Finite Group Databasebeta

← #29835244

Subgroups — #29835244

Order 15616.

564 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456555C211yesmaximal, characteristic, complemented
2#21454724C211yesmaximal, characteristic, complemented
3#21448032C212yesmaximal, complemented
4#21447991C211yesmaximal, characteristic, complemented
5#21447990C211yesmaximal, characteristic, complemented
6#21447892C211yesmaximal, characteristic, complemented
7#21447763C212yesmaximal, complemented
8#21447720C212yesmaximal, complemented
9#21447718C212yesmaximal, complemented
10#21447630C211yesmaximal, characteristic, complemented
11C61×#2213C211yesmaximal, characteristic, complemented
12#16840559C2211yescharacteristic
13#16840527—21no—
14#16840310—21no—
15#16840198C2212yes—
16#16840160C2211yescharacteristic, complemented
17#16840108—21no—
18#16840012—21no—
19#16840012C2212yes—
20#16839884C2211yescharacteristic
21#16839883C2211yescharacteristic
22#16839882—21no—
23#16839880C2211yescharacteristic
24#16839876—21no—
25#16839874—21no—
26#16839874C2212yescomplemented
27#16839809C2211yescharacteristic
28#104×D61—21no—
29C4×#11758189C2211yescharacteristic
30#16839586C2212yescomplemented
31#16839585C2212yescomplemented
32#16839583C2211yescharacteristic, complemented
33#16839575C2212yescomplemented
34#16839573C2212yes—
35#16839572C2212yes—
36#16839543C2212yes—
37#16839539C2212yescomplemented
38C61×#509C2212yes—
39C61×#493C2211yescharacteristic, complemented
40C61×#443C2211yescharacteristic
41C61×#364C2211yescharacteristic, complemented
42C61×#361C2212yescomplemented
43C2×#11758224—41no—
44#41×D61—21no—
45#14519809—21no—
46#14519806—21no—
47#14519804—21no—
48#14519801—21no—
49#14519799C2311yescharacteristic
50#14519797—21no—
51#14519785—21no—
52#14519785—41no—
53#14519784—21no—
54#14519783—21no—
55#14519783—41no—
56C2×#11758189—41no—
57C2×#11758189D411yescharacteristic
58C2×#11758189C2311yescharacteristic
59M16×D61—21no—
60M16×D61—41no—
61#14519777—21no—
62#14519764—41no—
63#14519748—22no—
64#14519748—41no—
65D4×Dic61—22no—
66D4×Dic61—41no—
67C42×D61D411yescharacteristic
68#14519721—22no—
69#14519718—22no—
70#14519717C2312yes—
71#14519713C2311yescharacteristic
72#14519712—21no—
73#14519709—21no—
74#14519709C2312yes—
75#14519705—21no—
76#14519704—21no—
77#14519703—21no—
78#14519698—22no—
79#14519698—22no—
80#14519687D411yescharacteristic
81#14519687C2311yescharacteristic
82C4×#92418D411yescharacteristic
83#135×C61C2311yescharacteristic
84C2×D8×C61—21no—
85C2×D8×C61C2312yes—
86#131×C61—21no—
87#127×C61—21no—
88SD32×C61—22no—
89D16×C61—22no—
90M32×C61C2312yes—
91#108×C61C2311yescharacteristic
92#104×C61—21no—
93C4×C8×C61C2311yescharacteristic
94#41×C61—21no—
95C2×D4×C61—22no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C2×C22⋊D61—42no—
99C22×Dic61—42no—
100#11758226—21no—
101Q8×D61—21no—
102#11758224—21no—
103#11758224—41no—
104#11758224—84no—
105D4×D61—21no—
106D4×D61—41no—
107#11758222—21no—
108#11758222—41no—
109C2×C4×D61—21no—
110C2×C4×D61—41no—
111C2×C4×D61C2×D411yescharacteristic
112C2×Dic122—41no—
113Q16×C61#4111yescharacteristic, complemented
114SD16×C61—21no—
115D8×C61—22no—
116D8×C61—22no—
117D8×C61—41no—
118D8×C61—41no—
119D8×C61#4111yescharacteristic, complemented
120M16×C61—21no—
121M16×C61—41no—
122C2×C8×C61—41no—
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C22⋊Dic61—42no—
127C22⋊Dic61—42no—
128#11758202—21no—
129#11758201—21no—
130#11758200—21no—
131#11758199—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61—41no—
134C4×Dic61—21no—
135C4×Dic61C2×D411yescharacteristic
136#11758194—21no—
137#11758194—41no—
138C2×#92418—41no—
139C2×#92418C2×D411yescharacteristic
140C2×#92418C2×D411yescharacteristic
141Dic244#4111yescharacteristic, complemented
142D488#4111yescharacteristic, complemented
143#11758190—21no—
144#11758189—21no—
145#11758189—21no—
146#11758189—21no—
147#11758189—21no—
148#11758189—41no—
149#11758189—41no—
150#11758189—41no—
151C8×D61—41no—
152C8×D61—41no—
153C8×D61—41no—
154C16×C61—22no—
155#11758185—22no—
156C23×C61—42no—
157C22×D61—41no—
158C22×D61AGL(1,ℤ/8)11yescharacteristic
159Q8×C61—21no—
160D4×C61—21no—
161D4×C61—41no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—82no—
165D4×C61—82no—
166C2×C4×C61—21no—
167C2×C4×C61—41no—
168C2×C4×C61C22×D411yescharacteristic
169C22⋊D61—41no—
170C22⋊D61—41no—
171C22⋊D61—41no—
172C22⋊D61—84no—
173C2×Dic61—21no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—84no—
177C2×Dic61AGL(1,ℤ/8)11yescharacteristic
178D244—21no—
179D244—41no—
180C4×D61—21no—
181C4×D61—21no—
182C4×D61—41no—
183C4×D61—42no—
184C4×D61—41no—
185C4×D61—41no—
186Dic122—21no—
187Dic122—41no—
188Dic122—82no—
189C8×C61—41no—
190C8×C61AGL(1,ℤ/8)11yescharacteristic
191C8×C61AGL(1,ℤ/8)11yescharacteristic
