Finite Group Databasebeta

← #30624329

Subgroups — #30624329

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462595C211yesmaximal, characteristic, complemented
2#21462582C211yesmaximal, characteristic, complemented
3#21462229C212yesmaximal, complemented
4#21462228C212yesmaximal, complemented
5#21462225C212yesmaximal, complemented
6#21462224C212yesmaximal, complemented
7D61⋊#365C211yesmaximal, characteristic, complemented
8#21462146C211yesmaximal, characteristic, complemented
9C2×#16840678C211yesmaximal, characteristic, complemented
10#21462039C211yesmaximal, characteristic, complemented
11#21458951C211yesmaximal, characteristic
12#16840840—21no—
13#16840831—21no—
14C2×#14519852C2211yescharacteristic
15#16840809C2212yes—
16#16840809C2212yes—
17#16840804—22no—
18#16840803C2212yescomplemented
19#16840802C2212yescomplemented
20SD16×#6923—22no—
21D61⋊#108C2211yescharacteristic
22C2×#14519845C2211yescharacteristic
23#16840760—21no—
24#16840753—21no—
25#16840725C2211yescharacteristic
26#16840687C2212yescomplemented
27#16840686C2212yescomplemented
28#16840681C2211yescharacteristic, complemented
29#16840681C2211yescharacteristic, complemented
30#16840680C2211yescharacteristic, complemented
31#16840680C2211yescharacteristic, complemented
32#16840678C2212yes—
33#16840678C2212yes—
34#16840653C2212yescomplemented
35#16840652C2212yescomplemented
36C2×#14519796C2211yescharacteristic
37#135×D61C2211yescharacteristic
38#16840205C411yescharacteristic, complemented
39#16840202C411yescharacteristic, complemented
40#16840201C2212yes—
41#16840198C411yescharacteristic, complemented
42#16840196C2212yes—
43#16840193C412yescomplemented
44#16840191C412yescomplemented
45M32×D61C2211yescharacteristic
46C61×#509C411yescharacteristic, complemented
47#14519864—22no—
48#14519864—22no—
49#14519864—41no—
50#14519864—41no—
51D61⋊M16—21no—
52D61⋊M16D411yescharacteristic
53C2×#11758217—42no—
54Q8×#6923—22no—
55D4×#6923—22no—
56D4×#6923—42no—
57#14519856—41no—
58#14519856D411yescharacteristic
59C2×C4⋊#6923—21no—
60C2×C4⋊#6923—41no—
61#14519852—21no—
62#14519852C2311yescharacteristic
63#14519852C2311yescharacteristic
64C2×#11758187—21no—
65#14519848—21no—
66#14519847C2311yescharacteristic
67#14519847C2311yescharacteristic
68#14519845—21no—
69#14519845C2312yes—
70#14519841—22no—
71#14519841—22no—
72#14519840—22no—
73#14519839—22no—
74#14519831C2312yes—
75C8×#6923D412yes—
76C2×#11758224—21no—
77#41×D61—21no—
78#14519800C2×C411yescharacteristic
79#14519799C2×C411yescharacteristic
80#14519798—21no—
81#14519797—21no—
82#14519796—21no—
83#14519796—21no—
84#14519796C2×C411yescharacteristic
85#14519796C2×C412yescomplemented
86#14519796C2312yes—
87C2×Dic244C2×C411yescharacteristic, complemented
88Q16×D61C2311yescharacteristic
89C2×#11758200—21no—
90SD16×D61—21no—
91D8×D61C2311yescharacteristic
92C2×C8×D61C2311yescharacteristic
93Dic488C2×C412yescomplemented
94#14519723C2×C412yescomplemented
95#14519722C2×C412yescomplemented
96#14519721C2×C412yescomplemented
97#14519721C2×C412yescomplemented
