Finite Group Databasebeta

← #29689163

Subgroups — #29689163

Order 15616.

1520 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459454C211yesmaximal, characteristic
2#21454277C212yesmaximal, complemented
3#21446358C214yesmaximal, complemented
4#21446224C214yesmaximal, complemented
5#21445925C212yesmaximal, complemented
6#21445734C211yesmaximal, characteristic, complemented
7C61×#1781C211yesmaximal, characteristic, complemented
8C2×#14519822C2211yescharacteristic
9#16840406—24no—
10D4×C22⋊D61—24no—
11#16840329C412yescomplemented
12#16840321C412yescomplemented
13#16840321C2212yes—
14#16840273C412yescomplemented
15C2×#14519746C412yescomplemented
16C2×#14519746C2212yes—
17#16840237C2211yescharacteristic
18#16840235C2214yescomplemented
19#16839734—24no—
20C2×#14519692C2212yes—
21C2×#14519692C2212yescomplemented
22C2×#14519691C2214yescomplemented
23#99×D61—24no—
24#16839618C2212yescomplemented
25#16839529C2214yescomplemented
26#16839528C2214yescomplemented
27C23×D4×C61C2211yescharacteristic
28C2×#99×C61C2212yes—
29C2×#99×C61C2212yescomplemented
30C61×#379C2212yescomplemented
31#14519822—42no—
32#14519822—42no—
33C2×#11758222C2×C412yes—
34C2×#11758224D414yes—
35C22×C22⋊D61—22no—
36C22×C22⋊D61—22no—
37C22×C22⋊D61C2×C412yes—
38C22×C22⋊D61C2312yes—
39C2×D4×D61D414yes—
40#14519773—22no—
41#14519764—22no—
42#14519760C2×C414yescomplemented
43#14519758—22no—
44#14519758—22no—
45#14519756—22no—
46#14519751—22no—
47#14519749—24no—
48#14519749—24no—
49#14519748—24no—
50#14519746—24no—
51#14519746—24no—
52#14519746C2×C414yescomplemented
53#14519746C2×C414yescomplemented
54#14519746C2×C414yescomplemented
55#14519745—22no—
56#14519745—22no—
57D4×Dic61—24no—
58C4×C22⋊D61—24no—
59C4×C22⋊D61—44no—
60#14519737—44no—
61#14519736C2×C412yescomplemented
62#14519733—24no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—42no—
66C2×C22⋊Dic61C2×C412yescomplemented
67C2×C22⋊Dic61C2314yes—
68C2×C22⋊Dic61C2312yes—
69C22⋊C4×D61—24no—
70#14519692—24no—
71#14519692—42no—
72#14519691—28no—
73C25×C61C2×C411yescharacteristic, complemented
74C25×C61C2311yescharacteristic
75C22×D4×C61—24no—
76C22×D4×C61—42no—
77C22×D4×C61—42no—
78C22×D4×C61C2×C411yescharacteristic, complemented
79C22×D4×C61C2×C411yescharacteristic, complemented
80C22×D4×C61C2311yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61C2×C411yescharacteristic, complemented
83C2×C22⋊C4×C61—42no—
84C2×C22⋊C4×C61C2314yes—
85#99×C61—24no—
86#99×C61—42no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—41no—
92C24×C61—41no—
93C24×C61—82no—
94C24×C61—82no—
95C24×C61C22×C411yescharacteristic
96C24×C61C22×C411yescharacteristic
97C24×C61C22×C411yescharacteristic
98C24×C61C22×C411yescharacteristic
99C24×C61C2411yescharacteristic
100C23×D61C22⋊C412yescomplemented
101C23×D61C2×D412yes—
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—416no—
114C2×D4×C61C2×D414yes—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—24no—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C2×C22⋊D61—84no—
129C2×C22⋊D61—84no—
130C2×C22⋊D61C22⋊C414yescomplemented
131C2×C22⋊D61C22×C414yes—
132C2×C22⋊D61C2×D414yes—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—24no—
136C22×Dic61C22⋊C412yescomplemented
137C22×Dic61C2×D412yes—
138#11758224—48no—
139D4×D61—48no—
140#11758222—42no—
141#11758222—42no—
142C2×D244C22⋊C412yescomplemented
143C2×C4×D61—44no—
144C2×C4×D61C22⋊C414yescomplemented
145C2×Dic122C22⋊C412yescomplemented
146C22⋊C4×C61—24no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—84no—
149C22⋊Dic61—22no—
150C22⋊Dic61—22no—
151C22⋊Dic61—24no—
152C22⋊Dic61—24no—
153C22⋊Dic61—24no—
154C22⋊Dic61—24no—
155C22⋊Dic61—24no—
156C22⋊Dic61—84no—
157C22⋊Dic61C22×C414yes—
158#11758198—24no—
159#11758198—44no—
160C4⋊Dic61—22no—
161#11758196—24no—
162#11758196—44no—
163C4×Dic61—22no—
164C4×Dic61—44no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—42no—