192C8×C61C2×#4111yescharacteristic
193C8×C61C2×#4111yescharacteristic
194#92418—21no—
195#92418—21no—
196#92418—41no—
197#33688—611nomaximal, Sylow
198C22×C61—41no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
203C2×D61—21no—
204C2×D61—41no—
205C2×D61—41no—
206C2×D61—161no—
207C4×C61—22no—
208C4×C61—41no—
209C4×C61#52511yescharacteristic
210C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
211Dic61—21no—
212Dic61—41no—
213Dic61—41no—
214Dic61—82no—
215#3184—611no—
216#2956—611no—
217#2238—612no—
218#2230—611no—
219#2230—611no—
220#2213—611no—
221#2213D6111yescharacteristic, complemented
222#2190—612no—
223AGL(1,ℤ/16)—612no—
224AGL(1,ℤ/16)—612no—
225#2165—611no—
226C2×C61—21no—
227C2×C61—81no—
228C2×C61—162no—
229C2×C61#310911yescharacteristic
230D61—41no—
231D61—81no—
232#575—611no—
233#568—1221no—
234#532—1221no—
235#509—612no—
236#509C2×D6112yes—
237#498—611no—
238#493C2×D6111yescharacteristic, complemented
239#486—1221no—
240#469—612no—
241#469—1221no—
242#444—611no—
243#444—611no—
244#444—1221no—
245#443—611no—
246#443—1221no—
247#443C2×D6111yescharacteristic
248#442—612no—
249#442—1221no—
250#430—611no—
251C2×#104—1221no—
252C4×M16—611no—
253#365—612no—
254#365—612no—
255#364—611no—
256#364C2×D6111yescharacteristic, complemented
257#361—612no—
258#361C2×D6112yescomplemented
259#360—612no—
260#360—612no—
261#349—612no—
262#347—612no—
263C61#3368811yescharacteristic, complemented, Sylow
264C2×#41—1221no—
265C2×#41—2441no—
266#137—1221no—
267#137—1221no—
268AGL(1,ℤ/8)—1221no—
269AGL(1,ℤ/8)—1221no—
270#135—611no—
271#135—1221no—
272#135C22×D6111yescharacteristic
273C2×D8—21no—
274C2×D8C22×D6112yes—
275#131—21no—
276#131—1221no—
277#131—1221no—
278#131—1221no—
279#131—2441no—
280#131—2441no—
281C2×M16—611no—
282C2×M16—611no—
283C2×M16—1221no—
284C2×M16—2441no—
285C2×M16—2441no—
286C4⋊Q8—1221no—
287#127—21no—
288#124—2441no—
289#121—1222no—
290#121—2441no—
291C4×D4—1222no—
292C4×D4—2441no—
293C2×C42—611no—
294SD32—22no—
295SD32—1222no—
296D16—22no—
297D16—1222no—
298M32—612no—
299M32C22×D6112yes—
300#108—611no—
301#108—1221no—
302#108C22×D6111yescharacteristic
303#106—612no—
304#106—1221no—
305#104—21no—
306#104—1221no—
307#104—1221no—
308#104—1221no—
309#102—1222no—
310#102—1222no—
311#97—611no—
312#97—611no—
313C4×C8—611no—
314C4×C8C22×D6111yescharacteristic
315#41—21no—
316#41—1221no—
317#41—1221no—
318#41—1221no—
319#41—4884no—
320#41—2441no—
321#41—2441no—
322C2×Q8—1221no—
323C2×Q8—2441no—
324C2×D4—22no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—1221no—
328C2×D4—2441no—
329C2×D4—2442no—
330C22×C4—611no—
331C22×C4—1221no—
332C22×C4—2441no—
333C22×C4—2442no—
334Q16—611no—
335Q16—1221no—
336Q16#1175822411yescharacteristic, complemented
337SD16—21no—
338SD16—1221no—
339SD16—1221no—
340SD16—1221no—
341D8—22no—
342D8—22no—
343D8—41no—
344D8—41no—
345D8—611no—
346D8—1221no—
347D8#1175822411yescharacteristic, complemented
348M16—21no—
349M16—41no—
350M16—1221no—
351M16—1221no—
352M16—1221no—
353M16—1221no—
354M16—1221no—
355M16—2441no—
356M16—2441no—
357M16—2441no—
358M16—2441no—
359C2×C8—41no—
360C2×C8—611no—
361C2×C8—611no—
362C2×C8—2441no—
363C2×C8—2441no—
364C2×C8—2441no—
365C2×C8—2441no—
366C2×C8D4×D6111yescharacteristic
367C2×C8C23×D6111yescharacteristic
368C4⋊C4—1222no—
369C4⋊C4—2441no—
370C22⋊C4—2442no—
371C22⋊C4—2442no—
372C42—611no—
373C42—1221no—
374C42D4×D6111yescharacteristic
375C16—22no—
376C16—1222no—
377C23—42no—
378C23—611no—
379C23—2441no—
380Q8—21no—
381Q8—1221no—
382Q8—4882no—
383Q8—2441no—
384D4—21no—
385D4—41no—
386D4—42no—
387D4—42no—
388D4—82no—
389D4—82no—
390D4—1221no—
391D4—4884no—
392D4—2441no—
393D4—2441no—
394D4—2441no—
395D4—2441no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—611no—
399C2×C4—1221no—
400C2×C4—1221no—
401C2×C4—1221no—
402C2×C4—4884no—
403C2×C4—2442no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2442no—
408C2×C4—2441no—
409C2×C4C2×D4×D6111yescharacteristic
410C8—41no—
411C8—1221no—
412C8—1221no—
413C8—2441no—
414C8#1451980111yescharacteristic
415C8#1451980111yescharacteristic
416C8#41×D6111yescharacteristic
417C8C2×#1175822411yescharacteristic
418C22—41no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—1221no—
423C22—9761no—
424C22—2441no—
425C22—2441no—
426C22AGL(1,ℤ/8)×D6111yescharacteristic
427C4—22no—
428C4—41no—
429C4—1221no—
430C4—4882no—
431C4—2441no—
432C4—2441no—
433C4#1684027211yescharacteristic
434C4C2×#1451980111yescharacteristic
435C2—21no—
436C2—81no—
437C2—162no—
438C2—4881no—
439C2—2441no—
440C2#2145590911yescharacteristic
subgroupG/H
centerC2#21455909
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2213C2
socleC2×C61#3109
hypercenter#2213D61
hypocenterC61#33688
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2213

