Finite Group Databasebeta

← #29662801

Subgroups — #29662801

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456361C211yesmaximal, characteristic, complemented
2#21455828C211yesmaximal, characteristic
3C2×#16839964C211yesmaximal, characteristic
4#21454554C211yesmaximal, characteristic
5#21445331C212yesmaximal, complemented
6#21445317C212yesmaximal, complemented
7#21445207C212yesmaximal, complemented
8#21445193C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10#21444510C211yesmaximal, characteristic, complemented
11C61×#1549C211yesmaximal, characteristic, complemented
12C22×#11758201—21no—
13C2×M16×D61—21no—
14#16840319C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16#16840168—22no—
17#16840165C2212yescomplemented
18#16840158C2212yescomplemented
19#16840093—21no—
20#16840053C2211yescharacteristic
21#16840053C2211yescharacteristic
22#16839968C2211yescharacteristic
23#16839968C2211yescharacteristic
24#16839964—21no—
25#16839964—21no—
26#16839964C2212yes—
27#16839964C2212yes—
28#16839915—21no—
29C4×#11758190—21no—
30C4×#11758190—21no—
31#16839839C2212yes—
32#16839832—22no—
33#16839827C2212yes—
34#16839802—21no—
35#16839802—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C2×#14519701—21no—
38#16839725—21no—
39C4×#11758194C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839510C2212yescomplemented
43#16839507C2212yescomplemented
44C2×C4⋊Q8×C61C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C61×#332C2212yescomplemented
48C61×#332C2212yescomplemented
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758190—21no—
52C2×#11758190—21no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758201—44no—
56C2×#11758194D411yescharacteristic
57C2×#11758194D411yescharacteristic
58C2×#11758189—22no—
59M16×D61—44no—
60C22×#92418—21no—
61C2×C8×D61—22no—
62#14519777C2311yescharacteristic
63#14519777C2311yescharacteristic
64#14519769—21no—
65#14519756—21no—
66#14519756—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C4×D244—21no—
70C4×D244—21no—
71C4×D244D411yescharacteristic
72C4×D244D411yescharacteristic
73C4×D244C2311yescharacteristic
74C4×D244C2311yescharacteristic
75#14519736—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×#11758198—21no—
78#14519710—22no—
79#14519710—22no—
80#14519709—21no—
81#14519709—21no—
82#14519708—22no—
83#14519708—22no—
84C4⋊#92418—21no—
85C4⋊#92418C2312yes—
86#14519702—21no—
87#14519702—21no—
88#14519701—22no—
89#14519701—22no—
90#14519699—21no—
91#14519699—21no—
92#14519689—22no—
93#14519689—22no—
94#14519688D412yes—
95C4×#92418C2312yes—
96C22×Q8×C61—21no—
97C2×M16×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61—21no—
101C4⋊Q8×C61C2312yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2311yescharacteristic
107C4⋊C8×C61C2311yescharacteristic
108C4⋊C8×C61C2312yes—
109C22⋊C8×C61—21no—
110C4×C8×C61—21no—
111C23×D61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—42no—
114C2×Q8×C61—42no—
115C2×Q8×C61—41no—
116C22×C4×C61—41no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×Dic61—21no—
121C2×D244—22no—
122C2×D244—21no—
123C2×D244—41no—
124C2×D244C2×D411yescharacteristic
125C2×D244C2×D411yescharacteristic
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134M16×C61—42no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—41no—
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61#4111yescharacteristic
145C42×C61C2411yescharacteristic
146C22⋊Dic61—41no—
147#11758201—44no—
148#11758201—44no—
149#11758198—21no—
150#11758198—21no—
