Finite Group Databasebeta

← #29665432

Subgroups — #29665432

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456737C211yesmaximal, characteristic, complemented
2#21455295C211yesmaximal, characteristic, complemented
3#21455276C211yesmaximal, characteristic, complemented
4#21454928C211yesmaximal, characteristic, complemented
5#21445262C212yesmaximal, complemented
6#21445255C212yesmaximal, complemented
7#21445178C212yesmaximal, complemented
8#21445173C212yesmaximal, complemented
9#21444593C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C2×C61×#331C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519799—21no—
14#16840312C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16#16840138C2212yes—
17#16840108C2212yes—
18#16839985—21no—
19#16839978—21no—
20#16839973C2211yescharacteristic, complemented
21#16839968C2211yescharacteristic, complemented
22#16839963C2211yescharacteristic, complemented
23#16839959—21no—
24#16839959C2212yes—
25#16839958C2211yescharacteristic, complemented
26#16839954—21no—
27#16839954—21no—
28#16839954—21no—
29#16839954C2212yes—
30#16839896—22no—
31#16839886C2212yescomplemented
32#16839827C2212yescomplemented
33C8×C22⋊D61—22no—
34#16839815C2211yescharacteristic
35#16839814—21no—
36#16839786C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839767—21no—
39#16839759—21no—
40#16839514C2212yescomplemented
41#16839514C2212yescomplemented
42#16839507C2212yescomplemented
43#16839505C2212yescomplemented
44C2×#127×C61C2211yescharacteristic
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#331C2212yes—
48C61×#331C2212yes—
49C2×#11758222—21no—
50C2×#11758222D411yescharacteristic
51#14519803—44no—
52#14519799—44no—
53C2×Dic244—21no—
54C2×#11758190—21no—
55C2×#11758190—21no—
56C2×#11758200—22no—
57C2×D488—21no—
58C2×#11758199—22no—
59C2×#11758199—22no—
60C2×#11758199—22no—
61C2×#11758194—21no—
62C2×C8×D61—22no—
63#14519777C2311yescharacteristic
64#14519774C2311yescharacteristic
65#14519765—21no—
66#14519751—21no—
67#14519751—21no—
68C4×Dic122D411yescharacteristic
69C4×Dic122C2311yescharacteristic
70C4×D244—21no—
71C4×D244—21no—
72C4×D244D411yescharacteristic
73C4×D244C2311yescharacteristic
74C4×C22⋊D61—21no—
75C4×C22⋊D61—21no—
76#14519736D411yescharacteristic
77#14519734—21no—
78C42×D61—21no—
79C4⋊#92418—21no—
80C4⋊#92418D411yescharacteristic
81C4⋊#92418D411yescharacteristic
82C4⋊#92418C2312yes—
83#14519706—22no—
84#14519702—21no—
85#14519702—21no—
86#14519699—21no—
87#14519699—21no—
88#14519698—22no—
89#14519698—22no—
90#14519698—22no—
91#14519698—22no—
92C22⋊#92418D412yes—
93#14519689—22no—
94C4×#92418C2312yes—
95C8×Dic61—22no—
96C22×D4×C61—21no—
97C22×D4×C61—21no—
98C22×D4×C61—21no—
99C22×C8×C61—21no—
100C22×C8×C61—21no—
101#127×C61—21no—
102#127×C61—21no—
103#127×C61—21no—
104#127×C61C2312yes—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2311yescharacteristic
109C4⋊C8×C61C2311yescharacteristic
110C4⋊C8×C61C2312yes—
111C24×C61—41no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—81no—
121C2×D4×C61—81no—
122C2×D4×C61—82no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C2×C22⋊D61—21no—
127C2×C22⋊D61—41no—
128#11758222—22no—
129#11758222—42no—
130#11758222—42no—
131#11758222D814yescomplemented
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244C2×D411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—41no—
140C2×Dic122—21no—
141C2×Dic122C2×D411yescharacteristic
142C2×C8×C61—22no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—42no—
146C2×C8×C61—81no—
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61#4111yescharacteristic
150C42×C61C2411yescharacteristic
151C22⋊Dic61—21no—
152#11758200—44no—
153#11758199—44no—
154#11758199—82no—
155#11758199—82no—
156#11758198—21no—
