Finite Group Databasebeta

← #510×#6923 #30624290

Subgroups — #510×#6923 #30624290

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×#6923C211yesmaximal, characteristic
2C2×Q16×#6923C211yesmaximal, characteristic, complemented
3#21462216C212yesmaximal, complemented
4#21462215C212yesmaximal, complemented
5Q32×#6923C212yesmaximal, complemented
6SD32×#6923C212yesmaximal, complemented
7#21462171C211yesmaximal, characteristic, complemented
8#21462129C211yesmaximal, characteristic, complemented
9#21462127C211yesmaximal, characteristic
10M32×#6923C211yesmaximal, characteristic, complemented
11#510×D61C211yesmaximal, characteristic
12#41×#6923—21no—
13C2×Q8×#6923—21no—
14D61⋊M32C2211yescharacteristic
15Q16×#6923—21no—
16Q16×#6923—21no—
17Q16×#6923C2212yes—
18Q16×#6923C2212yes—
19Q16×#6923C2211yescharacteristic
20Q16×#6923C2211yescharacteristic
21SD16×#6923—21no—
22SD16×#6923—21no—
23D8×#6923C2211yescharacteristic
24D8×#6923C2211yescharacteristic
25#16840772C2211yescharacteristic
26C2×#14519846C2211yescharacteristic
27#16840752—21no—
28C2×#14519840—21no—
29C2×C8×#6923C2211yescharacteristic
30#16840689C2212yes—
31#16840688C2212yes—
32#16840675C2212yes—
33#16840675C2212yes—
34#16840675C2212yes—
35#16840674C2212yes—
36#16840649C2212yes—
37C16×#6923C2212yes—
38#135×D61C2211yescharacteristic
39C2×Q16×D61C2211yescharacteristic
40#16840205C411yescharacteristic, complemented
41#16840204C412yescomplemented
42#16840203C411yescharacteristic, complemented
43#16840202C411yescharacteristic, complemented
44#16840201C412yescomplemented
45Q32×D61C2212yes—
46SD32×D61C2212yes—
47M32×D61C2211yescharacteristic
48C61×#510C411yescharacteristic, complemented
49C2×D61⋊C8D411yescharacteristic
50Q8×#6923—21no—
51Q8×#6923—21no—
52Q8×#6923—22no—
53Q8×#6923—22no—
54Q8×#6923—41no—
55Q8×#6923—41no—
56D4×#6923—21no—
57D4×#6923—21no—
58D4×#6923—41no—
59D4×#6923—41no—
60#14519856—21no—
61#14519856—41no—
62C2×C4⋊#6923—21no—
63C2×C4⋊#6923—41no—
64C2×C4×#6923—41no—
65C2×C4×#6923—41no—
66C2×C4×#6923D411yescharacteristic
67#14519852—21no—
68#14519851—21no—
69D61⋊C16C2312yes—
70#14519846—21no—
71#14519846C2312yes—
72#14519846C2311yescharacteristic
73#14519846C2311yescharacteristic
74#14519845—21no—
75#14519840—22no—
76#14519840—22no—
77#14519840—21no—
78#14519840—21no—
79#14519839—21no—
80#14519839—21no—
81C8×#6923D411yescharacteristic
82C8×#6923D411yescharacteristic
83C8×#6923C2311yescharacteristic
84C8×#6923C2311yescharacteristic
85#41×D61—21no—
86C2×Q8×D61—21no—
87#14519800C2×C411yescharacteristic
88#14519799C2×C411yescharacteristic
89#14519798—21no—
90#14519797—21no—
91#14519796C2×C411yescharacteristic
92C2×Dic244C2×C411yescharacteristic, complemented
93C2×#11758202—21no—
94Q16×D61—21no—
95Q16×D61—21no—
96Q16×D61C2×C412yescomplemented
97Q16×D61C2312yes—
98Q16×D61C2311yescharacteristic
99SD16×D61—21no—
100D8×D61C2311yescharacteristic
101C2×C8×D61C2311yescharacteristic
102Dic488C2×C412yescomplemented
103#14519723C2×C412yescomplemented
104#14519723C2×C412yescomplemented
105#14519722C2×C412yescomplemented
106#14519721C2×C412yescomplemented
107#14519720C2×C412yescomplemented
108#14519717C2×C411yescharacteristic, complemented
109#14519716C2×C412yescomplemented
110C16×D61C2312yes—
111#135×C61C2×C411yescharacteristic