151#11758198—21no—
152#11758198—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61C2×D411yescharacteristic
159C4⋊Dic61C2×D411yescharacteristic
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163#11758194—22no—
164#11758194SD1614yescomplemented
165C2×#92418—22no—
166C2×#92418—42no—
167C2×#92418C2×D412yes—
168C2×#92418C2×D412yes—
169#11758190—42no—
170#11758190—42no—
171#11758189—44no—
172C8×D61—44no—
173C23×C61#12111yescharacteristic
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—81no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—82no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—82no—
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61C2×SD1612yes—
191C2×C4×C61#13712yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200D244—21no—
201D244—21no—
202D244—21no—
203D244—42no—
204D244AGL(1,ℤ/8)12yes—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—41no—
208C4×D61—41no—
209C4×D61—42no—
210C4×D61—42no—
211Dic122—41no—
212Dic122—41no—
213Dic122—81no—
214C8×C61—42no—
215C8×C61—41no—
216C8×C61—41no—
217#92418—22no—
218#92418—42no—
219#92418C2×SD1614yescomplemented
220#19164—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61#46112yes—
224C22×C61C2×#12111yescharacteristic
225C2×D61—22no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—41no—
230C2×D61—41no—
231C2×D61—84no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—82no—
236C4×C61#46112yes—
237C4×C61C22×SD1611yescharacteristic
238C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
239C4×C61C2×#13711yescharacteristic
240C4×C61#57511yescharacteristic
241Dic61—41no—
242Dic61—41no—
243Dic61—81no—
244Dic61—81no—
245#3165—611no—
246#3101—611no—
247C2×#461—611no—
248D4×M16—611no—
249#1693—612no—
250#1691—612no—
251#1673—612no—
252#1671—612no—
253#1565—611no—
254#1550—611no—
255#1549D6111yescharacteristic, complemented
256C2×C61—22no—
257C2×C61C2×#46111yescharacteristic
258C2×C61#305511yescharacteristic
259C2×C61#306911yescharacteristic
260D61—42no—
261D61—81no—
262C22×SD16—1221no—
263C22×M16—1221no—
264#533—611no—
265C2×C4⋊Q8C2×D6111yescharacteristic
266C2×C4×D4—611no—
267#501—1222no—
268#500—612no—
269#498—612no—
270#483—1221no—
271#476—611no—
272#476—611no—
273#461—612no—
274#461—612no—
275#461—611no—
276#461—611no—
277#461—1221no—
278#461—1221no—
279#450—1221no—
280C4×SD16—1221no—
281C4×SD16—1221no—
282#436—612no—
283#435—1222no—
284C8×D4—612no—
285#431C2×D6111yescharacteristic
286#428—1221no—
287#428—1221no—
288#423C2×D6111yescharacteristic
289C2×C4⋊C8—611no—
290C2×#103—1221no—
291#407—1221no—
292C4×M16—611no—
293#335—612no—
294#334—612no—
295#333—612no—
296#332—612no—
297#332C2×D6112yescomplemented
298#332C2×D6112yescomplemented
299C61#1916411yescharacteristic, complemented, Sylow
300C22×Q8—21no—
301C22×D4—611no—
302C23×C4—1221no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×SD16—2444no—
308C2×M16—21no—
309C2×M16—611no—
310C2×M16—611no—
311C2×M16—1222no—
312C2×M16—2444no—
313C22×C8—1222no—
314C22×C8—1221no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—611no—
319C4⋊Q8—611no—
320C4⋊Q8C22×D6112yes—
321#125—1221no—
322C22⋊Q8—1221no—
323C22⋊Q8—1221no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×D4—611no—
327C4×D4—611no—
328C4×D4—611no—
329C4×D4—611no—
330C4×D4—1221no—
331C4×D4—1221no—
332#117—1221no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C22⋊C4—1221no—
337C2×C42C22×D6111yescharacteristic
338#107—1222no—
339#107—1222no—
340#106—1222no—
341#106—1222no—
342#106—1221no—
343#106—1221no—
