Finite Group Databasebeta

← #29718388

Subgroups — #29718388

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
1C22×#14519697C211yesmaximal, characteristic
2#21449471C211yesmaximal, characteristic
3#21448696C211yesmaximal, characteristic, complemented
4#21447202C212yesmaximal, complemented
5#21447057C212yesmaximal, complemented
6#21446748C212yesmaximal, complemented
7#21446438C212yesmaximal, complemented
8#21446243C211yesmaximal, characteristic, complemented
9#21445996C211yesmaximal, characteristic, complemented
10#21445751C211yesmaximal, characteristic
11C2×C61×#484C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519766C2212yes—
15C2×#14519706C2212yescomplemented
16C2×#14519701C2212yescomplemented
17C2×#14519699C2212yes—
18C2×#14519699C2212yes—
19C2×#14519697—24no—
20C2×#14519697C2212yes—
21C2×#14519697C2212yes—
22#16839679—22no—
23#16839663C2212yes—
24#16839661C2212yescomplemented
25#16839648—22no—
26#16839622—21no—
27#16839622C2211yescharacteristic
28#16839617—21no—
29#16839617—21no—
30C4⋊C4×Dic61C2212yescomplemented
31C4×C22⋊Dic61—21no—
32C4×C22⋊Dic61C2211yescharacteristic
33#16839524C2212yescomplemented
34#16839516C2212yescomplemented
35C2×C22⋊Q8×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#484—24no—
38C2×#107×C61C2211yescharacteristic, complemented
39C2×#107×C61C2211yescharacteristic, complemented
40C2×#103×C61C2211yescharacteristic, complemented
41C2×#103×C61C2211yescharacteristic, complemented
42C2×C22⋊C8×C61C2211yescharacteristic
43C24×D61—21no—
44C22×D244—22no—
45C22×D244—24no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244—42no—
49C22×D244C2312yes—
50C22×#92418—22no—
51C22×#92418—42no—
52C22×#92418C2312yes—
53#14519766—24no—
54#14519766—42no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61C2312yes—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61D411yescharacteristic
59C2×C22⋊Dic61D411yescharacteristic
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×C4×Dic61—22no—
65C2×C4×Dic61C2312yes—
66#14519706—42no—
67#14519701—42no—
68#14519699—24no—
69#14519699—42no—
70#14519697—24no—
71#14519697—24no—
72#14519684—21no—
73#14519684—22no—
74#14519684—22no—
75#14519684D412yes—
76C22×Q8×C61C2311yescharacteristic
77C23×C4×C61—21no—
78C23×C4×C61C2311yescharacteristic
79C22×C8×C61C2311yescharacteristic
80C22×C8×C61C2311yescharacteristic
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61D414yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61—21no—
91#107×C61—22no—
92#107×C61—22no—
93#103×C61—22no—
94#103×C61—22no—
95C22⋊C8×C61—22no—
96C24×C61C2×D411yescharacteristic
97C23×D61—22no—
98C23×D61—44no—
99C23×D61—42no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—82no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—24no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—82no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×C4×C61#4112yes—
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—22no—
120C22×Dic61C2×D412yes—
121C22×Dic61C2×D412yes—
122C2×D244—28no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—48no—
129C2×D244—44no—
130C2×D244—44no—
131C2×D244—42no—
132C2×D244#4114yes—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61C2×D412yes—
144C4⋊C4×C61C2×D412yes—
145C22⋊C4×C61—42no—
146C22⋊Dic61—22no—
147C22⋊Dic61—22no—
148C22⋊Dic61—42no—
149#11758198—44no—
150#11758198—44no—
151#11758198—82no—
152#11758198—82no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—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#12112yes—
166C23×C61#13512yes—
167C23×C61#13712yes—
168C23×C61C22×D411yescharacteristic
169C22×D61—24no—
170C22×D61—48no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—88no—
178C22×D61—88no—
179C22×D61—84no—
180C22×D61—84no—
181Q8×C61—41no—
182Q8×C61—41no—
183Q8×C61—42no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—84no—
197C2×C4×C61#12112yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4112yes—
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)12yes—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211D244—24no—
212D244—24no—
213D244—48no—
214D244—42no—
215D244—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#46414yes—
225C22×C61C2×#12111yescharacteristic
226C22×C61#54612yes—
227C22×C61C2×#13511yescharacteristic
228C22×C61C2×#13711yescharacteristic
229C22×C61#57612yes—
230C2×D61—44no—
231C2×D61—44no—
232C2×D61—44no—
233C2×D61—816no—
234C2×D61—82no—
235C2×D61—82no—
236C2×D61—82no—
237C2×D61—82no—
238C4×C61—22no—
239C4×C61—42no—
240C4×C61—42no—
241C4×C61—81no—
242C4×C61—81no—
243C4×C61#52511yescharacteristic
244C4×C61#53511yescharacteristic
245C4×C61#55112yes—
246Dic61—42no—
247Dic61—42no—
248Dic61—82no—
249C2×#484D6111yescharacteristic, complemented
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#1786—611no—
260C2×C61—24no—
261C2×C61#241111yescharacteristic
