Finite Group Databasebeta

← #29837055

Subgroups — #29837055

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458247C211yesmaximal, characteristic, complemented
2#21457498C211yesmaximal, characteristic
3#21448057C211yesmaximal, characteristic, complemented
4#21448025C211yesmaximal, characteristic, complemented
5#21448010C211yesmaximal, characteristic, complemented
6#21447964C211yesmaximal, characteristic, complemented
7#21447926C211yesmaximal, characteristic, complemented
8#21447915C211yesmaximal, characteristic, complemented
9#21447884C211yesmaximal, characteristic, complemented
10#21447759C211yesmaximal, characteristic, complemented
11#21447749C211yesmaximal, characteristic, complemented
12#21447680C211yesmaximal, characteristic, complemented
13#21447644C211yesmaximal, characteristic, complemented
14#21447533C211yesmaximal, characteristic, complemented
15C61×#2138C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840480—21no—
18#16840400—21no—
19C2×#14519723C2211yescharacteristic, complemented
20C2×#14519722C2211yescharacteristic
21C2×#14519721C2211yescharacteristic
22C2×#14519716C2211yescharacteristic
23#16840166C2211yescharacteristic, complemented
24#16840147C2211yescharacteristic
25#16840106—21no—
26#16840103—21no—
27#16840046—21no—
28#16840004C2211yescharacteristic
29#16840003C2211yescharacteristic, complemented
30#16840000C2211yescharacteristic
31#16839998C2211yescharacteristic
32#16839979—21no—
33#16839911—21no—
34#16839907—21no—
35#16839856C2211yescharacteristic, complemented
36D8×Dic61C2211yescharacteristic
37C8×C22⋊D61C2211yescharacteristic
38#16839799C2211yescharacteristic
39#16839590C2211yescharacteristic, complemented
40#16839589C2211yescharacteristic, complemented
41#16839587C2211yescharacteristic, complemented
42#16839580C2211yescharacteristic, complemented
43#16839567C2211yescharacteristic, complemented
44#16839566C2211yescharacteristic, complemented
45#16839564C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic, complemented
50#16839541C2211yescharacteristic
51#16839537C2211yescharacteristic, complemented
52C2×#135×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic, complemented
54C61×#427C2211yescharacteristic, complemented
55C61×#358C2211yescharacteristic, complemented
56C61×#358C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×D4×D61—21no—
60C2×Dic244C2311yescharacteristic
61C2×#11758201—21no—
62C2×D488C2311yescharacteristic
63C2×#11758199—21no—
64D8×D61—22no—
65D8×D61—41no—
66C2×C8×D61C2311yescharacteristic
67#14519776—21no—
68#14519771—41no—
69#14519770—41no—
70#14519766—21no—
71#14519763—41no—
72#14519760—41no—
73#14519757—21no—
74#14519756—21no—
75#14519753—21no—
76#14519751—21no—
77#14519749—41no—
78#14519745—41no—
79C4×Dic122—41no—
80#14519742—21no—
81D4×Dic61—21no—
82C4×C22⋊D61D411yescharacteristic
83#14519735—21no—
84C2×#11758198—41no—
85#14519723—41no—
86#14519722—41no—
87#14519721—22no—
88#14519716—41no—
89C2×#11758185C2311yescharacteristic
90C2×#11758185C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519708—21no—
95#14519706D411yescharacteristic
96#14519702—21no—
97#14519701—21no—
98#14519700—21no—
99#14519700—21no—
100#14519699—21no—
101#14519698—21no—
102C22⋊#92418D411yescharacteristic
103#14519689D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×#41×C61—21no—
106#135×C61—22no—
107#135×C61—41no—
108C2×Q16×C61C2311yescharacteristic
109C2×SD16×C61—21no—
110C2×D8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#117×C61—21no—
113C2×C16×C61—41no—
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#106×C61—21no—
119C23×D61—41no—
120#41×C61—42no—
121#41×C61—81no—
122C2×Q8×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—41no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C2×C22⋊D61—41no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61D811yescharacteristic
130C22×Dic61—41no—
131D4×D61—42no—
132D4×D61—81no—
133C2×D244—21no—
134C2×C4×D61C2×D411yescharacteristic
135C2×Dic122—21no—
136C2×Dic122—41no—
137Q16×C61—21no—
138Q16×C61—21no—
139SD16×C61—42no—