157#11758198—21no—
158#11758198—41no—
159#11758198—41no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61C2×D411yescharacteristic
163C4⋊Dic61C2×D411yescharacteristic
164#11758196—21no—
165#11758196—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169#11758194—42no—
170C2×#92418—22no—
171C2×#92418C2×D412yes—
172C2×#92418C2×D412yes—
173Dic244—42no—
174D488—42no—
175#11758190—42no—
176#11758190—42no—
177C8×D61—82no—
178C23×C61—42no—
179C23×C61—82no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61#12111yescharacteristic
183C22×D61—21no—
184C22×D61—41no—
185D4×C61—42no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—82no—
190D4×C61—82no—
191D4×C61—82no—
192D4×C61—82no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61C2×D812yes—
200C2×C4×C61#13512yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C22⋊D61—24no—
205C22⋊D61—82no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—41no—
210D244—21no—
211D244—41no—
212D244—41no—
213D244—41no—
214D244—41no—
215D244—81no—
216D244C2×D812yes—
217C4×D61—24no—
218C4×D61—41no—
219C4×D61—41no—
220C4×D61—81no—
221C4×D61—82no—
222Dic122—21no—
223Dic122—41no—
224Dic122—41no—
225Dic122C2×D812yes—
226C8×C61—41no—
227C8×C61—41no—
228C8×C61—42no—
229#92418—22no—
230#92418—22no—
231#92418—42no—
232#19410—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61—82no—
238C22×C61—164no—
239C22×C61#46612yes—
240C22×C61C2×#12111yescharacteristic
241C2×D61—22no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61#46612yes—
251C4×C61C22×D811yescharacteristic
252C4×C61C2×#13511yescharacteristic
253C4×C61#57611yescharacteristic
254C4×C61#57711yescharacteristic
255Dic61—22no—
256Dic61—41no—
257Dic61—41no—
258Dic61—81no—
259#3205—611no—
260#3038—611no—
261#3035—611no—
262#2988—611no—
263#1682—612no—
264#1681—612no—
265#1668—612no—
266#1667—612no—
267#1571—611no—
268#1547—611no—
269C2×#331D6111yescharacteristic, complemented
270C2×C61—22no—
271C2×C61—162no—
272C2×C61#304511yescharacteristic
273C2×C61C2×#46611yescharacteristic
274C2×C61#306811yescharacteristic
275D61—42no—
276D61—161no—
277C2×AGL(1,ℤ/8)—1221no—
278C2×#135—1221no—
279#532—611no—
280C2×#127C2×D6111yescharacteristic
281C4×#41—611no—
282#493—612no—
283#486—612no—
284#464—1221no—
285#463—1221no—
286#462—611no—
287#461—611no—
288#460—612no—
289#460—611no—
290#460—1221no—
291#459—612no—
292#459—611no—
293#459—1221no—
294#459—1221no—
295#459—1221no—
296#447—1222no—
297C8×Q8—612no—
298C8×D4—612no—
299C8×D4—1222no—
300#431—611no—
301#431—1221no—
302#423—611no—
303C2×C4⋊C8—1221no—
304C2×C4⋊C8C2×D6111yescharacteristic
305C2×C4⋊C8C2×D6111yescharacteristic
306#418—1221no—
307#416—1221no—
308#335—612no—
309#335—612no—
310#332—612no—
311#331—612no—
312#331C2×D6112yes—
313#331C2×D6112yes—
314C61#1941011yescharacteristic, complemented, Sylow
315C2×#41—611no—
316C2×#41—1221no—
317C22×D4—21no—
318C22×D4—21no—
319C22×D4—21no—
320AGL(1,ℤ/8)—2444no—
321#135—2444no—
322C2×Q16—1221no—
323C2×SD16—1221no—
324C2×SD16—1221no—
325C2×SD16—1222no—
326C2×D8—1221no—
327C2×D8—1222no—
328C2×D8—1222no—
329C2×D8—1222no—
330C2×M16—1221no—
331C22×C8—21no—
332C22×C8—21no—
333C22×C8—1222no—
334C4⋊Q8—611no—
335#127—21no—
336#127—21no—
337#127—21no—
338#127—611no—
339#127C22×D6112yes—
340#124—1221no—
341#121—1221no—
342#121—1221no—
343C4×Q8—611no—
344C4×Q8—611no—
345C4×D4—611no—
346C4×D4—611no—
347C4×D4—1221no—
348C4×D4—1221no—
349C4×D4—1221no—
350C4×D4—1221no—
351#117—611no—
352#117—1221no—
353C2×C42—1221no—
354C2×C42C22×D6111yescharacteristic
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—611no—
358C4⋊C8—611no—
359C4⋊C8—612no—
360C4⋊C8—1222no—
361C4⋊C8—1221no—
362C4⋊C8C22×D6112yes—
363C4⋊C8C22×D6111yescharacteristic
364C4⋊C8C22×D6111yescharacteristic
365#103—1221no—
