Finite Group Databasebeta

← #29834816

Subgroups — #29834816

Order 15616.

590 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#21455401C211yesmaximal, characteristic
2#427×D61C211yesmaximal, characteristic, complemented
3#21448030C211yesmaximal, characteristic, complemented
4#21448029C211yesmaximal, characteristic, complemented
5#21448023C211yesmaximal, characteristic, complemented
6#21448018C211yesmaximal, characteristic, complemented
7#21448013C211yesmaximal, characteristic, complemented
8#21448012C211yesmaximal, characteristic, complemented
9#21448006C211yesmaximal, characteristic, complemented
10#21448001C211yesmaximal, characteristic, complemented
11#21447601C211yesmaximal, characteristic, complemented
12#21447527C211yesmaximal, characteristic, complemented
13#21447524C211yesmaximal, characteristic, complemented
14C2×#16839541C211yesmaximal, characteristic, complemented
15C61×#2158C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18#16840274—21no—
19#117×D61—21no—
20C22×#11758185C2211yescharacteristic
21#16840004C2211yescharacteristic
22#16840004C2211yescharacteristic
23#16839999C2211yescharacteristic
24#16839999C2211yescharacteristic
25#16839992—21no—
26#16839992—21no—
27#16839800C2211yescharacteristic
28#16839799C2211yescharacteristic
29#16839799C2211yescharacteristic
30#16839798—21no—
31#16839797—21no—
32#107×D61C2211yescharacteristic
33#107×D61C2211yescharacteristic
34#106×D61—21no—
35#16839756—21no—
36#16839589C2211yescharacteristic, complemented
37#16839589C2211yescharacteristic, complemented
38#16839587C2211yescharacteristic, complemented
39#16839587C2211yescharacteristic, complemented
40#16839567C2211yescharacteristic, complemented
41#16839567C2211yescharacteristic, complemented
42#16839565C2211yescharacteristic, complemented
43#16839565C2211yescharacteristic, complemented
44#16839564C2211yescharacteristic, complemented
45#16839564C2211yescharacteristic, complemented
46#16839562C2211yescharacteristic, complemented
47#16839562C2211yescharacteristic, complemented
48#16839541C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839541C2211yescharacteristic
51#16839541C2211yescharacteristic
52C22×C16×C61C2211yescharacteristic
53C61×#427C2211yescharacteristic
54C61×#427C2211yescharacteristic
55C61×#367C2211yescharacteristic, complemented
56C61×#367C2211yescharacteristic, complemented
57C61×#366C2211yescharacteristic, complemented
58C61×#366C2211yescharacteristic, complemented
59C2×#11758222—21no—
60C22×C4×D61D411yescharacteristic
61#14519799—22no—
62#14519799—22no—
63C2×Dic244C2311yescharacteristic
64C2×#11758190—21no—
65C2×D488C2311yescharacteristic
66C22×#92418D411yescharacteristic
67C2×C8×D61—22no—
68C2×C8×D61—22no—
69C2×C8×D61D411yescharacteristic
70C2×C8×D61D411yescharacteristic
71C2×C8×D61C2311yescharacteristic
72C2×C8×D61C2311yescharacteristic
73#14519761—41no—
74#14519757—21no—
75#14519757—21no—
76#14519752—21no—
77#14519752—21no—
78#14519747—41no—
79C4×D244—41no—
80C4×D244—41no—
81#14519736—21no—
82#14519735—21no—
83#14519735—21no—
84#14519734—41no—
85#14519733—41no—
86C4⋊C4×D61—21no—
87C4⋊C4×D61—21no—
88C22⋊C4×D61—41no—
89C42×D61—41no—
90C2×#11758185—22no—
91C2×#11758185C2311yescharacteristic
92C2×#11758185C2311yescharacteristic
93#14519711C2311yescharacteristic
94#14519711C2311yescharacteristic
95#14519710—21no—
96#14519710—21no—
97#14519709—21no—
98#14519708—21no—
99#14519708—21no—
100#14519700—21no—
101#14519700—21no—
102#14519697—21no—
103#14519697—21no—
104C22×C8×C61C2311yescharacteristic
105#117×C61—21no—
106#117×C61—21no—
107C2×C16×C61—22no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#106×C61—21no—
115#106×C61—21no—
116C23×D61D811yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C2×C22⋊D61—41no—
119C22×Dic61D811yescharacteristic
120#11758222—42no—
121#11758222—42no—
122C2×D244—21no—
123C2×D244—41no—
124C2×D244—41no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—41no—
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—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61#4112yes—
137C2×C8×C61#4112yes—
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C22⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C42×C61—41no—
146C42×C61—41no—
147C22⋊Dic61—41no—
148#11758198—41no—
149#11758198—41no—
150#11758198—41no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—41no—
157C4⋊Dic61—41no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61—41no—
163C2×#92418—22no—
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166Dic244—21no—
167Dic244—21no—
168D488—21no—
169D488—21no—
170#11758190—42no—
171C8×D61—22no—
172C8×D61—22no—
173C8×D61—21no—
174C8×D61—21no—
175C8×D61—21no—
176C8×D61—21no—
177C16×C61—21no—
178C16×C61—21no—
179#11758185—21no—
180#11758185—21no—
181C23×C61C2×D811yescharacteristic
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61C2×D811yescharacteristic
187C22×D61C2×D811yescharacteristic
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61#12312yes—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61#13712yes—
195C2×C4×C61C22×D411yescharacteristic
196C22⋊D61—82no—
197C2×Dic61—22no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61C2×D811yescharacteristic
202C2×Dic61C2×D811yescharacteristic