262C2×C61C2×#46411yescharacteristic
263C2×C61#304111yescharacteristic
264C2×C61#329712yes—
265C2×C61#331412yes—
266D61—84no—
267C23×D4—611no—
268C23×C8—611no—
269C2×#124—612no—
270C2×C22⋊Q8C2×D6111yescharacteristic
271C22×C4⋊C4C2×D6111yescharacteristic
272#484—24no—
273C2×#107C2×D6111yescharacteristic, complemented
274C2×#107C2×D6111yescharacteristic, complemented
275C2×C4⋊C8—612no—
276C2×#103—612no—
277C2×#103C2×D6111yescharacteristic, complemented
278C2×#103C2×D6111yescharacteristic, complemented
279C2×#102—612no—
280C2×#102—612no—
281C2×#102—612no—
282C2×#102—612no—
283C2×#102—1224no—
284C2×C22⋊C8C2×D6111yescharacteristic
285#394—1222no—
286#390—612no—
287#389—612no—
288#387—1222no—
289#380—611no—
290#380—1221no—
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—1221no—
300C22×Q8C22×D6111yescharacteristic
301C22×D4—612no—
302C22×D4—1224no—
303C22×D4—1221no—
304C22×D4—1222no—
305C22×D4—1222no—
306C22×D4—2442no—
307C23×C4—21no—
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—22no—
317C22⋊Q8—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4C22⋊D6114yes—
324C2×C4⋊C4C22×D6111yescharacteristic
325C2×C4⋊C4C22×D6111yescharacteristic
326C2×C4⋊C4C22×D6111yescharacteristic
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—22no—
338#107—22no—
339C4⋊C8—2442no—
340#103—22no—
341#103—22no—
342#103—2442no—
343#102—1224no—
344#102—1224no—
345#102—1224no—
346#102—2442no—
347C22⋊C8—22no—
348#95—612no—
349#95—1222no—
350#95—1222no—
351#95—1221no—
352C24—1222no—
353C24—4882no—
354C24—2444no—
355C24—2442no—
356C24—2442no—
357C24—2442no—
358C24D4×D6111yescharacteristic
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×D4—614no—
364C2×D4—1228no—
365C2×D4—1222no—
366C2×D4—1222no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—2448no—
371C2×D4—2444no—
372C2×D4—2444no—
373C2×D4—2442no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—82no—
382C22×C4—612no—
383C22×C4—612no—
384C22×C4—1222no—
385C22×C4D4×D6111yescharacteristic
386C22×C4#1175822412yes—
387C22×C4C2×C22⋊D6112yes—
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—21no—
398C4⋊C4—21no—
399C4⋊C4—24no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—42no—
403C4⋊C4—1222no—
404C4⋊C4—1222no—
405C4⋊C4—4882no—
406C4⋊C4C2×C22⋊D6112yes—
407C4⋊C4C2×C22⋊D6112yes—
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—1224no—
422C23—4884no—
423C23—4888no—
424C23—4888no—
425C23—4884no—
426C23—2448no—
427C23—2442no—
428C23—2442no—
429C23—2442no—
430C23—2442no—
431C23—2442no—
432C23—2442no—
433C23#1451974912yes—
434C23#1451980012yes—
435C23#1451980912yes—
436C23C2×D4×D6111yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—42no—
440D4—1224no—
441D4—1224no—
442D4—2448no—
443D4—2442no—
444D4—2442no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—84no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4#1451974912yes—
468C2×C4C22×C22⋊D6111yescharacteristic
469C2×C4#41×D6112yes—
470C2×C4C2×#1175822411yescharacteristic
471C2×C4#1451982312yes—
472C8—41no—
473C8—41no—
474C8—2442no—
475C22—24no—
476C22—22no—
477C22—24no—
478C22—22no—
479C22—48816no—
480C22—4882no—
481C22—4882no—
482C22—4882no—
483C22—4882no—
484C22—2444no—
485C22—2444no—
486C22—2444no—
487C22#1683998614yes—
488C22C2×#1451974911yescharacteristic
489C22#1684040412yes—
490C22C2×#1451980011yescharacteristic
491C22C2×#1451980911yescharacteristic
492C22#1684056612yes—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—81no—
497C4—81no—
498C4—4882no—
499C4—2442no—
500C4—2442no—
501C4#1684026911yescharacteristic
502C4#1684032611yescharacteristic
503C4#1684043512yes—
504C2—24no—
505C2—4884no—
506C2#2144920511yescharacteristic
507C2C2×#1683998611yescharacteristic
508C2#2145532411yescharacteristic
509C2#2145766712yes—
510C2#2145789812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#484C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenterC2×#484D61
hypocenterC61#24479
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
3. #21448696C2semidirect
4. #21447202C2semidirect
5. #21447057C2semidirect
6. #21446748C2semidirect
7. #21446438C2semidirect
8. #21446243C2semidirect
9. #21445996C2semidirect
11. C2×C61×#484C2semidirect
15. C2×#14519706C22semidirect
16. C2×#14519701C22semidirect
24. #16839661C22semidirect
30. C4⋊C4×Dic61C22semidirect
33. #16839524C22semidirect
34. #16839516C22semidirect
38. C2×#107×C61C22semidirect
39. C2×#107×C61C22semidirect
40. C2×#103×C61C22semidirect
41. C2×#103×C61C22semidirect
249. C2×#484D61semidirect
273. C2×#107C2×D61semidirect
274. C2×#107C2×D61semidirect
277. C2×#103C2×D61semidirect
278. C2×#103C2×D61semidirect
298. C61#24479semidirect

Maximal subgroups

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

subgroupindex
C22×#145196972
#214494712
#214486962
#214472022
#214470572
#214467482
#214464382
#214462432
#214459962
#214457512
C2×C61×#4842
#2447961