Finite Group Databasebeta

← #30624333

Subgroups — #30624333

Order 15616.

632 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462598C211yesmaximal, characteristic
2#21462580C211yesmaximal, characteristic, complemented
3#21462227C212yesmaximal, complemented
4#21462226C212yesmaximal, complemented
5#21462225C212yesmaximal, complemented
6#21462224C212yesmaximal, complemented
7#21462167C211yesmaximal, characteristic, complemented
8C2×#16840681C211yesmaximal, characteristic, complemented
9#21462139C211yesmaximal, characteristic
10#21462039C211yesmaximal, characteristic, complemented
11#510×D61C211yesmaximal, characteristic
12#16840841—21no—
13#16840830—21no—
14C2×#14519852C2211yescharacteristic
15#16840809C2212yes—
16#16840808C2212yes—
17#16840805—22no—
18#16840802C2212yescomplemented
19#16840802C2212yescomplemented
20Q16×#6923—22no—
21C2×#14519847C2211yescharacteristic
22#16840772C2211yescharacteristic
23#16840759—21no—
24#16840755—21no—
25#16840725C2211yescharacteristic
26#16840687C2212yescomplemented
27#16840686C2212yescomplemented
28#16840681C2212yescomplemented
29#16840681C2212yescomplemented
30#16840679C2211yescharacteristic
31#16840679C2211yescharacteristic
32#16840678C2211yescharacteristic
33#16840678C2211yescharacteristic
34#16840653C2212yescomplemented
35#16840652C2212yescomplemented
36#135×D61C2211yescharacteristic
37C2×Q16×D61C2211yescharacteristic
38#16840205C411yescharacteristic, complemented
39#16840204C2212yes—
40#16840203C411yescharacteristic, complemented
41#16840202C411yescharacteristic, complemented
42#16840201C2212yes—
43Q32×D61C412yescomplemented
44SD32×D61C412yescomplemented
45M32×D61C2211yescharacteristic
46C61×#510C411yescharacteristic, complemented
47#14519865—22no—
48#14519864—22no—
49#14519864—42no—
50C2×#11758213—21no—
51C2×#11758213—41no—
52D61⋊M16—41no—
53D61⋊M16D411yescharacteristic
54Q8×#6923—22no—
55Q8×#6923—22no—
56Q8×#6923—41no—
57Q8×#6923—41no—
58#14519856—21no—
59#14519856—42no—
60#14519856D411yescharacteristic
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#14519840—22no—
73#14519840—22no—
74#14519831C2312yes—
75C8×#6923D412yes—
76#41×D61—21no—
77C2×Q8×D61—21no—
78#14519800C2311yescharacteristic
79#14519799C2×C411yescharacteristic
80#14519798—21no—
81#14519797—21no—
82#14519796C2311yescharacteristic
83C2×Dic244C2×C411yescharacteristic
84C2×#11758202—21no—
85Q16×D61—21no—
86Q16×D61—21no—
87Q16×D61C2×C412yes—
88Q16×D61C2×C411yescharacteristic
89Q16×D61C2312yes—
90SD16×D61—21no—
91D8×D61C2×C411yescharacteristic
92C2×C8×D61C2311yescharacteristic
93Dic488C2×C412yes—
94#14519723C2×C412yes—
95#14519723C2×C412yes—
96#14519722C2×C412yes—
97#14519721C2×C412yes—
98#14519720C2×C412yes—
99#14519717C2×C411yescharacteristic
100#14519716C2312yes—
101C16×D61C2×C412yes—
102#135×C61C2×C411yescharacteristic
103C2×Q16×C61C2×C411yescharacteristic
104Q32×C61C2×C412yes—
105SD32×C61C2×C412yes—
106M32×C61C2×C411yescharacteristic
107#41×C61—21no—
108C2×Q8×C61—21no—
109#11758226—21no—
110Q8×D61—22no—
111Q8×D61—21no—
112Q8×D61—22no—
113Q8×D61—41no—
114Q8×D61—41no—
115#11758224—21no—
116#11758224—41no—
117D4×D61—21no—
118D4×D61—41no—
119#11758222—21no—
120#11758222—41no—
121C2×C4×D61—41no—
122C2×C4×D61—41no—
123C2×C4×D61C2×D411yescharacteristic
124C2×Dic122—21no—
125C2×Dic122—41no—
126C22⋊#6923—21no—
127C22⋊#6923—41no—
128#11758217—21no—
129#11758217—42no—
130C2×#92420—42no—
131C2×#92420—41no—
132C4⋊#6923—21no—
133C4⋊#6923—21no—
134C4⋊#6923—21no—
135C4⋊#6923—42no—
136C4⋊#6923—42no—
137C4×#6923—42no—
138C4×#6923—42no—