203D244—41no—
204D244—41no—
205D244—82no—
206D244—82no—
207C4×D61—21no—
208C4×D61—21no—
209C4×D61—21no—
210C4×D61—21no—
211C4×D61—22no—
212C4×D61—22no—
213C4×D61—82no—
214C4×D61—82no—
215C4×D61—82no—
216Dic122—41no—
217Dic122—41no—
218C8×C61#12312yes—
219C8×C61C2×#4111yescharacteristic
220C8×C61C2×#4111yescharacteristic
221#92418—21no—
222#92418—21no—
223#33645—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61#48012yes—
227C22×C61C22×D811yescharacteristic
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—21no—
233C2×D61—44no—
234C2×D61—81no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61#50812yes—
238C4×C61—81no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61—81no—
242C4×C61#48012yes—
243C4×C61C2×#12311yescharacteristic
244C4×C61C2×#13711yescharacteristic
245Dic61—21no—
246Dic61—21no—
247Dic61—81no—
248Dic61—81no—
249Dic61—81no—
250#3050—611no—
251C2×#427—611no—
252#2237—611no—
253#2237—611no—
254#2236—611no—
255#2235—611no—
256#2234—611no—
257#2234—611no—
258#2233—611no—
259#2232—611no—
260#2158—611no—
261#2158D6111yescharacteristic, complemented
262#2138—611no—
263#2138—611no—
264C2×C22⋊C16—611no—
265C2×C61—22no—
266C2×C61C2×#48011yescharacteristic
267C2×C61C2×#50811yescharacteristic
268C2×C61#341411yescharacteristic
269D61—22no—
270D61—41no—
271D61—161no—
272C2×#135—611no—
273C23×C8—611no—
274#525—1221no—
275C2×#117—1221no—
276C22×C16—611no—
277C22×C16C2×D6111yescharacteristic
278#467—611no—
279#467—611no—
280#466—611no—
281#466—611no—
282#465—1221no—
283#465—1221no—
284#427—611no—
285#427—611no—
286#427—611no—
287#427—1221no—
288#427—1221no—
289#427C2×D6111yescharacteristic
290#427C2×D6111yescharacteristic
291C2×#107—611no—
292C2×#107—611no—
293C2×#106—1221no—
294#416—1221no—
295#367—611no—
296#367—611no—
297#367C2×D6111yescharacteristic, complemented
298#367C2×D6111yescharacteristic, complemented
299#366—611no—
300#366—611no—
301#366C2×D6111yescharacteristic, complemented
302#366C2×D6111yescharacteristic, complemented
303#358—611no—
304#358—611no—
305#358—611no—
306#358—611no—
307#357—611no—
308#357—611no—
309#357—611no—
310#357—611no—
311C22⋊C16—611no—
312C22⋊C16—611no—
313C22⋊C16—611no—
314C22⋊C16—611no—
315C61#3364511yescharacteristic, complemented, Sylow
316C2×#41—1221no—
317C23×C4—611no—
318#135—1222no—
319#135—1222no—
320C2×Q16—611no—
321C2×SD16—1221no—
322C2×D8—611no—
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—1221no—
335#121—1221no—
336#120—2441no—
337C4×D4—2441no—
338C4×D4—2441no—
339#117—21no—
340#117—21no—
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—
350C2×C16—22no—
351C2×C16—611no—
352C2×C16—611no—
353C2×C16—1222no—
354C2×C16C22×D6111yescharacteristic
355C2×C16C22×D6111yescharacteristic
356#107—611no—
357#107—611no—
358#107—1221no—
359#107—1221no—
360#107C22×D6111yescharacteristic
361#107C22×D6111yescharacteristic
362#107C22×D6111yescharacteristic
363#107C22×D6111yescharacteristic
364#106—21no—
365#106—21no—
366#106—1221no—
367#106—1221no—
368#106—1221no—
369#103—1221no—
370#103—1221no—
371#102—1221no—
372#102—1221no—
373C24—611no—
374#41—2442no—
375#41—2442no—
376C2×Q8—1221no—
377C2×D4—1221no—
378C2×D4—2441no—
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—1221no—
396D8—1221no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—1222no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1222no—
405C2×C8—1222no—
406C2×C8#1175822212yes—
407C2×C8D4×D6111yescharacteristic
408C2×C8#1175822612yes—
409C2×C8C23×D6111yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
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—41no—
431C42—2441no—
432C42—2441no—
433C42—2441no—
434C16—21no—
435C16—21no—
436C16—1221no—
437C16—1221no—
438C23—611no—
439C23—611no—
440C23—1222no—
441C23—1222no—
442C23—2441no—
443C23—2441no—
444C23D8×D6111yescharacteristic
445Q8—2441no—
446Q8—2441no—
447D4—4882no—
448D4—4882no—
449D4—4882no—
450D4—2441no—
451D4—2441no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—611no—
457C2×C4—611no—
458C2×C4—1222no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—4882no—
466C2×C4—4882no—
467C2×C4—4882no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4#1451976212yes—
472C2×C4D8×D6111yescharacteristic
473C2×C4#1451980912yes—
474C2×C4C2×D4×D6111yescharacteristic
475C8—1221no—
476C8—1221no—
477C8#1451976212yes—
478C8C2×#1175822211yescharacteristic
479C8C2×#1175822611yescharacteristic
480C22—21no—
481C22—21no—
482C22—612no—
483C22—1221no—
484C22—1221no—
485C22—1221no—
486C22—1221no—
487C22—1221no—
488C22—4881no—
489C22—4881no—
490C22—4881no—
491C22—2444no—
492C22#1684007712yes—
493C22C2×D8×D6111yescharacteristic
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—1221no—
499C4—1221no—
500C4—4881no—
501C4—4881no—
502C4—4881no—
503C4#1684007712yes—
504C4C2×#1451976211yescharacteristic
505C4C2×#1451980911yescharacteristic
506C2—22no—
507C2—1222no—
508C2—9761no—
509C2—2441no—
510C2C2×#1684007711yescharacteristic
511C2#508×D6111yescharacteristic
512C2#2145895211yescharacteristic
subgroupG/H
centerC2×C4D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2158C2
socleC22×C61C22×D8
prime radical#21455401C2
hypercenter#2158D61
hypocenterC61#33645
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