112C2×Q16×C61C2×C411yescharacteristic, complemented
113Q32×C61C2×C412yescomplemented
114SD32×C61C2×C412yescomplemented
115M32×C61C2×C411yescharacteristic, complemented
116C22×#6923—41no—
117C22×#6923—41no—
118C22×#6923—41no—
119C22×#6923D811yescharacteristic
120C22×#6923D811yescharacteristic
121#41×C61—21no—
122C2×Q8×C61—21no—
123#11758226—21no—
124Q8×D61—22no—
125Q8×D61—22no—
126Q8×D61—21no—
127Q8×D61—41no—
128Q8×D61—41no—
129#11758224—21no—
130#11758224—41no—
131D4×D61—21no—
132D4×D61—41no—
133#11758222—21no—
134#11758222—41no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61C2×D411yescharacteristic
138C2×Dic122—21no—
139C2×Dic122—41no—
140C22⋊#6923—41no—
141C22⋊#6923—41no—
142C22⋊#6923—41no—
143C2×#92420—21no—
144C4⋊#6923—21no—
145C4⋊#6923—21no—
146C4⋊#6923—22no—
147C4⋊#6923—42no—
148C4⋊#6923—42no—
149C4⋊#6923—41no—
150C4⋊#6923—41no—
151C4⋊#6923—41no—
152C4×#6923—41no—
153C4×#6923—41no—
154C4×#6923—42no—
155C4×#6923—42no—
156C4×#6923—41no—
157C4×#6923D811yescharacteristic
158C4×#6923D811yescharacteristic
159C4×#6923C2×D411yescharacteristic
160C4×#6923C2×D411yescharacteristic
161D61⋊C8—21no—
162D61⋊C8C2×D411yescharacteristic
163D61⋊C8C2×D411yescharacteristic
164Q16×C61—21no—
165Q16×C61C22×C412yes—
166Q16×C61C22×C411yescharacteristic
167SD16×C61—21no—
168D8×C61C22×C411yescharacteristic
169C2×C8×C61C22×C411yescharacteristic
170#11758202—22no—
171#11758202—22no—
172#11758202—21no—
173#11758201—21no—
174#11758200—21no—
175#11758199—21no—
176C2×#92418#4111yescharacteristic
177Dic244—21no—
178Dic244C22×C412yes—
179Dic244C22×C411yescharacteristic
180D488C22×C411yescharacteristic
181#11758190—21no—
182C8×D61C22×C411yescharacteristic
183C8×D61C2×D411yescharacteristic
184C8×D61#4111yescharacteristic
185C8×D61C2411yescharacteristic
186#11758187—22no—
187C16×C61C22×C412yes—
188#11758185C22×C412yes—
189C22×D61—41no—
190C22×D61C2×D811yescharacteristic
191C2×#6923—22no—
192C2×#6923—41no—
193C2×#6923—42no—
194C2×#6923—41no—
195C2×#6923—41no—
196C2×#6923—82no—
197C2×#6923—82no—
198C2×#6923—82no—
199C2×#6923C2×D811yescharacteristic
200C2×#6923C2×D811yescharacteristic
201C2×#6923C2×D811yescharacteristic
202C2×#6923C2×D811yescharacteristic
203Q8×C61—21no—
204Q8×C61—22no—
205Q8×C61—41no—
206D4×C61—21no—
207D4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61C4×D411yescharacteristic
211C22⋊D61—41no—
212C22⋊D61—41no—
213C22⋊D61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61#13511yescharacteristic
217D244—21no—
218D244—41no—
219C4×D61—41no—
220C4×D61—42no—
221C4×D61—42no—
222C4×D61—41no—
223C4×D61—41no—
224C4×D61C4×D411yescharacteristic
225C4×D61C2×D811yescharacteristic
226C4×D61#13511yescharacteristic
227C4×D61C22×D411yescharacteristic
228Dic122—21no—
229Dic122—22no—
230Dic122—41no—
231Dic122—42no—
232Dic122—41no—
233Dic122—42no—
234#92420—21no—
235#92420—21no—
236C8×C61C4×D411yescharacteristic
237C8×C61C23×C411yescharacteristic
238#92418C4×D411yescharacteristic
239#92418C2×#4111yescharacteristic
240C4×#510—611nomaximal, Sylow
241C22×C61—41no—
242C22×C61C4×D811yescharacteristic
243C2×D61—21no—
244C2×D61—21no—
245C2×D61—41no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61C4×D811yescharacteristic
249C2×D61C22×D811yescharacteristic
250#6923—21no—
251#6923—21no—
252#6923—21no—