Complements

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

NH ≅ G/Ntype
1. #21456555C2semidirect
2. #21454724C2semidirect
3. #21448032C2semidirect
4. #21447991C2semidirect
5. #21447990C2semidirect
6. #21447892C2semidirect
7. #21447763C2semidirect
8. #21447720C2semidirect
9. #21447718C2semidirect
10. #21447630C2semidirect
11. C61×#2213C2semidirect
16. #16840160C22semidirect
26. #16839874C22semidirect
30. #16839586C22semidirect
31. #16839585C22semidirect
32. #16839583C22semidirect
33. #16839575C22semidirect
37. #16839539C22semidirect
39. C61×#493C22semidirect
41. C61×#364C22semidirect
42. C61×#361C22semidirect
113. Q16×C61#41semidirect
119. D8×C61#41semidirect
141. Dic244#41semidirect
142. D488#41semidirect
221. #2213D61semidirect
238. #493C2×D61semidirect
256. #364C2×D61semidirect
258. #361C2×D61semidirect
263. C61#33688semidirect
336. Q16#11758224semidirect
347. D8#11758224semidirect

Maximal subgroups

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

subgroupindex
#214565552
#214547242
#214480322
#214479912
#214479902
#214478922
#214477632
#214477202
#214477182
#214476302
C61×#22132
#3368861