Finite Group Databasebeta

← #29718389

Subgroups — #29718389

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840098C211yesmaximal, characteristic, complemented
2C22×#14519698C211yesmaximal, characteristic
3#21449466C211yesmaximal, characteristic
4#21448698C211yesmaximal, characteristic, complemented
5#21447205C212yesmaximal, complemented
6#21447058C212yesmaximal, complemented
7#21446750C212yesmaximal, complemented
8#21446441C212yesmaximal, complemented
9#21446240C211yesmaximal, characteristic, complemented
10#21445996C211yesmaximal, characteristic, complemented
11C61×#1786C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×#14519764C2212yes—
14C2×#14519756C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16840098—24no—
17C2×#14519711C2211yescharacteristic, complemented
18C2×#14519711C2211yescharacteristic, complemented
19C2×#14519706C2212yescomplemented
20C2×#14519702C2211yescharacteristic, complemented
21C2×#14519702C2211yescharacteristic, complemented
22C2×#14519700C2212yescomplemented
23C2×#14519698—24no—
24C2×#14519698C2212yes—
25C2×#14519698C2212yes—
26#16839676—22no—
27#16839665C2212yes—
28#16839658C2212yescomplemented
29#16839647—22no—
30#16839620—21no—
31#16839618—21no—
32#16839618—21no—
33C4⋊C4×Dic61C2212yescomplemented
34C22⋊C4×Dic61—21no—
35C4×C22⋊Dic61C2211yescharacteristic
36#16839524C2212yescomplemented
37#16839516C2212yescomplemented
38C23×D4×C61C2211yescharacteristic
39C2×#102×C61C2212yes—
40C2×#102×C61C2212yes—
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C22×Dic122C2311yescharacteristic
44C23×Dic61—21no—
45C22×#92418—22no—
46C22×#92418—42no—
47C22×#92418C2312yes—
48#14519764—24no—
49#14519764—42no—
50#14519756—22no—
51#14519756—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61D414yes—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2311yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—22no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61D411yescharacteristic
66C2×C22⋊Dic61D411yescharacteristic
67C2×C4×Dic61—22no—
68C2×C4×Dic61C2312yes—
69#14519711—22no—
70#14519711—22no—
71#14519706—42no—
72#14519702—22no—
73#14519702—22no—
74#14519700—42no—
75#14519698—24no—
76#14519698—24no—
77#14519684—22no—
78#14519684—22no—
79#14519684D412yes—
80C25×C61—21no—
81C22×D4×C61—24no—
82C22×D4×C61—22no—
83C22×D4×C61—21no—
84C22×D4×C61—22no—
85C22×D4×C61—42no—
86C22×D4×C61C2312yes—
87C23×C4×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2312yes—
91C2×C22⋊C4×C61—21no—
92#102×C61—24no—
93#102×C61—42no—
94C22⋊C8×C61—22no—
95#95×C61—21no—
96C24×C61—22no—
97C24×C61—44no—
98C24×C61—42no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61—82no—
102C24×C61C2×D411yescharacteristic
103C2×D4×C61—28no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—48no—
110C2×D4×C61—44no—
111C2×D4×C61—44no—
112C2×D4×C61—42no—
113C2×D4×C61#4114yes—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D412yes—
119C22×C4×C61#4112yes—
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—24no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—82no—
127C22×Dic61C2×D412yes—
128C22×Dic61C2×D412yes—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C22⋊C4×C61—42no—
140C22⋊Dic61—22no—
141C22⋊Dic61—22no—
142C22⋊Dic61—44no—
143C22⋊Dic61—44no—
144C22⋊Dic61—42no—
145C22⋊Dic61—82no—
146C22⋊Dic61—82no—
147C4⋊Dic61—24no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61C2×D412yes—
153C4⋊Dic61C2×D412yes—
154#11758196—42no—
155#11758196—82no—
