Finite Group Databasebeta

← #29834827

Subgroups — #29834827

Order 15616.

584 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455398C211yesmaximal, characteristic
2#427×D61C211yesmaximal, characteristic, complemented
3#21448030C211yesmaximal, characteristic, complemented
4#21448029C211yesmaximal, characteristic, complemented
5#21448028C211yesmaximal, characteristic, complemented
6#21448027C211yesmaximal, characteristic, complemented
7#21448009C211yesmaximal, characteristic, complemented
8#21448008C211yesmaximal, characteristic, complemented
9#21448007C211yesmaximal, characteristic, complemented
10#21447606C211yesmaximal, characteristic, complemented
11#21447604C211yesmaximal, characteristic, complemented
12#21447528C211yesmaximal, characteristic, complemented
13#21447528C211yesmaximal, characteristic, complemented
14#21447428C211yesmaximal, characteristic, complemented
15C61×#2234C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18#16840272—21no—
19#117×D61—21no—
20C2×#14519716C2211yescharacteristic
21C2×#14519716C2211yescharacteristic
22#16840004C2211yescharacteristic
23#16840003C2211yescharacteristic
24#16839995C2211yescharacteristic
25#16839990—21no—
26#16839989—21no—
27#16839800C2211yescharacteristic
28#16839799C2211yescharacteristic
29#16839799C2211yescharacteristic
30#16839799C2211yescharacteristic
31#16839798—21no—
32#16839797—21no—
33#107×D61C2211yescharacteristic
34#107×D61C2211yescharacteristic
35#106×D61—21no—
36#16839755—21no—
37#16839590C2211yescharacteristic, complemented
38#16839589C2211yescharacteristic, complemented
39#16839589C2211yescharacteristic, complemented
40#16839588C2211yescharacteristic, complemented
41#16839587C2211yescharacteristic, complemented
42#16839587C2211yescharacteristic, complemented
43#16839567C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839565C2211yescharacteristic, complemented
46#16839565C2211yescharacteristic, complemented
47#16839563C2211yescharacteristic, complemented
48#16839563C2211yescharacteristic, complemented
49C22⋊#11758185C2211yescharacteristic
50#16839541C2211yescharacteristic
51#16839541C2211yescharacteristic
52C61×#466C2211yescharacteristic
53C61×#427C2211yescharacteristic
54C61×#367C2211yescharacteristic, complemented
55C61×#366C2211yescharacteristic, complemented
56C61×#357C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic
59C2×#11758224—21no—
60C22×C4×D61D411yescharacteristic
61#14519796—22no—
62#14519796—22no—
63C2×Dic244C2311yescharacteristic
64C2×#11758200—21no—
65C22×#92418D411yescharacteristic
66C2×C8×D61—22no—
67C2×C8×D61—22no—
68C2×C8×D61D411yescharacteristic
69C2×C8×D61D411yescharacteristic
70C2×C8×D61C2311yescharacteristic
71C2×C8×D61C2311yescharacteristic
72#14519759—41no—
73#14519757—21no—
74#14519756—21no—
75#14519748—21no—
76#14519745—41no—
77D4×Dic61—41no—
78#14519737—41no—
79#14519736—21no—
80#14519735—21no—
81#14519735—21no—
82#14519735—21no—
83#14519734—41no—
84#14519733—41no—
85C4⋊C4×D61—21no—
86C4⋊C4×D61—21no—
87C22⋊C4×D61—41no—
88C42×D61—41no—
89#14519716—22no—
90#14519716—22no—
91C2×#11758185C2311yescharacteristic
92C2×#11758185C2311yescharacteristic
93#14519711C2311yescharacteristic
94#14519711C2311yescharacteristic
95#14519711C2311yescharacteristic
96#14519710—21no—
97#14519710—21no—
98#14519709—21no—
99#14519708—21no—
100#14519708—21no—
101#14519708—21no—
102#14519702—21no—
103#14519700—21no—
104#14519698—21no—
105C2×D8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107#121×C61—21no—
108#117×C61—21no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#106×C61—21no—
115#102×C61—21no—
116C23×D61D811yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—41no—
123C22×Dic61D811yescharacteristic
124#11758224—42no—
125#11758224—42no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61D811yescharacteristic
131C2×C4×D61D811yescharacteristic
132C2×C4×D61C2×D411yescharacteristic
133C2×C4×D61C2×D411yescharacteristic
134C2×Dic122—21no—
