Finite Group Databasebeta

← #30624336

Subgroups — #30624336

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462598C211yesmaximal, characteristic, complemented
2#21462578C211yesmaximal, characteristic, complemented
3#21462231C212yesmaximal, complemented
4#21462230C212yesmaximal, complemented
5#21462229C212yesmaximal, complemented
6#21462228C212yesmaximal, complemented
7#21462167C211yesmaximal, characteristic, complemented
8C2×#16840680C211yesmaximal, characteristic, complemented
9#21462139C211yesmaximal, characteristic, complemented
10#21462039C211yesmaximal, characteristic, complemented
11#509×D61C211yesmaximal, characteristic
12#16840841—21no—
13#16840829—21no—
14C2×#14519852C2211yescharacteristic
15#16840809C2212yescomplemented
16#16840808C2212yescomplemented
17#16840805—22no—
18#16840803C2212yescomplemented
19#16840803C2212yescomplemented
20D8×#6923—22no—
21C2×#14519847C2211yescharacteristic
22#16840772C2211yescharacteristic
23#16840759—21no—
24#16840754—21no—
25#16840725C2211yescharacteristic
26#16840687C2212yescomplemented
27#16840686C2212yescomplemented
28#16840680C2212yescomplemented
29#16840680C2212yescomplemented
30#16840679C2211yescharacteristic, complemented
31#16840679C2211yescharacteristic, complemented
32#16840678C2211yescharacteristic, complemented
33#16840678C2211yescharacteristic, complemented
34#16840653C2212yescomplemented
35#16840652C2212yescomplemented
36#135×D61C2211yescharacteristic
37C2×D8×D61C2211yescharacteristic
38#16840200C411yescharacteristic, complemented
39#16840199C2212yes—
40#16840198C411yescharacteristic, complemented
41#16840197C411yescharacteristic, complemented
42#16840196C2212yes—
43SD32×D61C412yescomplemented
44D16×D61C412yescomplemented
45M32×D61C2211yescharacteristic
46C61×#509C411yescharacteristic, complemented
47#14519865—22no—
48#14519864—22no—
49#14519864—42no—
50C2×#11758213—21no—
51C2×#11758213—41no—
52D61⋊M16—41no—
53D61⋊M16D411yescharacteristic
54D4×#6923—22no—
55D4×#6923—22no—
56D4×#6923—41no—
57D4×#6923—41no—
58#14519856—21no—
59#14519856D411yescharacteristic
60C2×C22⋊#6923—42no—
61#14519852—21no—
62#14519852C2311yescharacteristic
63#14519852C2311yescharacteristic
64C2×#11758187—21no—
65#14519847—21no—
66#14519847C2312yes—
67#14519846—21no—
68#14519845C2311yescharacteristic
69#14519845C2311yescharacteristic
70#14519842—22no—
71#14519841—22no—
72#14519839—22no—
73#14519839—22no—
74#14519831C2312yes—
75C8×#6923D412yes—
76#41×D61—21no—
77C2×D4×D61—21no—
78#14519800C2311yescharacteristic
79#14519799C2×C411yescharacteristic, complemented
80#14519798—21no—
81#14519797—21no—
82#14519796C2311yescharacteristic
83Q16×D61C2×C411yescharacteristic, complemented
84SD16×D61—21no—
85C2×D488C2×C411yescharacteristic
86C2×#11758199—21no—
87D8×D61—21no—
88D8×D61—21no—
89D8×D61C2×C412yes—
90D8×D61C2×C411yescharacteristic, complemented
91D8×D61C2312yes—
92C2×C8×D61C2311yescharacteristic
93#14519722C2×C412yescomplemented
94#14519721C2×C412yescomplemented
95#14519720C2×C412yescomplemented
96D976C2×C412yescomplemented
