#6923—44no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61C22×C412yes—
169C42×C61—21no—
170C22⋊Dic61—22no—
171C22⋊Dic61—22no—
172C22⋊Dic61—22no—
173C22⋊Dic61C22×C412yes—
174#11758198—22no—
175#11758198—22no—
176#11758198—22no—
177#11758198—22no—
178#11758198—22no—
179C4⋊Dic61—21no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182#11758196—22no—
183#11758196—22no—
184#11758196—22no—
185C4×Dic61—21no—
186C23×C61—21no—
187C23×C61—22no—
188C23×C61—81no—
189C23×C61C2×C22⋊C412yes—
190C23×C61C23×C411yescharacteristic
191C22×D61—22no—
192C22×D61—24no—
193C22×D61—42no—
194C22×D61—81no—
195C22×D61—81no—
196C22×D61—82no—
197C22×D61—82no—
198C22×D61C2×C22⋊C412yes—
199C22×D61C2×C22⋊C412yes—
200C22×D61C22×D412yes—
201C22×D61H(5,𝔽2)14yes—
202C2×#6923—48no—
203C2×#6923—88no—
204C2×#6923—164no—
205Q8×C61—41no—
206D4×C61—24no—
207D4×C61—41no—
208D4×C61—82no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C411yescharacteristic
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C2×C22⋊C411yescharacteristic
219C22⋊D61—24no—
220C22⋊D61—24no—
221C22⋊D61—42no—
222C22⋊D61—44no—
223C22⋊D61—44no—
224C22⋊D61—82no—
225C2×Dic61—22no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61C2×C22⋊C412yes—
231D244—24no—
232D244—24no—
233D244—24no—
234D244—41no—
235D244—42no—
236C4×D61—24no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—44no—
240C4×D61—44no—
241C4×D61—82no—
242C4×D61—82no—
243Dic122—41no—
244Dic122—42no—
245Dic122—42no—
246#21777—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—41no—
249C22×C61—42no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—82no—
254C22×C61C22×C22⋊C411yescharacteristic
255C22×C61#51812yes—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—42no—
259C2×D61—44no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—84no—
263C2×D61—82no—
264C2×D61—162no—
265C2×D61#53414yes—
266#6923—88no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—22no—
270C4×C61—42no—
271C4×C61—41no—
272Dic61—22no—
273Dic61—42no—
274Dic61—41no—
275Dic61—41no—
276#3485—611no—
277#2883—612no—
278#1913—614no—
279#1890—614no—
280#1828—612no—
281#1782—612no—
282C2×C61—21no—
283C2×C61—22no—
284C2×C61—42no—
285C2×C61—81no—
286C2×C61#230811yescharacteristic
287D61—24no—
288D61—42no—
289D61—42no—
290C2×H(5,𝔽2)—611no—
291C22×#41—611no—
292#547—1222no—
293#547—1222no—
294#546—1222no—
295#546—1222no—
296#536—611no—
297#536—611no—
298#534—612no—
299#534—612no—
300#525—611no—
301#525#692311yescharacteristic, complemented
302C2×#123—612no—
303C2×#123—612no—
304#518—611no—
305#518—614no—
306#410—1222no—
307#410—1222no—
308C2×#99—614no—
309C2×#99—614no—
310C2×#99—614no—
311C2×#99—1222no—
312C2×#99—1222no—
313#381—614no—
314#342—618no—
315C61#2177711yescharacteristic, complemented, Sylow
316H(5,𝔽2)—2444no—
317C2×#41—612no—
318C2×#41—612no—
319C2×#41—611no—
320C2×#41—1222no—
321C2×#41—1222no—
322C2×#41—2444no—
323C2×#41C2×#692311yescharacteristic
324C22×Q8—611no—
325C22×D4—612no—
326C22×D4—612no—
327C22×D4—612no—
328C22×D4—611no—
329C22×D4—611no—
330C22×D4—612no—
331C22×D4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4C2×#692311yescharacteristic, complemented
335C4⋊Q8—1221no—
336C4⋊Q8—1221no—
337#124—1221no—
338#124—1221no—
339#123—612no—
340#123—612no—
341#123—612no—
342#123—612no—
343#123—612no—
344#123—1222no—
345#123—1222no—
346#123—1222no—
347#123—1222no—
348#123—1222no—
349#123—1222no—
350#123C2×#692312yescomplemented
351C22⋊Q8—21no—
352C22⋊Q8—1222no—
353C22⋊Q8—1221no—
354C22⋊Q8—1222no—
355#121—21no—
356#121—1222no—
357#121—1222no—
358#121—1221no—
359#121—1222no—
360#121—1222no—
361#120—612no—
362#120—1222no—
363#120—1222no—
364#120C2×#692312yescomplemented
365C4×D4—22no—
366C4×D4—1222no—
367C4×D4—1222no—
368C4×D4—1222no—
369C4×D4—2448no—
370#117—611no—
371#117—1224no—
372#117C2×#692311yescharacteristic, complemented
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C22⋊C4—612no—
376C2×C22⋊C4—612no—
377C2×C22⋊C4—618no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1224no—
381#99—1224no—
382#99—1224no—
383#99—1224no—
384#99—1224no—
385#99—2444no—
386#95—2444no—
387C24—611no—