344C4⋊C8—21no—
345C4⋊C8—612no—
346C4⋊C8—1221no—
347C4⋊C8C22×D6112yes—
348C4⋊C8C22×D6111yescharacteristic
349C4⋊C8C22×D6111yescharacteristic
350#103—1221no—
351#103—1221no—
352#103—1222no—
353#103—1222no—
354#102—1221no—
355#102—1221no—
356C22⋊C8—21no—
357C22⋊C8—1222no—
358C22⋊C8—1222no—
359#97—612no—
360C4×C8—21no—
361C4×C8—612no—
362C24—1221no—
363C2×Q8—22no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—41no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×D4—611no—
370C2×D4—611no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—2441no—
374C22×C4—41no—
375C22×C4—1221no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—2442no—
381C22×C4—2442no—
382C22×C4#1175822211yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4D4×D6111yescharacteristic
385SD16—2442no—
386SD16—2442no—
387SD16—2444no—
388SD16—2444no—
389M16—42no—
390M16—614no—
391M16—1222no—
392M16—2444no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—612no—
397C2×C8—612no—
398C2×C8—1222no—
399C2×C8—2444no—
400C2×C8—2442no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—611no—
406C4⋊C4—611no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C22⋊C4—1221no—
417C22⋊C4—1221no—
418C22⋊C4—2441no—
419C42—2441no—
420C42#1175822211yescharacteristic
421C42D4×D6111yescharacteristic
422C42D4×D6111yescharacteristic
423C42C23×D6111yescharacteristic
424C23—1221no—
425C23—1221no—
426C23—4881no—
427C23—2442no—
428C23—2442no—
429C23#1451975011yescharacteristic
430Q8—42no—
431Q8—42no—
432Q8—82no—
433Q8—4881no—
434Q8—2441no—
435Q8—2441no—
436D4—612no—
437D4—1221no—
438D4—1221no—
439D4—1221no—
440D4—2442no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4#1451975011yescharacteristic
458C2×C4#1451975011yescharacteristic
459C2×C4#1451975011yescharacteristic
460C2×C4SD16×D6112yes—
461C2×C4#1451980912yes—
462C2×C4C2×D4×D6111yescharacteristic
463C2×C4C2×D4×D6111yescharacteristic
464C2×C4C2×#1175822211yescharacteristic
465C8—42no—
466C8—41no—
467C8—41no—
468C8—614no—
469C8—1222no—
470C8—2442no—
471C22—21no—
472C22—21no—
473C22—1222no—
474C22—4884no—
475C22—2441no—
476C22—2441no—
477C22—2441no—
478C22—2441no—
479C22—2441no—
480C22#1683996612yes—
481C22C2×#1451975011yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—82no—
486C4—4881no—
487C4—4881no—
488C4—2441no—
489C4—2441no—
490C4#1683996612yes—
491C4C2×SD16×D6111yescharacteristic
492C4AGL(1,ℤ/8)×D6111yescharacteristic
493C4C2×#1451980911yescharacteristic
494C4#1684055811yescharacteristic
495C2—22no—
496C2—4881no—
497C2—2442no—
498C2C2×#1683996611yescharacteristic
499C2#2145543611yescharacteristic
500C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1549D61
hypocenterC61#19164
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. #21456361C2semidirect
5. #21445331C2semidirect
6. #21445317C2semidirect
7. #21445207C2semidirect
8. #21445193C2semidirect
9. #21444572C2semidirect
10. #21444510C2semidirect
11. C61×#1549C2semidirect
17. #16840165C22semidirect
18. #16840158C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839510C22semidirect
43. #16839507C22semidirect
47. C61×#332C22semidirect
48. C61×#332C22semidirect
164. #11758194SD16semidirect
219. #92418C2×SD16semidirect
255. #1549D61semidirect
297. #332C2×D61semidirect
298. #332C2×D61semidirect
299. C61#19164semidirect

Maximal subgroups

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

subgroupindex
#214563612
#214558282
C2×#168399642
#214545542
#214453312
#214453172
#214452072
#214451932
#214445722
#214445102
C61×#15492
#1916461