97#14519718C2×C412yescomplemented
98#14519718C2×C412yescomplemented
99#14519717C2×C411yescharacteristic, complemented
100#14519716C2312yes—
101C16×D61C2×C412yescomplemented
102#135×C61C2×C411yescharacteristic, complemented
103C2×D8×C61C2×C411yescharacteristic
104SD32×C61C2×C412yescomplemented
105D16×C61C2×C412yescomplemented
106M32×C61C2×C411yescharacteristic, complemented
107C23×D61—41no—
108C22×#6923—42no—
109C22×#6923—42no—
110#41×C61—21no—
111C2×D4×C61—21no—
112C2×C22⋊D61—41no—
113#11758226—21no—
114Q8×D61—21no—
115#11758224—21no—
116#11758224—41no—
117D4×D61—22no—
118D4×D61—21no—
119D4×D61—22no—
120D4×D61—41no—
121D4×D61—41no—
122D4×D61—41no—
123#11758222—21no—
124#11758222—41no—
125C2×D244—21no—
126C2×C4×D61—41no—
127C2×C4×D61C2×D411yescharacteristic
128C22⋊#6923—21no—
129C22⋊#6923—42no—
130C22⋊#6923—42no—
131C22⋊#6923—41no—
132#11758217—21no—
133#11758217—42no—
134C2×#92420—42no—
135C2×#92420—41no—
136C4⋊#6923—21no—
137C4⋊#6923—21no—
138C4⋊#6923—21no—
139C4×#6923—41no—
140C4×#6923C2×D412yes—
141#11758213—21no—
142#11758213—22no—
143#11758213—42no—
144#11758213—41no—
145D61⋊C8—42no—
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#11758199—22no—
157#11758199—21no—
158#11758199—22no—
159C2×#92418#4111yescharacteristic
160Dic244C22×C411yescharacteristic
161D488—21no—
162D488C22×C411yescharacteristic
163D488C22×C412yes—
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×D61—42no—
176C22×D61—41no—
177C22×D61—42no—
178C22×D61—161no—
179C22×D61—161no—
180C22×D61AGL(1,ℤ/8)11yescharacteristic
181C2×#6923—22no—
182C2×#6923—41no—
183C2×#6923—41no—
184C2×#6923—162no—
185C2×#6923—162no—
186Q8×C61—21no—
187D4×C61—22no—
188D4×C61—21no—
189D4×C61—41no—
190D4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61C4×D411yescharacteristic
193C22⋊D61—41no—
194C22⋊D61—41no—
195C22⋊D61—42no—
196C22⋊D61—41no—
197C22⋊D61—42no—
198C2×Dic61—41no—
199C2×Dic61AGL(1,ℤ/8)11yescharacteristic
200D244—21no—
201D244—22no—
202D244—41no—
203D244—41no—
204C4×D61—41no—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61C4×D411yescharacteristic
208C4×D61AGL(1,ℤ/8)11yescharacteristic
209C4×D61AGL(1,ℤ/8)11yescharacteristic
210C4×D61C22×D411yescharacteristic
211Dic122—21no—
212Dic122—41no—
213#92420—22no—
214#92420—42no—
215C8×C61C4×D411yescharacteristic
216C8×C61C23×C411yescharacteristic
217#92418C4×D411yescharacteristic
218#92418C2×#4111yescharacteristic
219#33471—611nomaximal, Sylow
220C22×C61—41no—
221C22×C61—42no—
222C22×C61—161no—
223C22×C61#44211yescharacteristic
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—42no—
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—161no—
230C2×D61—162no—
231C2×D61—161no—
232C2×D61#44211yescharacteristic
233C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
234#6923—42no—
235#6923—81no—
236#6923—81no—
237C4×C61—41no—
238C4×C61#44211yescharacteristic
239C4×C61C2×C4×D411yescharacteristic
240Dic61—41no—
241Dic61#44211yescharacteristic
242Dic61C2×AGL(1,ℤ/8)11yescharacteristic
243C2×#509—611no—
244#2957—611no—
245#2948—611no—