253#6923—81no—
254#6923—82no—
255#6923—81no—
256#6923#51012yesdirect product factor
257C4×C61—42no—
258C4×C61—41no—
259C4×C61C4×D811yescharacteristic
260C4×C61C2×C4×D411yescharacteristic
261Dic61—42no—
262Dic61—41no—
263Dic61C4×D811yescharacteristic
264Dic61C2×#13511yescharacteristic
265C2×#510—611no—
266C4×#135—611no—
267C2×C4×Q16—611no—
268#2176—612no—
269#2175—612no—
270C4×Q32—612no—
271C4×SD32—612no—
272#2159—611no—
273#2141—611no—
274#2139—611no—
275C4×M32—611no—
276C2×C61—21no—
277C2×C61—81no—
278C2×C61C2×C4×D811yescharacteristic
279D61—21no—
280D61—81no—
281D61C2×#51011yescharacteristic
282D61#341411yescharacteristic
283C2×#135—611no—
284C22×Q16—611no—
285C4×#41—1221no—
286C2×C4×Q8—1221no—
287#510—611no—
288#510—611no—
289#510—612no—
290#510—612no—
291#510—611no—
292#510#692311yescharacteristic, direct product factor
293C2×Q32—612no—
294C2×SD32—612no—
295C2×M32—611no—
296C2×M32—611no—
297C4×Q16—612no—
298C4×Q16—612no—
299C4×Q16—611no—
300C4×Q16—611no—
301C4×Q16—1221no—
302C4×Q16—1221no—
303C4×SD16—1221no—
304C4×SD16—1221no—
305C4×D8—611no—
306C4×D8—611no—
307#427—611no—
308C2×#107—611no—
309#416—1221no—
310C2×#103—1221no—
311C2×C4×C8—611no—
312#367—612no—
313#366—612no—
314#358—612no—
315#358—612no—
316#358—612no—
317#357—612no—
318#346—612no—
319C4×C16—612no—
320C61C4×#51011yescharacteristic, complemented, Sylow
321C2×#41—1221no—
322C22×Q8—1221no—
323#135—611no—
324#135—611no—
325#135—611no—
326#135—1221no—
327#135—1221no—
328#135C2×#692311yescharacteristic
329C2×Q16—612no—
330C2×Q16—612no—
331C2×Q16—611no—
332C2×Q16—611no—
333C2×Q16—1221no—
334C2×Q16—1221no—
335C2×Q16—1221no—
336C2×Q16C2×#692311yescharacteristic, complemented
337C2×SD16—1221no—
338C2×D8—611no—
339C22×C8—611no—
340C22×C8—611no—
341C4×Q8—1222no—
342C4×Q8—1222no—
343C4×Q8—1221no—
344C4×Q8—1221no—
345C4×Q8—2441no—
346C4×Q8—2441no—
347C4×D4—1221no—
348C4×D4—1221no—
349C4×D4—2441no—
350C4×D4—2441no—
351#117—1221no—
352#117—2441no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—2441no—
355C2×C42—611no—
356C2×C42—2441no—
357C2×C42—2441no—
358Q32—612no—
359Q32—612no—
360Q32—612no—
361Q32C2×#692312yescomplemented
362SD32—612no—
363SD32—612no—
364SD32—612no—
365SD32C2×#692312yescomplemented
366M32—611no—
367M32—612no—
368M32—1221no—
369M32—1221no—
370M32C2×#692311yescharacteristic, complemented
371C2×C16—612no—
372C2×C16—612no—
373#107—612no—
374#107—611no—
375#107—611no—
376#107—1221no—
377#106—1221no—
378#103—1222no—
379#103—1222no—
380#103—1221no—
381#103—1221no—
382#102—1221no—
383#102—1221no—
384C4×C8—611no—
385C4×C8—611no—
386C4×C8—611no—
387C4×C8—611no—
388#41—21no—
389#41—1221no—
390#41—1221no—
391#41—1221no—
392#41—2441no—
393#41—2441no—
394C2×Q8—21no—
395C2×Q8—1221no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—1221no—
399C2×Q8—2441no—
400C2×Q8—2441no—
401C2×Q8—2441no—
402C2×D4—1221no—
403C2×D4—2441no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—2441no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4—2441no—
412Q16—21no—
413Q16—612no—
414Q16—611no—
415Q16—1222no—
416Q16—1222no—
417Q16—1221no—
418Q16—1221no—
419Q16C22×#692312yes—
420Q16C22×#692311yescharacteristic
421SD16—21no—
422SD16—1221no—
423SD16—1221no—
424SD16—1221no—
425D8—611no—
426D8—1221no—
427D8C22×#692311yescharacteristic