135D8×C61—21no—
136D8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2411yescharacteristic
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C22⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C42×C61—41no—
147C22⋊Dic61—41no—
148C22⋊Dic61—41no—
149C22⋊Dic61—41no—
150#11758200—42no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154#11758198—41no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—41no—
159#11758196—41no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C2×#92418—22no—
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169Dic244—21no—
170Dic244—21no—
171C8×D61—22no—
172C8×D61—21no—
173C8×D61—21no—
174C8×D61—22no—
175C8×D61#4112yes—
176C8×D61#4112yes—
177C16×C61—21no—
178C16×C61—21no—
179#11758185—21no—
180#11758185—21no—
181C23×C61—41no—
182C23×C61C2×D811yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61C2×D811yescharacteristic
187C22×D61C2×D811yescharacteristic
188D4×C61—41no—
189D4×C61—41no—
190D4×C61—82no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—82no—
199C22⋊D61—82no—
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—82no—
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208C4×D61—22no—
209C4×D61—21no—
210C4×D61—21no—
211C4×D61—22no—
212C4×D61—82no—
213C4×D61—82no—
214C4×D61#12312yes—
215C4×D61#13712yes—
216Dic122—41no—
217Dic122—41no—
218C8×C61—21no—
219C8×C61C2×#4111yescharacteristic
220C8×C61C2×#4111yescharacteristic
221#92418—21no—
222#92418#12312yes—
223#33647—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61C22×D811yescharacteristic
231C2×D61—21no—
232C2×D61—21no—
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—21no—
236C2×D61—44no—
237C2×D61#48012yes—
238C4×C61—21no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61—81no—
242C4×C61C2×#12311yescharacteristic
243C4×C61C2×#13711yescharacteristic
244Dic61—21no—
245Dic61—81no—
246Dic61—81no—
247Dic61—81no—
248Dic61—81no—
249Dic61#48012yes—
250#3050—611no—
251C2×#427—611no—
252#2237—611no—
253#2237—611no—
254#2237—611no—
255#2237—611no—
256#2234—611no—
257#2234—611no—
258#2234—611no—
259#2234D6111yescharacteristic, complemented
260#2159—611no—
261#2159—611no—
262#2139—611no—
263#2139—611no—
264#2112—611no—
265C2×C61—41no—
266C2×C61—161no—
267C2×C61C2×#48011yescharacteristic
268C2×C61#341411yescharacteristic
269C2×C61#341411yescharacteristic
270D61—22no—
271D61—41no—
272C2×#135—611no—
273C23×C8—611no—
274#525—1221no—
275C2×#117—1221no—
276C2×M32—611no—
277C2×M32—611no—
278#467—611no—
279#467—611no—
280#466—611no—
281#466C2×D6111yescharacteristic
282#465—1221no—
283#465—1221no—
284#427—611no—
285#427—611no—
286#427—611no—
287#427—611no—
288#427—1221no—
289#427—1221no—
290#427C2×D6111yescharacteristic
291C2×#107—611no—
292C2×#107—611no—
293C2×#106—1221no—
294#416—1221no—
295#367—611no—
296#367—611no—
297#367—611no—
298#367C2×D6111yescharacteristic, complemented
299#366—611no—
300#366—611no—
301#366—611no—
302#366C2×D6111yescharacteristic, complemented
303#358—611no—
304#358—611no—
305#358—611no—
306#358—611no—
307#357—611no—
308#357—611no—
309#357C2×D6111yescharacteristic, complemented
310#357C2×D6111yescharacteristic, complemented
311C22⋊C16—611no—
312C22⋊C16—611no—
313C22⋊C16—611no—
314C22⋊C16C2×D6111yescharacteristic
315C61#3364711yescharacteristic, complemented, Sylow
316C2×#41—1221no—
317C23×C4—611no—
318#135—1222no—
319#135—1222no—
320C2×Q16—611no—
321C2×SD16—1221no—
322C2×D8C22×D6111yescharacteristic
323C22×C8—611no—
324C22×C8—611no—
325C22×C8—611no—
326C22×C8—611no—
327C22×C8—611no—
328C22×C8—1222no—
329C22×C8—1222no—
330C22×C8C22×D6111yescharacteristic
331#123—2441no—
332C22⋊Q8—1221no—
333C22⋊Q8—1221no—
334#121—21no—
335#121—1221no—
336#120—2441no—
337C4×D4—2441no—
338C4×D4—2441no—
339#117—21no—
340#117—1221no—
341#117—1221no—
342#117—1221no—
343#117—1221no—
344#117—2441no—
345#117—2441no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C22⋊C4—2441no—
349C2×C42—2441no—
350M32—1222no—
351M32—1222no—