140D8×C61—21no—
141D8×C61#4112yes—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2411yescharacteristic
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C22⋊C4×C61—41no—
149C42×C61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61—41no—
152C22⋊Dic61D811yescharacteristic
153#11758201—81no—
154#11758199—42no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198D811yescharacteristic
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—41no—
165#11758196—41no—
166#11758196—41no—
167#11758196—41no—
168#11758196—41no—
169#11758196D811yescharacteristic
170C4×Dic61—41no—
171C4×Dic61—41no—
172C4×Dic61C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175Dic244—21no—
176Dic244—21no—
177D488—21no—
178D488—21no—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61—41no—
186C23×C61C2×D811yescharacteristic
187C22×D61—41no—
188C22×D61—82no—
189C22×D61—82no—
190C22×D61—161no—
191C22×D61C2×D811yescharacteristic
192Q8×C61—41no—
193Q8×C61—41no—
194D4×C61—22no—
195D4×C61—41no—
196D4×C61—82no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—82no—
203C2×C4×C61C2×D811yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C22⋊D61—41no—
206C22⋊D61—82no—
207C22⋊D61—161no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—161no—
212C2×Dic61C2×D811yescharacteristic
213C2×Dic61C2×D811yescharacteristic
214C2×Dic61C2×D811yescharacteristic
215D244—41no—
216D244—41no—
217C4×D61—21no—
218C4×D61—21no—
219Dic122—41no—
220Dic122—41no—
221Dic122—82no—
222C8×C61—21no—
223C8×C61C2×#4111yescharacteristic
224C8×C61H(5,𝔽2)11yescharacteristic
225#92418—21no—
226#92418—21no—
227#33878—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—42no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61C22×D811yescharacteristic
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—164no—
242C4×C61—21no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61—81no—
246C4×C61#54611yescharacteristic
247C4×C61#57711yescharacteristic
248Dic61—21no—
249Dic61—21no—
250Dic61—21no—
251Dic61—81no—
252Dic61—81no—
253Dic61—81no—
254#3342—611no—
255#3280—611no—
256#2243—611no—
257#2237—611no—
258#2234—611no—
259#2226—611no—
260#2219—611no—
261#2217—611no—
262#2211—611no—
263#2189—611no—
264#2187—611no—
265#2175—611no—
266#2168—611no—
267#2140—611no—
268#2138D6111yescharacteristic, complemented
269C2×C61—41no—
270C2×C61—82no—
271C2×C61#329411yescharacteristic
272C2×C61#341411yescharacteristic
273C2×C61#341611yescharacteristic
274D61—41no—
275D61—161no—
276C2×#135C2×D6111yescharacteristic
277C22×D8—611no—
278#556—1221no—
279#546—1221no—
280C2×Q32—611no—
281C2×SD32—611no—
282C2×SD32—611no—
283C2×M32—611no—
284C22×C16C2×D6111yescharacteristic, complemented
285#500—611no—
286#495—611no—
287#485—1221no—
288#485—1221no—
289#475—1221no—
290#467—611no—
291#467—611no—
292#467—611no—
293#466—611no—
294#463—1221no—
295#450—1221no—
296#449—1221no—
297C4×Q16—611no—
298C4×D8—611no—
299C8×D4—611no—
300#427—611no—
301#427C2×D6111yescharacteristic, complemented
302#367—611no—
303#367—611no—
304#366—611no—
305C4⋊C16—611no—
306#358—611no—
307#358—611no—
308#358C2×D6111yescharacteristic, complemented
309#358C2×D6111yescharacteristic, complemented
310#357—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357C2×D6111yescharacteristic
315#357C2×D6111yescharacteristic
316C22⋊C16—611no—
317C22⋊C16—611no—
318#346—611no—
319C61#3387811yescharacteristic, complemented, Sylow
320C2×#41—21no—
321C22×D4—1221no—
322#135—22no—
323#135—41no—
324C2×Q16—611no—
325C2×Q16C22×D6111yescharacteristic
326C2×SD16—21no—
327C2×SD16—1221no—
328C2×D8—611no—
329C2×D8—1221no—
330C2×D8—1222no—
331C2×D8—2441no—
332C2×D8C22×D6111yescharacteristic
333C22×C8—611no—
334C22×C8C22×D6111yescharacteristic
335C4⋊Q8—1221no—
336#126—2441no—
337#126—2441no—
338#124—1221no—
339#124—2441no—
340#123—2441no—
341C22⋊Q8—1221no—
342C22⋊Q8—1221no—
343C22⋊Q8—1221no—
344#121—1221no—
345#121—2441no—
346#120—2441no—
347C4×Q8—1221no—
348C4×Q8—2441no—
349C4×D4—611no—
350C4×D4—1221no—
351#117—21no—
352#117—1221no—
353C2×C22⋊C4—2441no—