98#14519718C2×C412yescomplemented
99#14519717C2×C411yescharacteristic, complemented
100#14519716C2312yes—
101C16×D61C2×C412yescomplemented
102#135×C61C2×C411yescharacteristic
103C2×D8×C61C2×C411yescharacteristic, complemented
104SD32×C61C2×C412yescomplemented
105D16×C61C2×C412yescomplemented
106M32×C61C2×C411yescharacteristic, complemented
107C22×#6923—42no—
108C22×#6923—41no—
109#41×C61—21no—
110C2×D4×C61—21no—
111C2×C22⋊D61—41no—
112C22×Dic61—41no—
113#11758226—21no—
114Q8×D61—21no—
115#11758224—21no—
116#11758224—22no—
117#11758224—22no—
118#11758224—41no—
119#11758224—41no—
120#11758224—41no—
121D4×D61—21no—
122D4×D61—41no—
123#11758222—21no—
124#11758222—41no—
125C2×C4×D61—41no—
126C2×C4×D61C2×D411yescharacteristic
127C2×Dic122—21no—
128C22⋊#6923—21no—
129C22⋊#6923—42no—
130#11758217—21no—
131#11758217—42no—
132#11758217—42no—
133#11758217—41no—
134C2×#92420—42no—
135C2×#92420—42no—
136C4⋊#6923—21no—
137C4⋊#6923—22no—
138C4⋊#6923—42no—
139C4⋊#6923—41no—
140C4×#6923—42no—
141C4×#6923C2×D412yes—
142#11758213—21no—
143#11758213—21no—
144#11758213—21no—
145D61⋊C8—41no—
146D61⋊C8C2×D412yes—
147Q16×C61C22×C411yescharacteristic
148SD16×C61—21no—
149D8×C61—21no—
150D8×C61C22×C411yescharacteristic
151D8×C61C22×C412yes—
152C2×C8×C61C22×C411yescharacteristic
153#11758202—21no—
154#11758201—21no—
155#11758200—21no—
156#11758200—22no—
157#11758200—22no—
158#11758199—21no—
159C2×#92418#4111yescharacteristic
160Dic244—21no—
161Dic244C22×C411yescharacteristic
162Dic244C22×C412yes—
163D488C22×C411yescharacteristic
164#11758190—21no—
165C8×D61C22×C411yescharacteristic
166C8×D61C2×D411yescharacteristic
167C8×D61#4111yescharacteristic
168C8×D61C2411yescharacteristic
169#11758187—21no—
170#11758187—21no—
171C16×C61C22×C412yes—
172#11758185C22×C412yes—
173C23×C61—41no—
174C22×D61—41no—
175C22×D61AGL(1,ℤ/8)11yescharacteristic
176C2×#6923—22no—
177C2×#6923—42no—
178C2×#6923—162no—
179C2×#6923—161no—
180Q8×C61—21no—
181D4×C61—21no—
182D4×C61—22no—
183D4×C61—41no—
184D4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61C4×D411yescharacteristic
187C22⋊D61—41no—
188C22⋊D61—42no—
189C22⋊D61—41no—
190C22⋊D61—42no—
191C22⋊D61—41no—
192C2×Dic61—41no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61AGL(1,ℤ/8)11yescharacteristic
197D244—21no—
198D244—41no—
199C4×D61—41no—
200C4×D61—41no—
201C4×D61—41no—
202C4×D61C4×D411yescharacteristic
203C4×D61AGL(1,ℤ/8)11yescharacteristic
204C4×D61AGL(1,ℤ/8)11yescharacteristic
205C4×D61C22×D411yescharacteristic
206Dic122—21no—
207Dic122—22no—
208Dic122—41no—
209Dic122—41no—
210#92420—22no—
211#92420—41no—
212#92420—41no—
213C8×C61C4×D411yescharacteristic
214C8×C61C23×C411yescharacteristic
215#92418C4×D411yescharacteristic
216#92418C2×#4111yescharacteristic
217#33469—611nomaximal, Sylow
218C22×C61—42no—
219C22×C61—41no—
220C22×C61—161no—
221C22×C61#44211yescharacteristic
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61#44211yescharacteristic
227C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
228#6923—42no—
229#6923—82no—
230C4×C61—41no—