246AGL(1,ℤ/16)—612no—
247AGL(1,ℤ/16)—612no—
248AGL(1,ℤ/16)—612no—
249AGL(1,ℤ/16)—612no—
250#2155—611no—
251C2×#361—611no—
252#2145—611no—
253#2110—611no—
254C2×C61—21no—
255C2×C61—82no—
256C2×C61—81no—
257C2×C61C2×#44211yescharacteristic
258D61—21no—
259D61—21no—
260D61—82no—
261D61—81no—
262C2×#135—611no—
263C22×D8—611no—
264#568—1221no—
265#518—1221no—
266#509—611no—
267#509—611no—
268#509—612no—
269#509—612no—
270#509—611no—
271#509#692311yescharacteristic, complemented
272C2×SD32—612no—
273C2×D16—612no—
274C2×M32—611no—
275C2×M32—611no—
276#444—612no—
277#444—612no—
278#443—1222no—
279#442—612no—
280#442—612no—
281C4×D8—1222no—
282C2×#108—611no—
283#427—611no—
284#421—1221no—
285#418—1221no—
286#405—611no—
287#365—612no—
288#365—612no—
289#361—612no—
290#361—612no—
291#360—611no—
292#360—611no—
293#360—611no—
294#360—611no—
295#347—612no—
296#347—612no—
297C61#3347111yescharacteristic, complemented, Sylow
298C2×#41—1221no—
299C22×D4—1221no—
300#135—611no—
301#135—611no—
302#135—611no—
303#135—1221no—
304#135—1221no—
305#135C2×#692311yescharacteristic, complemented
306C2×Q16—611no—
307C2×SD16—1221no—
308C2×D8—612no—
309C2×D8—612no—
310C2×D8—611no—
311C2×D8—611no—
312C2×D8—1221no—
313C2×D8—1221no—
314C2×D8—1221no—
315C2×D8C2×#692311yescharacteristic
316#131—1222no—
317#131—1222no—
318#131—2442no—
319C2×M16—611no—
320C2×M16—1221no—
321C2×M16—2441no—
322C2×M16—2441no—
323C22×C8—611no—
324C4×D4—1222no—
325C4×D4—1222no—
326C4×D4—2441no—
327C4×D4—2441no—
328#117—611no—
329#117—1221no—
330C2×C22⋊C4—2442no—
331SD32—612no—
332SD32—612no—
333SD32—612no—
334SD32C2×#692312yescomplemented
335D16—612no—
336D16—612no—
337D16—612no—
338D16C2×#692312yescomplemented
339M32—611no—
340M32—612no—
341M32—611no—
342M32—611no—
343M32—1221no—
344M32C2×#692311yescharacteristic, complemented
345C2×C16—612no—
346C2×C16—1221no—
347#108—612no—
348#108—1221no—
349#107—1221no—
350#106—611no—
351#106—611no—
352#104—1222no—
353#104—1222no—
354#102—1222no—
355#102—1222no—
356#97—612no—
357C4×C8—612no—
358C24—2441no—
359#41—21no—
360#41—1221no—
361#41—1221no—
362#41—1221no—
363#41—2441no—
364#41—2441no—
365C2×Q8—1221no—
366C2×D4—21no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2441no—
374C2×D4—2441no—
375C22×C4—611no—
376C22×C4—2441no—
377C22×C4—2442no—
378C22×C4—2442no—
379Q16—611no—
380Q16—1221no—
381Q16C22×#692311yescharacteristic
382SD16—21no—
383SD16—1221no—
384SD16—1221no—
385SD16—1221no—
386D8—21no—
387D8—611no—
388D8—612no—
389D8—1222no—
390D8—1222no—
391D8—1221no—
392D8—1221no—
393D8C22×#692311yescharacteristic
394D8C22×#692312yes—
395M16—1221no—
396M16—1221no—
397M16—1222no—
398M16—2442no—
399M16—2442no—
400M16—2441no—
401C2×C8—611no—
402C2×C8—611no—
403C2×C8—611no—
404C2×C8—611no—
405C2×C8—611no—
406C2×C8—612no—
407C2×C8—2442no—
408C2×C8—2442no—
409C2×C8—2441no—
410C2×C8C22×#692311yescharacteristic
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C22⋊C4—1221no—
415C22⋊C4—2442no—
416C22⋊C4—2442no—
417C22⋊C4—2441no—
418C42—612no—
419C42—2441no—
420C16—612no—
421C16—1221no—