156C4×Dic61—22no—
157C4×Dic61—22no—
158C4×Dic61—82no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—44no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—24no—
166C23×C61—48no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—88no—
174C23×C61—88no—
175C23×C61—84no—
176C23×C61—84no—
177C23×C61#12112yes—
178C23×C61#13512yes—
179C23×C61#13712yes—
180C23×C61C22×D411yescharacteristic
181D4×C61—24no—
182D4×C61—24no—
183D4×C61—48no—
184D4×C61—42no—
185D4×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—24no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61#12112yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4112yes—
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)12yes—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—84no—
213Dic122—41no—
214Dic122—41no—
215Dic122—42no—
216C8×C61—41no—
217C8×C61—41no—
218#92418—42no—
219#24479—611nomaximal, Sylow
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—24no—
223C22×C61—24no—
224C22×C61—44no—
225C22×C61—44no—
226C22×C61—44no—
227C22×C61—816no—
228C22×C61—82no—
229C22×C61—82no—
230C22×C61—82no—
231C22×C61—82no—
232C22×C61#46414yes—
233C22×C61C2×#12111yescharacteristic
234C22×C61#54612yes—
235C22×C61C2×#13511yescharacteristic
236C22×C61C2×#13711yescharacteristic
237C22×C61#57612yes—
238C4×C61—22no—
239C4×C61—82no—
240C4×C61#52511yescharacteristic
241C4×C61#53511yescharacteristic
242C4×C61#55112yes—
243Dic61—42no—
244Dic61—42no—
245Dic61—42no—
246Dic61—42no—
247Dic61—81no—
248Dic61—81no—
249C2×#484—611no—
250C22×#102—611no—
251#2435—611no—
252#2345—611no—
253#2067—612no—
254#2042—612no—
255#1983—612no—
256#1925—612no—
257#1892—611no—
258#1850—611no—
259#1786D6111yescharacteristic
260C2×C61—24no—
261C2×C61—84no—
262C2×C61#241111yescharacteristic
263C2×C61C2×#46411yescharacteristic
264C2×C61#304111yescharacteristic
265C2×C61#329712yes—
266C2×C61#331412yes—
267C23×D4C2×D6111yescharacteristic
268C23×C8—611no—
269C2×#124—612no—
270C2×C22⋊Q8—611no—
271C22×C4⋊C4—611no—
272#484—1224no—
273C2×#107—611no—
274C2×#107—611no—
275C2×C4⋊C8—612no—
276C2×#103—611no—
277C2×#103—611no—
278C2×#103—612no—
279C2×#102—612no—
280C2×#102—612no—
281C2×#102—1224no—
282C2×#102C2×D6112yes—
283C2×#102C2×D6112yes—
284C2×C22⋊C8C2×D6111yescharacteristic
285#394—1222no—
286#390—612no—
287#389—612no—
288#387—1222no—
289#380—1221no—
290#380C2×D6111yescharacteristic
291#379—1221no—
292#379—1221no—
293C4×C4⋊C4—612no—
294C4×C22⋊C4—611no—
295C4×C22⋊C4—1221no—
296#340—612no—
297#336—612no—
298C61#2447911yescharacteristic, complemented, Sylow
299C25—21no—
300C22×Q8—611no—
301C22×D4—24no—
302C22×D4—22no—
303C22×D4—22no—
304C22×D4—21no—
305C22×D4—42no—
306C22×D4C22×D6112yes—
307C23×C4—1221no—
308C23×C4C22×D6111yescharacteristic
309C22×C8—612no—
310C22×C8—1222no—
311C22×C8—2442no—
312C22×C8C22×D6111yescharacteristic
313C22×C8C22×D6111yescharacteristic
314#124—1224no—
315#124—2442no—
316C22⋊Q8—1222no—
317C22⋊Q8—1222no—
318C2×C4⋊C4—614no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—611no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4C22×D6112yes—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—611no—
329C2×C22⋊C4—611no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4—1222no—
332C2×C22⋊C4—1222no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1221no—
335C2×C42—612no—
336C2×C42—1222no—
337#107—1222no—
338#107—1222no—
339C4⋊C8—2442no—
340#103—1222no—
341#103—1222no—
342#103—2442no—
343#102—24no—
344#102—42no—