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×C61—21no—
148Q16×C61C22×C412yes—
149Q16×C61C22×C411yescharacteristic
150SD16×C61—21no—
151D8×C61C22×C411yescharacteristic
152C2×C8×C61C22×C411yescharacteristic
153#11758202—22no—
154#11758202—21no—
155#11758202—22no—
156#11758201—21no—
157#11758200—21no—
158#11758199—21no—
159C2×#92418#4111yescharacteristic
160Dic244—21no—
161Dic244C22×C412yes—
162Dic244C22×C411yescharacteristic
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—
173C22×D61—41no—
174C22×D61AGL(1,ℤ/8)11yescharacteristic
175C2×#6923—22no—
176C2×#6923—41no—
177C2×#6923—41no—
178Q8×C61—21no—
179Q8×C61—22no—
180Q8×C61—41no—
181D4×C61—21no—
182D4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61C4×D411yescharacteristic
186C22⋊D61—41no—
187C22⋊D61—41no—
188C22⋊D61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61AGL(1,ℤ/8)11yescharacteristic
192D244—21no—
193D244—41no—
194C4×D61—42no—
195C4×D61—41no—
196C4×D61—42no—
197C4×D61—41no—
198C4×D61—41no—
199C4×D61C4×D411yescharacteristic
200C4×D61AGL(1,ℤ/8)11yescharacteristic
201C4×D61AGL(1,ℤ/8)11yescharacteristic
202C4×D61C22×D411yescharacteristic
203Dic122—21no—
204Dic122—22no—
205Dic122—41no—
206Dic122—42no—
207Dic122—42no—
208Dic122—41no—
209#92420—22no—
210#92420—42no—
211C8×C61C4×D411yescharacteristic
212C8×C61C23×C411yescharacteristic
213#92418C4×D411yescharacteristic
214#92418C2×#4111yescharacteristic
215#33470—611nomaximal, Sylow
216C22×C61—41no—
217C22×C61#44211yescharacteristic
218C2×D61—41no—
219C2×D61—41no—
220C2×D61—161no—
221C2×D61#44211yescharacteristic
222C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
223#6923—42no—
224#6923—81no—
225#6923—81no—
226C4×C61—42no—
227C4×C61—41no—
228C4×C61#44211yescharacteristic
229C4×C61C2×C4×D411yescharacteristic
230Dic61—42no—
231Dic61—41no—
232Dic61#44211yescharacteristic
233Dic61C2×AGL(1,ℤ/8)11yescharacteristic
234C2×#510—611no—
235#2957—611no—
236#2950—611no—
237#2180—612no—
238#2180—612no—
239#2180—612no—
240#2180—612no—
241#2155—611no—
242C2×#362—611no—
243#2145—611no—
244#2110—611no—
245C2×C61—21no—
246C2×C61—81no—
247C2×C61C2×#44211yescharacteristic
248D61—21no—
249D61—21no—
250D61—81no—
251C2×#135—611no—
252C22×Q16—611no—
253#568—1221no—
254#519—1221no—
255#510—611no—
256#510—611no—
257#510—612no—
258#510—612no—
259#510—611no—
260#510#692311yescharacteristic, complemented
261C2×Q32—612no—
262C2×SD32—612no—
263C2×M32—611no—
264C2×M32—611no—
265#444—612no—
266#444—612no—
267#443—1222no—
268#441—612no—
269#441—612no—
270C4×Q16—1222no—
271C2×#108—611no—
272#427—611no—
273#421—1221no—
274#419—1221no—
275#405—611no—
276#365—612no—
277#365—612no—
278#362—612no—
279#362—612no—
280#360—611no—
281#360—611no—
282#360—611no—
283#360—611no—
284#347—612no—
285#347—612no—
286C61#3347011yescharacteristic, complemented, Sylow
287C2×#41—1221no—
288C22×Q8—1221no—
289#135—611no—
290#135—611no—
291#135—611no—
292#135—1221no—
293#135—1221no—
294#135C2×#692311yescharacteristic
295C2×Q16—612no—
296C2×Q16—612no—
297C2×Q16—611no—
298C2×Q16—611no—
299C2×Q16—1221no—
300C2×Q16—1221no—
301C2×Q16—1221no—
302C2×Q16C2×#692311yescharacteristic
303C2×SD16—1221no—
304C2×D8—611no—
305#131—1222no—
306#131—1222no—
307#131—2442no—
308C2×M16—611no—
309C2×M16—1221no—
310C2×M16—2441no—
311C2×M16—2441no—
312C22×C8—611no—
313C4×Q8—1222no—
314C4×Q8—1222no—
315C4×Q8—2441no—
316C4×Q8—2441no—
317#117—611no—
318#117—1221no—
319#117—2442no—
320Q32—612no—
321Q32—612no—
322Q32—612no—
323Q32C2×#692312yes—
324SD32—612no—
325SD32—612no—
326SD32—612no—
327SD32C2×#692312yes—
328M32—611no—
329M32—612no—