366#103—1221no—
367#102—1221no—
368#102—1221no—
369#102—1222no—
370#102—1222no—
371#102—1222no—
372#102—1222no—
373C22⋊C8—612no—
374C22⋊C8—1222no—
375C4×C8—612no—
376C4×C8—1222no—
377C24—41no—
378#41—614no—
379#41—1222no—
380#41—2442no—
381#41—2442no—
382C2×Q8—611no—
383C2×Q8—1221no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—82no—
395C2×D4—611no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—2441no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—2441no—
405C22×C4#1175822211yescharacteristic
406C22×C4D4×D6111yescharacteristic
407C22×C4D4×D6111yescharacteristic
408Q16—2442no—
409SD16—2442no—
410SD16—2442no—
411SD16—2444no—
412D8—4882no—
413D8—4882no—
414D8—2442no—
415D8—2444no—
416M16—2442no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—42no—
421C2×C8—81no—
422C2×C8—612no—
423C2×C8—612no—
424C2×C8—1222no—
425C2×C8—4882no—
426C4⋊C4—611no—
427C4⋊C4—611no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C22⋊C4—1221no—
433C22⋊C4—1221no—
434C22⋊C4—1221no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C42—1221no—
438C42—1221no—
439C42—1221no—
440C42#1175822211yescharacteristic
441C42D4×D6111yescharacteristic
442C42D4×D6111yescharacteristic
443C42C23×D6111yescharacteristic
444C23—42no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—1221no—
449C23—2441no—
450C23#1451975011yescharacteristic
451Q8—612no—
452Q8—1221no—
453Q8—2441no—
454Q8—2441no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—82no—
460D4—82no—
461D4—82no—
462D4—82no—
463D4—612no—
464D4—1224no—
465D4—1221no—
466D4—4881no—
467D4—4882no—
468D4—2441no—
469D4—2441no—
470D4—2441no—
471D4—2441no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—1221no—
476C2×C4—1224no—
477C2×C4—1221no—
478C2×C4—1221no—
479C2×C4—4881no—
480C2×C4—4882no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4#1451975011yescharacteristic
485C2×C4#1451975011yescharacteristic
486C2×C4#1451975011yescharacteristic
487C2×C4D8×D6112yes—
488C2×C4#1451979812yes—
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4C2×#1175822211yescharacteristic
492C8—41no—
493C8—41no—
494C8—42no—
495C8—1222no—
496C8—1222no—
497C8—2442no—
498C22—21no—
499C22—21no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—164no—
504C22—1222no—
505C22—4881no—
506C22—4881no—
507C22—4881no—
508C22—2441no—
509C22—2441no—
510C22#1683999712yes—
511C22C2×#1451975011yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—1222no—
516C4—4881no—
517C4—2441no—
518C4—2441no—
519C4#1683999712yes—
520C4C2×D8×D6111yescharacteristic
521C4C2×#1451979811yescharacteristic
522C4#1684056311yescharacteristic
523C4#1684056811yescharacteristic
524C2—22no—
525C2—162no—
526C2—9761no—
527C2—2442no—
528C2#2145536311yescharacteristic
529C2C2×#1683999711yescharacteristic
530C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#331C2
socleC22×C61C2×#121
hypercenterC2×#331D61
hypocenterC61#19410
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
1. #21456737C2semidirect
2. #21455295C2semidirect
3. #21455276C2semidirect
4. #21454928C2semidirect
5. #21445262C2semidirect
6. #21445255C2semidirect
7. #21445178C2semidirect
8. #21445173C2semidirect
9. #21444593C2semidirect
10. #21444494C2semidirect
11. C2×C61×#331C2semidirect
20. #16839973C22semidirect
21. #16839968C22semidirect
22. #16839963C22semidirect
25. #16839958C22semidirect
31. #16839886C22semidirect
32. #16839827C22semidirect
40. #16839514C22semidirect
41. #16839514C22semidirect
42. #16839507C22semidirect
43. #16839505C22semidirect
131. #11758222D8semidirect
269. C2×#331D61semidirect
314. C61#19410semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214552952
#214552762
#214549282
#214452622
#214452552
#214451782
#214451732
#214445932
#214444942
C2×C61×#3312
#1941061