352C2×C16—611no—
353C2×C16—611no—
354C2×C16C22×D6111yescharacteristic
355C2×C16C22×D6111yescharacteristic
356#107—611no—
357#107—611no—
358#107—611no—
359#107—1221no—
360#107—1221no—
361#107C22×D6111yescharacteristic
362#107C22×D6111yescharacteristic
363#107C22×D6111yescharacteristic
364#106—21no—
365#106—1221no—
366#106—1221no—
367#106—1221no—
368#106—1221no—
369#103—1221no—
370#103—1221no—
371#102—21no—
372#102—1221no—
373C24—611no—
374#41—2442no—
375#41—2442no—
376C2×Q8—1221no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—2441no—
380C2×D4—2441no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—2441no—
389C22×C4—2441no—
390C22×C4—2441no—
391C22×C4D4×D6111yescharacteristic
392Q16—1221no—
393Q16—1221no—
394SD16—2442no—
395D8—21no—
396D8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—611no—
400C2×C8—612no—
401C2×C8—612no—
402C2×C8—611no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—1222no—
406C2×C8—1221no—
407C2×C8—1221no—
408C2×C8D4×D6111yescharacteristic
409C2×C8C23×D6111yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—1221no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C4⋊C4—2441no—
419C4⋊C4—2441no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C42—41no—
430C42—2441no—
431C42—2441no—
432C42—2441no—
433C42—2441no—
434C16—21no—
435C16—21no—
436C16—1221no—
437C16—1221no—
438C23—41no—
439C23—611no—
440C23—611no—
441C23—1222no—
442C23—2441no—
443C23—2441no—
444C23D8×D6111yescharacteristic
445Q8—2441no—
446Q8—2441no—
447D4—41no—
448D4—41no—
449D4—82no—
450D4—4882no—
451D4—4882no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—611no—
458C2×C4—612no—
459C2×C4—612no—
460C2×C4—611no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—4882no—
467C2×C4—4882no—
468C2×C4—4882no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4D8×D6111yescharacteristic
474C2×C4C2×D4×D6111yescharacteristic
475C8—21no—
476C8—612no—
477C8—1221no—
478C8#41×D6111yescharacteristic
479C8#41×D6111yescharacteristic
480C22—21no—
481C22—21no—
482C22—21no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—612no—
487C22—1221no—
488C22—1221no—
489C22—1221no—
490C22—1221no—
491C22—1221no—
492C22—2444no—
493C22C2×D8×D6111yescharacteristic
494C4—21no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—612no—
499C4—1221no—
500C4—4881no—
501C4—4881no—
502C4—4881no—
503C4—4881no—
504C4#123×D6111yescharacteristic
505C4#137×D6111yescharacteristic
506C2—41no—
507C2—161no—
508C2—1222no—
509C2—2441no—
510C2#480×D6111yescharacteristic
511C2#2145895111yescharacteristic
512C2#2145895211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2234C2
socleC22×C61C22×D8
prime radical#21455398C2
hypercenter#2234D61
hypocenterC61#33647
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
2. #427×D61C2semidirect
3. #21448030C2semidirect
4. #21448029C2semidirect
5. #21448028C2semidirect
6. #21448027C2semidirect
7. #21448009C2semidirect
8. #21448008C2semidirect
9. #21448007C2semidirect
10. #21447606C2semidirect
11. #21447604C2semidirect
12. #21447528C2semidirect
13. #21447528C2semidirect
14. #21447428C2semidirect
15. C61×#2234C2semidirect
37. #16839590C22semidirect
38. #16839589C22semidirect
39. #16839589C22semidirect
40. #16839588C22semidirect
41. #16839587C22semidirect
42. #16839587C22semidirect
43. #16839567C22semidirect
44. #16839567C22semidirect
45. #16839565C22semidirect
46. #16839565C22semidirect
47. #16839563C22semidirect
48. #16839563C22semidirect
54. C61×#367C22semidirect
55. C61×#366C22semidirect
56. C61×#357C22semidirect
57. C61×#357C22semidirect
259. #2234D61semidirect
298. #367C2×D61semidirect
302. #366C2×D61semidirect
309. #357C2×D61semidirect
310. #357C2×D61semidirect
315. C61#33647semidirect

Maximal subgroups

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

subgroupindex
#214553982
#427×D612
#214480302
#214480292
#214480282
#214480272
#214480092
#214480082
#214480072
#214476062
#214476042
#214475282
#214474282
C61×#22342
#3364761