Finite Group Databasebeta

← #29718383

Subgroups — #29718383

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×#16840095C211yesmaximal, characteristic, complemented
2C22×#14519698C211yesmaximal, characteristic
3#21448694C211yesmaximal, characteristic, complemented
4#21447207C212yesmaximal, complemented
5#21447055C212yesmaximal, complemented
6#21446752C212yesmaximal, complemented
7#21446436C212yesmaximal, complemented
8#21446243C211yesmaximal, characteristic, complemented
9#21445999C211yesmaximal, characteristic, complemented
10#21445750C211yesmaximal, characteristic
11C61×#2435C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14#16840095—24no—
15C2×#14519710C2211yescharacteristic, complemented
16C2×#14519710C2211yescharacteristic, complemented
17C2×C4⋊#92418C2212yescomplemented
18C2×#14519701C2211yescharacteristic, complemented
19C2×#14519701C2211yescharacteristic, complemented
20C2×#14519701C2212yescomplemented
21C2×#14519698—24no—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×#14519698C2212yes—
25C2×#14519698C2212yes—
26C2×C22⋊#92418C2211yescharacteristic
27#16839660C2212yescomplemented
28#16839645—22no—
29#16839622C2211yescharacteristic
30#16839620—21no—
31C4×C4⋊Dic61C2212yescomplemented
32C22⋊C4×Dic61—21no—
33#16839524C2212yescomplemented
34#16839517C2212yescomplemented
35C23×D4×C61C2211yescharacteristic
36C2×#124×C61C2212yes—
37C2×C22⋊Q8×C61C2211yescharacteristic
38C61×#394—22no—
39C61×#390C2212yes—
40C61×#379—21no—
41C61×#379—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C23×Dic61—21no—
44C22×#92418—22no—
45C22×#92418—42no—
46C22×#92418C2311yescharacteristic
47C22×#92418C2311yescharacteristic
48C22×#92418C2312yes—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61D414yes—
52C2×C4⋊Dic61C2312yes—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×#11758196—22no—
56C2×C22⋊Dic61—21no—
57C2×C4×Dic61—22no—
58#14519710—22no—
59#14519710—22no—
60C4⋊#92418—42no—
61#14519701—22no—
62#14519701—22no—
63#14519701—42no—
64#14519698—24no—
65#14519698—24no—
66#14519698—24no—
67#14519698—42no—
68C22⋊#92418—22no—
69#14519684—22no—
70#14519684—22no—
71#14519684—21no—
72C25×C61—21no—
73C22×Q8×C61C2311yescharacteristic
74C22×D4×C61—24no—
75C22×D4×C61—22no—
76C22×D4×C61—21no—
77C22×D4×C61—22no—
78C22×D4×C61—42no—
79C22×D4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81#124×C61—24no—
82#124×C61—42no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C22⋊C4×C61—21no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61D411yescharacteristic
93C2×C22⋊C4×C61D411yescharacteristic
94C2×C42×C61C2312yes—
95#95×C61D412yes—
96C24×C61—22no—
97C24×C61—44no—
98C24×C61—42no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61—82no—
102C24×C61C2×D411yescharacteristic
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×D4×C61—28no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—48no—
114C2×D4×C61—44no—
115C2×D4×C61—44no—
116C2×D4×C61—42no—
117C2×D4×C61#4114yes—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61#4112yes—
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—24no—
131C22×Dic61—22no—
132C22×Dic61—82no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—42no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—44no—
139C22⋊C4×C61—44no—
140C22⋊C4×C61—42no—
141C22⋊C4×C61—82no—
142C22⋊C4×C61—82no—
143C42×C61—22no—
144C42×C61—22no—
145C22⋊Dic61—42no—
146C4⋊Dic61—24no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61C2×D412yes—
152C4⋊Dic61C2×D412yes—
153#11758196—82no—
154C4×Dic61—82no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—22no—
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
181Q8×C61—41no—
182Q8×C61—41no—
183Q8×C61—42no—
184D4×C61—24no—
185D4×C61—24no—
186D4×C61—48no—
187D4×C61—42no—
188D4×C61—42no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—24no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61#12112yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4112yes—
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)12yes—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—84no—
216#92418—41no—
217#92418—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—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×#484—611no—
250C22×#102—611no—
251#2435D6111yescharacteristic
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—84no—
262C2×C61#241111yescharacteristic
263C2×C61C2×#46411yescharacteristic
264C2×C61#304111yescharacteristic
265C2×C61#329712yes—