330M32—611no—
331M32—611no—
332M32—1221no—
333M32C2×#692311yescharacteristic
334C2×C16—612no—
335C2×C16—1221no—
336#108—612no—
337#108—1221no—
338#107—1221no—
339#106—611no—
340#106—611no—
341#104—1222no—
342#104—1222no—
343#103—1222no—
344#103—1222no—
345#97—612no—
346C4×C8—612no—
347#41—21no—
348#41—1221no—
349#41—1221no—
350#41—1221no—
351#41—2441no—
352#41—2441no—
353C2×Q8—21no—
354C2×Q8—1222no—
355C2×Q8—1221no—
356C2×Q8—1222no—
357C2×Q8—1221no—
358C2×Q8—2441no—
359C2×Q8—2441no—
360C2×Q8—2441no—
361C2×D4—1221no—
362C2×D4—2441no—
363C22×C4—611no—
364C22×C4—2441no—
365C22×C4—2441no—
366Q16—21no—
367Q16—612no—
368Q16—611no—
369Q16—1222no—
370Q16—1222no—
371Q16—1221no—
372Q16—1221no—
373Q16C22×#692312yes—
374Q16C22×#692311yescharacteristic
375SD16—21no—
376SD16—1221no—
377SD16—1221no—
378SD16—1221no—
379D8—611no—
380D8—1221no—
381D8C22×#692311yescharacteristic
382M16—1221no—
383M16—1221no—
384M16—1222no—
385M16—2442no—
386M16—2442no—
387M16—2441no—
388C2×C8—611no—
389C2×C8—611no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8—612no—
394C2×C8—2442no—
395C2×C8—2442no—
396C2×C8—2441no—
397C2×C8C22×#692311yescharacteristic
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—2442no—
402C4⋊C4—2442no—
403C22⋊C4—1221no—
404C22⋊C4—2441no—
405C42—612no—
406C42—2441no—
407C42—2442no—
408C42—2442no—
409C16—612no—
410C16—1221no—
411C16—1221no—
412C16C22×#692312yes—
413C23—611no—
414C23—2441no—
415Q8—21no—
416Q8—22no—
417Q8—41no—
418Q8—1221no—
419Q8—1222no—
420Q8—2442no—
421Q8—2442no—
422Q8—2441no—
423Q8—2441no—
424D4—21no—
425D4—41no—
426D4—1221no—
427D4—2441no—
428D4—2441no—
429D4—2441no—
430D4—2441no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—611no—
434C2×C4—611no—
435C2×C4—611no—
436C2×C4—611no—
437C2×C4—611no—
438C2×C4—1222no—
439C2×C4—2441no—
440C2×C4—2442no—
441C2×C4—2441no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4D4×#692311yescharacteristic
449C8—611no—
450C8—611no—
451C8—1222no—
452C8—2442no—
453C8D4×#692311yescharacteristic
454C8C23×#692311yescharacteristic
455C22—41no—
456C22—611no—
457C22—611no—
458C22—9761no—
459C22—2441no—
460C22—2441no—
461C22#1684080311yescharacteristic
462C4—41no—
463C4—42no—
464C4—611no—
465C4—611no—
466C4—4881no—
467C4—4881no—
468C4—2442no—
469C4—2441no—
470C4—2442no—
471C4#1684080311yescharacteristic
472C4C2×D4×#692311yescharacteristic
473C2—21no—
474C2—81no—
475C2—1221no—
476C2—1221no—
477C2—4881no—
478C2C2×#1684080311yescharacteristic
subgroupG/H
centerC2C2×#16840803
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#510C4
socleC2×C61C2×#442
prime radical#135×D61C22
cosocleC8×D61C24
hypercenter#510#6923
hypocenterC61#33470
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #21462580C2semidirect
3. #21462227C2semidirect
4. #21462226C2semidirect
5. #21462225C2semidirect
6. #21462224C2semidirect
7. #21462167C2semidirect
8. C2×#16840681C2semidirect
10. #21462039C2semidirect
18. #16840802C22semidirect
19. #16840802C22semidirect
26. #16840687C22semidirect
27. #16840686C22semidirect
28. #16840681C22semidirect
29. #16840681C22semidirect
34. #16840653C22semidirect
35. #16840652C22semidirect
38. #16840205C4semidirect
40. #16840203C4semidirect
41. #16840202C4semidirect
43. Q32×D61C4semidirect
44. SD32×D61C4semidirect
46. C61×#510C4semidirect
260. #510#6923semidirect
286. C61#33470semidirect

Maximal subgroups

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

subgroupindex
#214625982
#214625802
#214622272
#214622262
#214622252
#214622242
#214621672
C2×#168406812
#214621392
#214620392
#510×D612
#3347061