428C2×C8—611no—
429C2×C8—611no—
430C2×C8—611no—
431C2×C8—611no—
432C2×C8—611no—
433C2×C8—611no—
434C2×C8—611no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8C22×#692311yescharacteristic
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1222no—
441C4⋊C4—2441no—
442C4⋊C4—2442no—
443C4⋊C4—2442no—
444C4⋊C4—2441no—
445C4⋊C4—2441no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C42—611no—
450C42—611no—
451C42—611no—
452C42—611no—
453C42—2441no—
454C42—2441no—
455C42—2441no—
456C42—2442no—
457C42—2442no—
458C16—612no—
459C16—1222no—
460C16C22×#692312yes—
461C23—611no—
462C23—2441no—
463Q8—21no—
464Q8—22no—
465Q8—41no—
466Q8—1221no—
467Q8—1222no—
468Q8—2442no—
469Q8—2442no—
470Q8—2441no—
471Q8—2441no—
472D4—21no—
473D4—41no—
474D4—1221no—
475D4—2441no—
476D4—2441no—
477D4—2441no—
478D4—2441no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—611no—
482C2×C4—611no—
483C2×C4—611no—
484C2×C4—611no—
485C2×C4—611no—
486C2×C4—611no—
487C2×C4—611no—
488C2×C4—611no—
489C2×C4—611no—
490C2×C4—1222no—
491C2×C4—4882no—
492C2×C4—4882no—
493C2×C4—4882no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2442no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4D4×#692311yescharacteristic
506C8—611no—
507C8—611no—
508C8—1221no—
509C8—1221no—
510C8D4×#692311yescharacteristic
511C8C23×#692311yescharacteristic
512C22—41no—
513C22—611no—
514C22—611no—
515C22—1221no—
516C22—1221no—
517C22—4881no—
518C22—4881no—
519C22—2441no—
520C22D8×#692311yescharacteristic
521C4—41no—
522C4—42no—
523C4—612no—
524C4—611no—
525C4—611no—
526C4—1221no—
527C4—1221no—
528C4—1221no—
529C4—4881no—
530C4—4881no—
531C4—4882no—
532C4—2441no—
533C4—2442no—
534C4D8×#692311yescharacteristic
535C4C2×D4×#692311yescharacteristic
536C2—21no—
537C2—81no—
538C2—611no—
539C2—611no—
540C2—1221no—
541C2—4881no—
542C2C2×D8×#692311yescharacteristic
subgroupG/H
centerC2C2×D8×#6923
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#510C4
socleC2×C61C2×C4×D8
prime radical#135×D61C22
cosocleC8×D61C24
hypercenter#510#6923
hypocenterC61C4×#510
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. C2×Q16×#6923C2semidirect
3. #21462216C2semidirect
4. #21462215C2semidirect
5. Q32×#6923C2semidirect
6. SD32×#6923C2semidirect
7. #21462171C2semidirect
8. #21462129C2semidirect
10. M32×#6923C2semidirect
40. #16840205C4semidirect
41. #16840204C4semidirect
42. #16840203C4semidirect
43. #16840202C4semidirect
44. #16840201C4semidirect
48. C61×#510C4semidirect
92. C2×Dic244C2×C4semidirect
96. Q16×D61C2×C4semidirect
102. Dic488C2×C4semidirect
103. #14519723C2×C4semidirect
104. #14519723C2×C4semidirect
105. #14519722C2×C4semidirect
106. #14519721C2×C4semidirect
107. #14519720C2×C4semidirect
108. #14519717C2×C4semidirect
109. #14519716C2×C4semidirect
112. C2×Q16×C61C2×C4semidirect
113. Q32×C61C2×C4semidirect
114. SD32×C61C2×C4semidirect
115. M32×C61C2×C4semidirect
256. #6923#510direct (N × H)
292. #510#6923direct (N × H)
320. C61C4×#510semidirect
336. C2×Q16C2×#6923semidirect
361. Q32C2×#6923semidirect
365. SD32C2×#6923semidirect
370. M32C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#135×#69232
C2×Q16×#69232
#214622162
#214622152
Q32×#69232
SD32×#69232
#214621712
#214621292
#214621272
M32×#69232
#510×D612
C4×#51061