177C23×C61—44no—
178C23×C61—44no—
179C23×C61—44no—
180C23×C61—44no—
181C23×C61—44no—
182C23×C61—44no—
183C23×C61—82no—
184C23×C61—82no—
185C23×C61—81no—
186C23×C61—81no—
187C23×C61—82no—
188C23×C61—82no—
189C23×C61—82no—
190C23×C61—82no—
191C23×C61—81no—
192C23×C61—81no—
193C23×C61—82no—
194C23×C61—82no—
195C23×C61C2×C22⋊C412yes—
196C23×C61C23×C411yescharacteristic
197C23×C61C22×D412yes—
198C23×C61H(5,𝔽2)12yes—
199C23×C61H(5,𝔽2)12yes—
200C22×D61—24no—
201C22×D61—42no—
202C22×D61—82no—
203C22×D61—82no—
204C22×D61C2×C22⋊C414yes—
205D4×C61—28no—
206D4×C61—44no—
207D4×C61—44no—
208D4×C61—44no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—44no—
215C2×C4×C61—44no—
216C2×C4×C61—84no—
217C2×C4×C61C2×C22⋊C412yes—
218C22⋊D61—24no—
219C22⋊D61—24no—
220C22⋊D61—28no—
221C22⋊D61—44no—
222C22⋊D61—48no—
223C22⋊D61—84no—
224C22⋊D61—84no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—42no—
228C2×Dic61—44no—
229C2×Dic61—82no—
230C2×Dic61—82no—
231C2×Dic61—84no—
232C2×Dic61C2×C22⋊C414yes—
233D244—42no—
234C4×D61—44no—
235C4×D61—164no—
236Dic122—42no—
237#21776—611nomaximal, Sylow
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—41no—
246C22×C61—42no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—88no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—88no—
263C22×C61C22×C22⋊C411yescharacteristic
264C22×C61#51811yescharacteristic
265C22×C61#51811yescharacteristic
266C22×C61#53414yes—
267C2×D61—24no—
268C2×D61—44no—
269C2×D61—84no—
270C2×D61—162no—
271C4×C61—22no—
272C4×C61—41no—
273C4×C61—84no—
274Dic61—24no—
275Dic61—44no—
276Dic61—84no—
277#3484—611no—
278#2883—612no—
279#1913—614no—
280#1889—614no—
281#1828—612no—
282#1781—611no—
283#1781D6111yescharacteristic, complemented
284C2×C61—21no—
285C2×C61—24no—
286C2×C61—21no—
287C2×C61—21no—
288C2×C61—42no—
289C2×C61—42no—
290C2×C61—81no—
291C2×C61—81no—
292C2×C61#230811yescharacteristic
293D61—44no—
294C2×H(5,𝔽2)—611no—
295C23×D4C2×D6111yescharacteristic
296#546—1224no—
297D42—1224no—
298#535—612no—
299#534—612no—
300#534—612no—
301#525—612no—
302C2×#120—612no—
303C2×#120—612no—
304#518—611no—
305#518—614no—
306#410—1224no—
307C2×#99—612no—
308C2×#99—614no—
309C2×#99—612no—
310C2×#99—1224no—
311C2×#99C2×D6112yes—
312C2×#99C2×D6112yescomplemented
313#379—612no—
314#379C2×D6112yescomplemented
315#342—614no—
316#342—614no—
317C61#2177611yescharacteristic, complemented, Sylow
318C25C4×D6111yescharacteristic, complemented
319C25C22×D6111yescharacteristic
320H(5,𝔽2)—2442no—
321H(5,𝔽2)—2442no—
322C2×#41—614no—
323C2×#41—612no—
324C22×D4—24no—
325C22×D4—42no—
326C22×D4—42no—
327C22×D4—612no—
328C22×D4—612no—
329C22×D4—614no—
330C22×D4—1222no—
331C22×D4—1222no—
332C22×D4C4×D6111yescharacteristic, complemented
333C22×D4C4×D6111yescharacteristic, complemented
334C22×D4C22×D6111yescharacteristic
335C22×D4C22×D6111yescharacteristic
336C23×C4C4×D6111yescharacteristic, complemented
337#127—1222no—
338#124—1222no—
339#123—614no—
340#123—1222no—
341#123—1222no—
342C22⋊Q8—1222no—
343#121—1222no—
344#121—1224no—
345#121—1224no—
346#121—1224no—
347#120—614no—
348#120—614no—
349#120—614no—
350#120—1222no—
351#120—1222no—
352#120—1224no—
353#120—1224no—
354C4×D4—1224no—
355C4×D4—1224no—
356C4×D4—2444no—
357C4×D4—2444no—
358#117—612no—
359#117—1224no—
360C2×C22⋊C4—42no—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4—614no—
364C2×C22⋊C4—1224no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—2442no—
368C2×C22⋊C4C22×D6114yes—
369#99—24no—
370#99—42no—
371#99—1228no—
372#99—1224no—
373#99—2442no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—41no—
379C24—41no—
380C24—82no—
381C24—82no—
382C24—612no—
383C24—612no—
384C24C2×C4×D6111yescharacteristic
385C24C2×C4×D6111yescharacteristic
386C24C2×C4×D6111yescharacteristic
387C24C2×C4×D6111yescharacteristic
388C24C23×D6111yescharacteristic