231C4×C61#44211yescharacteristic
232C4×C61C2×C4×D411yescharacteristic
233Dic61—41no—
234Dic61—42no—
235Dic61#44211yescharacteristic
236Dic61C2×AGL(1,ℤ/8)11yescharacteristic
237#3414—611no—
238#2957—611no—
239#2952—611no—
240AGL(1,ℤ/16)—612no—
241AGL(1,ℤ/16)—612no—
242#2180—612no—
243#2180—612no—
244C2×#365—611no—
245#2148—611no—
246C2×#360—611no—
247#2110—611no—
248C2×C61—21no—
249C2×C61—81no—
250C2×C61—82no—
251C2×C61C2×#44211yescharacteristic
252D61—21no—
253D61—21no—
254D61—81no—
255C2×#135—611no—
256C2×#135—611no—
257#568—1221no—
258#520—1221no—
259#510—611no—
260#510—612no—
261#510—611no—
262#509—611no—
263#509—612no—
264#509#692311yescharacteristic, complemented
265#508—612no—
266#508—612no—
267C2×M32—611no—
268C2×M32—611no—
269#444—612no—
270#444—612no—
271#443—1222no—
272#442—612no—
273#441—612no—
274C4×SD16—1222no—
275C2×#108—611no—
276C2×#106—611no—
277#421—1221no—
278#417—1221no—
279#405—611no—
280#365—612no—
281#365—612no—
282#362—611no—
283#362—611no—
284#361—611no—
285#361—611no—
286#360—612no—
287#360—612no—
288#347—612no—
289#347—612no—
290C61#3346911yescharacteristic, complemented, Sylow
291C2×#41—1221no—
292C2×#41—1221no—
293#135—611no—
294#135—612no—
295#135—611no—
296#135—612no—
297#135—611no—
298#135—1221no—
299#135—1221no—
300#135—1221no—
301#135—1221no—
302#135C2×#692311yescharacteristic
303C2×Q16—611no—
304C2×Q16—611no—
305C2×SD16—1221no—
306C2×SD16—1221no—
307C2×D8—611no—
308C2×D8C2×#692311yescharacteristic, complemented
309#131—1222no—
310#131—1222no—
311#131—2441no—
312#131—2441no—
313C2×M16—611no—
314C2×M16—1221no—
315C2×M16—2442no—
316C22×C8—611no—
317C4×Q8—1222no—
318C4×D4—1222no—
319C4×D4—2442no—
320#117—611no—
321#117—2441no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—2441no—
324Q32—612no—
325Q32—612no—
326SD32—612no—
327SD32—612no—
328SD32—612no—
329SD32C2×#692312yescomplemented
330D16—612no—
331D16C2×#692312yescomplemented
332M32—611no—
333M32—612no—
334M32—611no—
335M32—611no—
336M32—1221no—
337M32C2×#692311yescharacteristic, complemented
338C2×C16—612no—
339C2×C16—1221no—
340#108—611no—
341#108—611no—
342#108—1221no—
343#106—612no—
344#106—1221no—
345#104—1222no—
346#104—1222no—
347#103—1222no—
348#102—1222no—
349#97—612no—
350C4×C8—612no—
351#41—21no—
352#41—1222no—
353#41—1222no—
354#41—1221no—
355#41—1221no—
356#41—1221no—
357#41—2441no—
358#41—2441no—
359#41—2441no—
360#41—2441no—
361C2×Q8—1221no—
362C2×Q8—1221no—
363C2×D4—21no—
364C2×D4—1221no—
365C2×D4—2441no—
366C2×D4—2441no—
367C22×C4—611no—
368C22×C4—2441no—
369C22×C4—2441no—
370C22×C4—2442no—
371C22×C4—2441no—
372Q16—611no—
373Q16—612no—
374Q16—1221no—
375Q16—1221no—
376Q16C22×#692311yescharacteristic
377SD16—21no—
378SD16—1222no—
379SD16—1221no—
380SD16—1222no—
381SD16—1221no—
382SD16—1221no—
383D8—21no—
384D8—611no—
385D8—1221no—
386D8C22×#692312yes—
387D8C22×#692311yescharacteristic
388M16—1221no—
389M16—1221no—
390M16—1221no—
391M16—1221no—
392M16—2442no—
393M16—2442no—
394M16—2441no—
395C2×C8—611no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—611no—