354Q32—2441no—
355SD32—1222no—
356SD32—2441no—
357M32—2441no—
358C2×C16—41no—
359C2×C16—611no—
360C2×C16—611no—
361C2×C16C22×D6111yescharacteristic
362C2×C16C22×D6111yescharacteristic
363#107—611no—
364#107—611no—
365#107—1221no—
366#107C22×D6111yescharacteristic
367#107C22×D6111yescharacteristic
368#106—21no—
369#106—1221no—
370C4⋊C8—611no—
371#103—1221no—
372#103—1221no—
373#103—1221no—
374#103—1221no—
375#102—1221no—
376#102—1221no—
377C22⋊C8—611no—
378C22⋊C8—611no—
379C4×C8—611no—
380C24—2441no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—1221no—
385C2×Q8—2441no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—611no—
389C2×D4—1221no—
390C2×D4—4881no—
391C2×D4—2441no—
392C2×D4—2442no—
393C2×D4—2441no—
394C22×C4—41no—
395C22×C4—611no—
396C22×C4—2441no—
397C22×C4D4×D6111yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1221no—
401Q16—1221no—
402SD16—42no—
403SD16—4881no—
404D8—21no—
405D8—1221no—
406D8—1221no—
407D8—2442no—
408D8#1175822212yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—611no—
412C2×C8—611no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8D4×D6111yescharacteristic
416C2×C8C23×D6111yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—611no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—2441no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C22⋊C4—41no—
428C22⋊C4—611no—
429C22⋊C4—611no—
430C22⋊C4—2441no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C42—41no—
439C42—611no—
440C42—2441no—
441C42—2441no—
442C16—21no—
443C16—21no—
444C16—1221no—
445C16—1221no—
446C23—41no—
447C23—611no—
448C23—9761no—
449C23—4882no—
450C23—4882no—
451C23—2441no—
452C23D8×D6111yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—4882no—
456Q8—2441no—
457Q8—2441no—
458D4—22no—
459D4—41no—
460D4—82no—
461D4—9761no—
462D4—4882no—
463D4—2441no—
464D4—2441no—
465D4—2441no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—9761no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4D8×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C8—21no—
484C8—1221no—
485C8—1221no—
486C8C2×#1175822211yescharacteristic
487C8#1451982211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—1221no—
495C22—1221no—
496C22—1221no—
497C22—9764no—
498C22—4881no—
499C22—4881no—
500C22—4881no—
501C22C2×D8×D6111yescharacteristic
502C4—21no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—1221no—
507C4—1221no—
508C4—1221no—
509C4—4881no—
510C4—4881no—
511C4—4881no—
512C4#1684040211yescharacteristic
513C4#1684056811yescharacteristic
514C2—41no—
515C2—82no—
516C2—9761no—
517C2—2441no—
518C2#2145762811yescharacteristic
519C2#2145895211yescharacteristic
520C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radical#21457498C2
hypercenter#2138D61
hypocenterC61#33878
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #21458247C2semidirect
3. #21448057C2semidirect
4. #21448025C2semidirect
5. #21448010C2semidirect
6. #21447964C2semidirect
7. #21447926C2semidirect
8. #21447915C2semidirect
9. #21447884C2semidirect
10. #21447759C2semidirect
11. #21447749C2semidirect
12. #21447680C2semidirect
13. #21447644C2semidirect
14. #21447533C2semidirect
15. C61×#2138C2semidirect
19. C2×#14519723C22semidirect
23. #16840166C22semidirect
29. #16840003C22semidirect
35. #16839856C22semidirect
39. #16839590C22semidirect
40. #16839589C22semidirect
41. #16839587C22semidirect
42. #16839580C22semidirect
43. #16839567C22semidirect
44. #16839566C22semidirect
49. C22⋊#11758185C22semidirect
51. #16839537C22semidirect
53. C22×C16×C61C22semidirect
54. C61×#427C22semidirect
55. C61×#358C22semidirect
56. C61×#358C22semidirect
268. #2138D61semidirect
284. C22×C16C2×D61semidirect
301. #427C2×D61semidirect
308. #358C2×D61semidirect
309. #358C2×D61semidirect
319. C61#33878semidirect

Maximal subgroups

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

subgroupindex
#214582472
#214574982
#214480572
#214480252
#214480102
#214479642
#214479262
#214479152
#214478842
#214477592
#214477492
#214476802
#214476442
#214475332
C61×#21382
#3387861