388C24—612no—
389C24—611no—
390C24—612no—
391C24C22×#692311yescharacteristic
392#41—42no—
393#41—2442no—
394#41—2444no—
395#41—2448no—
396#41—2448no—
397#41—2444no—
398C2×Q8—611no—
399C2×Q8—1221no—
400C2×Q8—1221no—
401C2×Q8—2442no—
402C2×Q8C22⋊#692311yescharacteristic
403C2×D4—22no—
404C2×D4—612no—
405C2×D4—612no—
406C2×D4—612no—
407C2×D4—612no—
408C2×D4—612no—
409C2×D4—612no—
410C2×D4—612no—
411C2×D4—611no—
412C2×D4—1224no—
413C2×D4—1228no—
414C2×D4—1222no—
415C2×D4—1224no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—2444no—
419C2×D4—2442no—
420C2×D4—2444no—
421C2×D4—2444no—
422C2×D4C22⋊#692311yescharacteristic
423C2×D4C22×#692312yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1224no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1228no—
438C22×C4—2442no—
439C22×C4—2444no—
440C22×C4C22⋊#692312yes—
441C22×C4C22×#692311yescharacteristic
442C22×C4C22×#692311yescharacteristic
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4—1221no—
451C4⋊C4—2444no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—612no—
455C22⋊C4—1222no—
456C22⋊C4—1222no—
457C22⋊C4—1222no—
458C22⋊C4—1222no—
459C22⋊C4—1222no—
460C22⋊C4—1222no—
461C22⋊C4—1222no—
462C22⋊C4—1222no—
463C22⋊C4—1228no—
464C22⋊C4—1228no—
465C22⋊C4—2444no—
466C22⋊C4C22×#692312yes—
467C42—21no—
468C42—1221no—
469C42—2444no—
470C23—21no—
471C23—22no—
472C23—81no—
473C23—612no—
474C23—614no—
475C23—612no—
476C23—612no—
477C23—1224no—
478C23—1222no—
479C23—4882no—
480C23—4881no—
481C23—4882no—
482C23—4881no—
483C23—2442no—
484C23C2×C22⋊#692312yes—
485C23C23×#692311yescharacteristic
486Q8—41no—
487Q8—2441no—
488Q8—2442no—
489Q8—2442no—
490D4—24no—
491D4—41no—
492D4—82no—
493D4—1224no—
494D4—1224no—
495D4—1224no—
496D4—1224no—
497D4—1224no—
498D4—4882no—
499D4—2441no—
500D4—2442no—
501D4—2442no—
502D4—2444no—
503D4—2444no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—612no—
512C2×C4—1224no—
513C2×C4—1222no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—9764no—
519C2×C4—4882no—
520C2×C4—4882no—
521C2×C4—4888no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2444no—
525C2×C4—2444no—
526C2×C4—2448no—
527C2×C4C2×C22⋊#692311yescharacteristic
528C2×C4C2×C22⋊#692311yescharacteristic
529C2×C4C2×C22⋊#692312yes—
530C22—22no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—82no—
537C22—614no—
538C22—1222no—
539C22—1222no—
540C22—9762no—
541C22—4882no—
542C22—4884no—
543C22—2444no—
544C22—2442no—
545C22—2442no—
546C22—2442no—
547C22C22×C22⋊#692311yescharacteristic
548C22#1684082912yes—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—42no—
553C4—41no—
554C4—1222no—
555C4—4888no—
556C4—2442no—
557C4—2441no—
558C4—2441no—
559C2—21no—
560C2—22no—
561C2—42no—
562C2—81no—
563C2—1224no—
564C2—2442no—
565C2—2442no—
566C2#2146229111yescharacteristic
subgroupG/H
centerC2#21462291
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC61×#525C4
socleC2×C61#2308
prime radical#21459461C2
cosocleC23×D61C24
hypercenter#525#6923
hypocenterC61#21777
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #21462419C2semidirect
2. #21461868C2semidirect
3. #21461829C2semidirect
4. #21461731C2semidirect
5. #21461648C2semidirect
14. C2×#14519835C22semidirect
15. #16840705C22semidirect
16. #16840639C22semidirect
23. #16840338C4semidirect
24. #16840333C4semidirect
25. #16840322C4semidirect
27. #16840273C4semidirect
28. #123×D61C4semidirect
31. C61×#525C4semidirect
49. C22×Dic122C2×C4semidirect
52. C22×C22⋊D61C2×C4semidirect
57. C22×C4×D61C2×C4semidirect
65. #14519761C2×C4semidirect
66. #14519761C2×C4semidirect
67. #14519760C2×C4semidirect
72. #14519758C2×C4semidirect
73. #14519758C2×C4semidirect
82. #14519746C2×C4semidirect
88. #14519736C2×C4semidirect
93. C22⋊C4×D61C2×C4semidirect
96. C23×C4×C61C2×C4semidirect
97. #123×C61C2×C4semidirect
100. #120×C61C2×C4semidirect
102. #117×C61C2×C4semidirect
301. #525#6923semidirect
315. C61#21777semidirect
334. C23×C4C2×#6923semidirect
350. #123C2×#6923semidirect
364. #120C2×#6923semidirect
372. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624192
#214618682
#214618292
#214617312
#214616482
#214594612
#2177761