345#102—1224no—
346#102—1224no—
347C22⋊C8—22no—
348#95—21no—
349#95—612no—
350#95—1222no—
351#95—1222no—
352C24—22no—
353C24—44no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—82no—
358C24D4×D6111yescharacteristic
359C2×Q8—1221no—
360C2×Q8—1222no—
361C2×Q8—1221no—
362C2×Q8—1222no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—42no—
373C2×D4#1175822414yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—612no—
378C22×C4—612no—
379C22×C4—1224no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—4882no—
385C22×C4#1175822212yes—
386C22×C4D4×D6112yes—
387C22×C4D4×D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—1222no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—2444no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—612no—
400C4⋊C4—612no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—1224no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—4882no—
407C4⋊C4—2442no—
408C22⋊C4—42no—
409C22⋊C4—1222no—
410C22⋊C4—1222no—
411C22⋊C4—4882no—
412C22⋊C4—4882no—
413C22⋊C4—2444no—
414C22⋊C4—2444no—
415C22⋊C4—2442no—
416C42—1222no—
417C42—1222no—
418C42—4882no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—48no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—88no—
430C23—88no—
431C23—84no—
432C23—84no—
433C23#1451975012yes—
434C23#1451979612yes—
435C23#1451980612yes—
436C23C2×D4×D6111yescharacteristic
437Q8—2441no—
438Q8—2441no—
439Q8—2442no—
440D4—24no—
441D4—24no—
442D4—48no—
443D4—42no—
444D4—42no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—4884no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4#1451975012yes—
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C2×#1175822412yes—
470C2×C4C2×#1175822211yescharacteristic
471C2×C4#1451982212yes—
472C8—41no—
473C8—41no—
474C8—2442no—
475C22—24no—
476C22—22no—
477C22—24no—
478C22—22no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—816no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22#1683998314yes—
488C22C2×#1451975011yescharacteristic
489C22#1684039912yes—
490C22C2×#1451979611yescharacteristic
491C22C2×#1451980611yescharacteristic
492C22#1684056612yes—
493C4—22no—
494C4—82no—
495C4—4881no—
496C4—4881no—
497C4—2442no—
498C4—2442no—
499C4—2442no—
500C4—2442no—
501C4#1684026811yescharacteristic
502C4#1684032711yescharacteristic
503C4#1684043812yes—
504C2—24no—
505C2—84no—
506C2#2144920611yescharacteristic
507C2C2×#1683998311yescharacteristic
508C2#2145532111yescharacteristic
509C2#2145767312yes—
510C2#2145790412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1786C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
hypercenter#1786D61
hypocenterC61#24479
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
1. C2×#16840098C2semidirect
4. #21448698C2semidirect
5. #21447205C2semidirect
6. #21447058C2semidirect
7. #21446750C2semidirect
8. #21446441C2semidirect
9. #21446240C2semidirect
10. #21445996C2semidirect
17. C2×#14519711C22semidirect
18. C2×#14519711C22semidirect
19. C2×#14519706C22semidirect
20. C2×#14519702C22semidirect
21. C2×#14519702C22semidirect
22. C2×#14519700C22semidirect
28. #16839658C22semidirect
33. C4⋊C4×Dic61C22semidirect
36. #16839524C22semidirect
37. #16839516C22semidirect
298. C61#24479semidirect

Maximal subgroups

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

subgroupindex
C2×#168400982
C22×#145196982
#214494662
#214486982
#214472052
#214470582
#214467502
#214464412
#214462402
#214459962
C61×#17862
#2447961