400C2×C8—612no—
401C2×C8—2442no—
402C2×C8—2442no—
403C2×C8—2441no—
404C2×C8C22×#692311yescharacteristic
405C4⋊C4—1221no—
406C4⋊C4—1222no—
407C4⋊C4—2441no—
408C4⋊C4—2442no—
409C22⋊C4—1221no—
410C22⋊C4—2442no—
411C42—612no—
412C42—2442no—
413C16—612no—
414C16—1221no—
415C16—1221no—
416C16C22×#692312yes—
417C23—41no—
418C23—611no—
419C23—2441no—
420Q8—21no—
421Q8—1221no—
422Q8—1222no—
423Q8—2441no—
424Q8—2441no—
425D4—22no—
426D4—21no—
427D4—41no—
428D4—41no—
429D4—1221no—
430D4—2441no—
431D4—2441no—
432D4—2441no—
433D4—2442no—
434D4—2442no—
435D4—2441no—
436C2×C4—41no—
437C2×C4—611no—
438C2×C4—611no—
439C2×C4—611no—
440C2×C4—611no—
441C2×C4—611no—
442C2×C4—1222no—
443C2×C4—9761no—
444C2×C4—9762no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4D4×#692311yescharacteristic
454C8—611no—
455C8—611no—
456C8—1222no—
457C8—2441no—
458C8—2441no—
459C8D4×#692311yescharacteristic
460C8C23×#692311yescharacteristic
461C22—41no—
462C22—42no—
463C22—161no—
464C22—611no—
465C22—611no—
466C22—4881no—
467C22—4881no—
468C22—2441no—
469C22—2441no—
470C22#1684080311yescharacteristic
471C4—41no—
472C4—611no—
473C4—611no—
474C4—4882no—
475C4—2442no—
476C4—2441no—
477C4—2442no—
478C4#1684080311yescharacteristic
479C4C2×D4×#692311yescharacteristic
480C2—21no—
481C2—82no—
482C2—81no—
483C2—1221no—
484C2—1221no—
485C2—4881no—
486C2C2×#1684080311yescharacteristic
subgroupG/H
centerC2C2×#16840803
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#509C4
socleC2×C61C2×#442
prime radical#21458951C2
cosocleC8×D61C24
hypercenter#509#6923
hypocenterC61#33469
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #21462595C2semidirect
2. #21462582C2semidirect
3. #21462229C2semidirect
4. #21462228C2semidirect
5. #21462225C2semidirect
6. #21462224C2semidirect
7. D61⋊#365C2semidirect
8. #21462146C2semidirect
9. C2×#16840678C2semidirect
10. #21462039C2semidirect
18. #16840803C22semidirect
19. #16840802C22semidirect
26. #16840687C22semidirect
27. #16840686C22semidirect
28. #16840681C22semidirect
29. #16840681C22semidirect
30. #16840680C22semidirect
31. #16840680C22semidirect
34. #16840653C22semidirect
35. #16840652C22semidirect
38. #16840205C4semidirect
39. #16840202C4semidirect
41. #16840198C4semidirect
43. #16840193C4semidirect
44. #16840191C4semidirect
46. C61×#509C4semidirect
85. #14519796C2×C4semidirect
87. C2×Dic244C2×C4semidirect
93. Dic488C2×C4semidirect
94. #14519723C2×C4semidirect
95. #14519722C2×C4semidirect
96. #14519721C2×C4semidirect
97. #14519721C2×C4semidirect
98. #14519718C2×C4semidirect
99. #14519717C2×C4semidirect
101. C16×D61C2×C4semidirect
103. C2×D8×C61C2×C4semidirect
104. SD32×C61C2×C4semidirect
105. D16×C61C2×C4semidirect
106. M32×C61C2×C4semidirect
264. #509#6923semidirect
290. C61#33469semidirect
308. C2×D8C2×#6923semidirect
329. SD32C2×#6923semidirect
331. D16C2×#6923semidirect
337. M32C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625952
#214625822
#214622292
#214622282
#214622252
#214622242
D61⋊#3652
#214621462
C2×#168406782
#214620392
#214589512
#3346961