422C16—1221no—
423C16C22×#692312yes—
424C23—41no—
425C23—611no—
426C23—9761no—
427C23—9761no—
428C23—2441no—
429C23—2442no—
430C23—2442no—
431C23—2441no—
432Q8—21no—
433Q8—1221no—
434Q8—2441no—
435D4—22no—
436D4—21no—
437D4—41no—
438D4—41no—
439D4—1222no—
440D4—1221no—
441D4—2441no—
442D4—2442no—
443D4—2442no—
444D4—2441no—
445D4—2441no—
446D4—2441no—
447D4—2441no—
448C2×C4—41no—
449C2×C4—611no—
450C2×C4—611no—
451C2×C4—611no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—1222no—
455C2×C4—9762no—
456C2×C4—9762no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4D4×#692311yescharacteristic
464C8—611no—
465C8—611no—
466C8—1222no—
467C8—2442no—
468C8D4×#692311yescharacteristic
469C8C23×#692311yescharacteristic
470C22—41no—
471C22—42no—
472C22—161no—
473C22—611no—
474C22—611no—
475C22—9761no—
476C22—9762no—
477C22—9761no—
478C22—4882no—
479C22—4882no—
480C22—2441no—
481C22—2441no—
482C22—2442no—
483C22#1684080311yescharacteristic
484C4—41no—
485C4—611no—
486C4—611no—
487C4—4881no—
488C4—4881no—
489C4—2442no—
490C4—2441no—
491C4#1684080311yescharacteristic
492C4C2×D4×#692311yescharacteristic
493C2—21no—
494C2—82no—
495C2—81no—
496C2—1221no—
497C2—1221no—
498C2—4882no—
499C2—4881no—
500C2C2×#1684080311yescharacteristic
subgroupG/H
centerC2C2×#16840803
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#509C4
socleC2×C61C2×#442
prime radical#509×D61C2
cosocleC8×D61C24
hypercenter#509#6923
hypocenterC61#33471
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. #21462598C2semidirect
2. #21462578C2semidirect
3. #21462231C2semidirect
4. #21462230C2semidirect
5. #21462229C2semidirect
6. #21462228C2semidirect
7. #21462167C2semidirect
8. C2×#16840680C2semidirect
9. #21462139C2semidirect
10. #21462039C2semidirect
15. #16840809C22semidirect
16. #16840808C22semidirect
18. #16840803C22semidirect
19. #16840803C22semidirect
26. #16840687C22semidirect
27. #16840686C22semidirect
28. #16840680C22semidirect
29. #16840680C22semidirect
30. #16840679C22semidirect
31. #16840679C22semidirect
32. #16840678C22semidirect
33. #16840678C22semidirect
34. #16840653C22semidirect
35. #16840652C22semidirect
38. #16840200C4semidirect
40. #16840198C4semidirect
41. #16840197C4semidirect
43. SD32×D61C4semidirect
44. D16×D61C4semidirect
46. C61×#509C4semidirect
79. #14519799C2×C4semidirect
83. Q16×D61C2×C4semidirect
90. D8×D61C2×C4semidirect
93. #14519722C2×C4semidirect
94. #14519721C2×C4semidirect
95. #14519720C2×C4semidirect
96. D976C2×C4semidirect
97. #14519718C2×C4semidirect
98. #14519718C2×C4semidirect
99. #14519717C2×C4semidirect
101. C16×D61C2×C4semidirect
102. #135×C61C2×C4semidirect
104. SD32×C61C2×C4semidirect
105. D16×C61C2×C4semidirect
106. M32×C61C2×C4semidirect
271. #509#6923semidirect
297. C61#33471semidirect
305. #135C2×#6923semidirect
334. SD32C2×#6923semidirect
338. D16C2×#6923semidirect
344. M32C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625982
#214625782
#214622312
#214622302
#214622292
#214622282
#214621672
C2×#168406802
#214621392
#214620392
#509×D612
#3347161