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

NH ≅ G/Ntype
2. #427×D61C2semidirect
3. #21448030C2semidirect
4. #21448029C2semidirect
5. #21448023C2semidirect
6. #21448018C2semidirect
7. #21448013C2semidirect
8. #21448012C2semidirect
9. #21448006C2semidirect
10. #21448001C2semidirect
11. #21447601C2semidirect
12. #21447527C2semidirect
13. #21447524C2semidirect
14. C2×#16839541C2semidirect
15. C61×#2158C2semidirect
36. #16839589C22semidirect
37. #16839589C22semidirect
38. #16839587C22semidirect
39. #16839587C22semidirect
40. #16839567C22semidirect
41. #16839567C22semidirect
42. #16839565C22semidirect
43. #16839565C22semidirect
44. #16839564C22semidirect
45. #16839564C22semidirect
46. #16839562C22semidirect
47. #16839562C22semidirect
55. C61×#367C22semidirect
56. C61×#367C22semidirect
57. C61×#366C22semidirect
58. C61×#366C22semidirect
261. #2158D61semidirect
297. #367C2×D61semidirect
298. #367C2×D61semidirect
301. #366C2×D61semidirect
302. #366C2×D61semidirect
315. C61#33645semidirect

Maximal subgroups

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

subgroupindex
#214554012
#427×D612
#214480302
#214480292
#214480232
#214480182
#214480132
#214480122
#214480062
#214480012
#214476012
#214475272
#214475242
C2×#168395412
C61×#21582
#3364561