266C2×C61#331412yes—
267C23×D4C2×D6111yescharacteristic
268C23×C8—611no—
269C2×#124C2×D6112yes—
270C2×C22⋊Q8C2×D6111yescharacteristic
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—612no—
282C2×#102—612no—
283C2×#102—1224no—
284C2×C22⋊C8—611no—
285#394—22no—
286#390C2×D6112yes—
287#389—612no—
288#387—1222no—
289#380—611no—
290#380—1221no—
291#379—21no—
292#379—21no—
293C4×C4⋊C4—612no—
294C4×C22⋊C4—1221no—
295C4×C22⋊C4C2×D6111yescharacteristic
296#340—612no—
297#336—612no—
298C61#2447911yescharacteristic, complemented, Sylow
299C25—21no—
300C22×Q8C22×D6111yescharacteristic
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—611no—
311C22×C8—611no—
312C22×C8—1222no—
313C22×C8—2442no—
314#124—24no—
315#124—42no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—614no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—611no—
322C2×C4⋊C4—612no—
323C2×C4⋊C4—1222no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1222no—
326C2×C4⋊C4C22×D6111yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—1221no—
333C2×C22⋊C4C22⋊D6111yescharacteristic
334C2×C22⋊C4C22⋊D6111yescharacteristic
335C2×C42—1222no—
336C2×C42C22×D6112yes—
337#107—1222no—
338#107—1222no—
339C4⋊C8—2442no—
340#103—1222no—
341#103—1222no—
342#103—2442no—
343#102—1224no—
344#102—1224no—
345#102—1224no—
346#102—2442no—
347C22⋊C8—1222no—
348#95—1222no—
349#95—1222no—
350#95—1221no—
351#95C22⋊D6112yes—
352C24—22no—
353C24—44no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—82no—
358C24C2×C22⋊D6111yescharacteristic
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—22no—
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—21no—
377C22×C4—21no—
378C22×C4—1224no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1222no—
382C22×C4—4882no—
383C22×C4D4×D6112yes—
384C22×C4#1175822412yes—
385C22×C4C2×C22⋊D6112yes—
386C22×C4C2×C22⋊D6112yes—
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389C2×C8—1221no—
390C2×C8—1221no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—1222no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—2444no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—612no—
401C4⋊C4—612no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1224no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—4882no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—2442no—
416C42—22no—
417C42—22no—
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#1451974812yes—
434C23#1451979712yes—
435C23#1451980712yes—
436C23C22×C22⋊D6111yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—42no—
440D4—24no—
441D4—24no—
442D4—48no—
443D4—42no—
444D4—42no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—4884no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4#1451974812yes—
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C2×#1175822412yes—
470C2×C4C2×#1175822411yescharacteristic
471C2×C4#1451982212yes—
472C8—2441no—
473C8—2441no—
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#1683998114yes—
488C22C2×#1451974811yescharacteristic
489C22#1684040612yes—
490C22C2×#1451979711yescharacteristic
491C22C2×#1451980711yescharacteristic
492C22#1684056512yes—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—81no—
497C4—81no—
498C4—4882no—
499C4—2442no—
500C4—2442no—
501C4#1684026811yescharacteristic
502C4#1684032711yescharacteristic
503C4#1684044312yes—
504C2—24no—
505C2—84no—
506C2#2144920011yescharacteristic
507C2C2×#1683998111yescharacteristic
508C2#2145531811yescharacteristic
509C2#2145766912yes—
510C2#2145790012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2435C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
hypercenter#2435D61
hypocenterC61#24479
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
1. C2×#16840095C2semidirect
3. #21448694C2semidirect
4. #21447207C2semidirect
5. #21447055C2semidirect
6. #21446752C2semidirect
7. #21446436C2semidirect
8. #21446243C2semidirect
9. #21445999C2semidirect
15. C2×#14519710C22semidirect
16. C2×#14519710C22semidirect
17. C2×C4⋊#92418C22semidirect
18. C2×#14519701C22semidirect
19. C2×#14519701C22semidirect
20. C2×#14519701C22semidirect
27. #16839660C22semidirect
31. C4×C4⋊Dic61C22semidirect
33. #16839524C22semidirect
34. #16839517C22semidirect
298. C61#24479semidirect

Maximal subgroups

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

subgroupindex
C2×#168400952
C22×#145196982
#214486942
#214472072
#214470552
#214467522
#214464362
#214462432
#214459992
#214457502
C61×#24352
#2447961