389#41—2442no—
390#41—2442no—
391#41—2448no—
392C2×Q8—612no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—416no—
405C2×D4—614no—
406C2×D4—614no—
407C2×D4—614no—
408C2×D4—612no—
409C2×D4—1224no—
410C2×D4—1224no—
411C2×D4—1224no—
412C2×D4—1224no—
413C2×D4—1224no—
414C2×D4—4884no—
415C2×D4—4884no—
416C2×D4—2448no—
417C2×D4—2442no—
418C2×D4—2442no—
419C2×D4D4×D6114yes—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—612no—
425C22×C4—612no—
426C22×C4—614no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1224no—
430C22×C4—2444no—
431C22×C4C2×C4×D6111yescharacteristic
432C22×C4C2×C4×D6111yescharacteristic
433C4⋊C4—1224no—
434C4⋊C4—1222no—
435C4⋊C4—2444no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—84no—
439C22⋊C4—614no—
440C22⋊C4—1224no—
441C22⋊C4—1224no—
442C22⋊C4—1224no—
443C22⋊C4—1224no—
444C22⋊C4—1222no—
445C22⋊C4—1222no—
446C22⋊C4—1224no—
447C22⋊C4—1224no—
448C22⋊C4—4884no—
449C22⋊C4—2444no—
450C42—1222no—
451C42—2444no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—82no—
471C23—82no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—614no—
483C23—1224no—
484C23—4882no—
485C23—4882no—
486C23—2442no—
487C23C22⋊C4×D6112yes—
488C23C22×C4×D6111yescharacteristic
489C23C2×D4×D6112yes—
490C23#1451982212yes—
491C23#1451982212yes—
492Q8—2442no—
493D4—28no—
494D4—44no—
495D4—44no—
496D4—44no—
497D4—1224no—
498D4—1224no—
499D4—1228no—
500D4—4884no—
501D4—4884no—
502D4—2442no—
503D4—2444no—
504D4—2448no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—84no—
513C2×C4—614no—
514C2×C4—1224no—
515C2×C4—1224no—
516C2×C4—9764no—
517C2×C4—4884no—
518C2×C4—4882no—
519C2×C4—4882no—
520C2×C4—2444no—
521C2×C4—2442no—
522C2×C4—2444no—
523C2×C4C22⋊C4×D6112yes—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—81no—
544C22—81no—
545C22—88no—
546C22—81no—
547C22—81no—
548C22—88no—
549C22—1224no—
550C22—9762no—
551C22—4884no—
552C22—2444no—
553C22C2×C22⋊C4×D6111yescharacteristic
554C22#1684023511yescharacteristic
555C22#1684023511yescharacteristic
556C22#1684032214yes—
557C4—22no—
558C4—41no—
559C4—84no—
560C4—1224no—
561C4—4884no—
562C4—2444no—
563C2—24no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—42no—
568C2—42no—
569C2—81no—
570C2—81no—
571C2—2444no—
572C2#2144839511yescharacteristic
subgroupG/H
centerC2#21448395
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1781C2
socleC2×C61#2308
prime radical#21459454C2
cosocleC24×C61C24
hypercenter#1781D61
hypocenterC61#21776
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C24 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #21454277C2semidirect
3. #21446358C2semidirect
4. #21446224C2semidirect
5. #21445925C2semidirect
6. #21445734C2semidirect
7. C61×#1781C2semidirect
11. #16840329C4semidirect
12. #16840321C4semidirect
14. #16840273C4semidirect
15. C2×#14519746C4semidirect
18. #16840235C22semidirect
21. C2×#14519692C22semidirect
22. C2×#14519691C22semidirect
24. #16839618C22semidirect
25. #16839529C22semidirect
26. #16839528C22semidirect
29. C2×#99×C61C22semidirect
30. C61×#379C22semidirect
42. #14519760C2×C4semidirect
52. #14519746C2×C4semidirect
53. #14519746C2×C4semidirect
54. #14519746C2×C4semidirect
61. #14519736C2×C4semidirect
66. C2×C22⋊Dic61C2×C4semidirect
73. C25×C61C2×C4semidirect
78. C22×D4×C61C2×C4semidirect
79. C22×D4×C61C2×C4semidirect
82. C23×C4×C61C2×C4semidirect
100. C23×D61C22⋊C4semidirect
130. C2×C22⋊D61C22⋊C4semidirect
136. C22×Dic61C22⋊C4semidirect
142. C2×D244C22⋊C4semidirect
144. C2×C4×D61C22⋊C4semidirect
145. C2×Dic122C22⋊C4semidirect
283. #1781D61semidirect
312. C2×#99C2×D61semidirect
314. #379C2×D61semidirect
317. C61#21776semidirect
318. C25C4×D61semidirect
332. C22×D4C4×D61semidirect
333. C22×D4C4×D61semidirect
336. C23×C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214594542
#214542772
#214463582
#214462242
#214459252
#214457342
C61×#17812
#2177661