Finite Group Databasebeta

← #482980

Subgroups — #482980

Order 512.

799 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62251C211yesmaximal, characteristic
2#33488C211yesmaximal, characteristic, complemented
3#22707C211yesmaximal, characteristic, complemented
4#11890C212yesmaximal, complemented
5#11886C212yesmaximal, complemented
6#11860C211yesmaximal, characteristic, complemented
7#11856C211yesmaximal, characteristic, complemented
8#11744C211yesmaximal, characteristic, complemented
9#11741C211yesmaximal, characteristic, complemented
10#11705C212yesmaximal, complemented
11#11703C212yesmaximal, complemented
12#3519—21no—
13#3341C411yescharacteristic, complemented
14#3339C412yescomplemented
15#3338C411yescharacteristic, complemented
16#3305—21no—
17#3297—22no—
18#3295—21no—
19#3287C411yescharacteristic, complemented
20#3280C412yescomplemented
21D4×D8C411yescharacteristic, complemented
22#3247—21no—
23#3188—22no—
24#3169C2211yescharacteristic
25#3168C2211yescharacteristic
26#3050C2212yes—
27C2×#466C2212yes—
28C2×C8×D4C2211yescharacteristic
29#2354—21no—
30#2169—21no—
31#2168—22no—
32D4×C16—21no—
33#2163C2211yescharacteristic, complemented
34#2162C2211yescharacteristic, complemented
35#2112C2212yescomplemented
36C2×C22⋊C16C2212yescomplemented
37#1947—21no—
38#1854—22no—
39#1834C2211yescharacteristic, complemented
40#1808C2212yescomplemented
41C4×#106C2211yescharacteristic, complemented
42#1735C2212yescomplemented
43#1348C2212yescomplemented
44#1348C2211yescharacteristic, complemented
45#1348C2211yescharacteristic, complemented
46#1347C2212yes—
47#1347C2211yescharacteristic
48#1347C2211yescharacteristic
49#1347C2212yescomplemented
50#1347C2212yescomplemented
51#1347C2212yescomplemented
52#1347C2212yescomplemented
53C2×#135C2×C412yes—
54C22×D8C2×C412yes—
55C23×C8C2312yes—
56#555—21no—
57#554—21no—
58#553—22no—
59#553—41no—
60#550—21no—
61#546—22no—
62#546—42no—
63D42—21no—
64D42—41no—
65#540—41no—
66#538—42no—
67#532—21no—
68#529—21no—
69#525—22no—
70C2×#121—22no—
71#518—41no—
72C2×C4×D4D411yescharacteristic
73C2×#117—22no—
74C2×M32—22no—
75C22×C16—22no—
76#500C2×C411yescharacteristic
77#499C2×C411yescharacteristic
78#495C2×C411yescharacteristic
79#493C2×C411yescharacteristic
80#488—21no—
81#486—21no—
82#484—22no—
83#482—22no—
84#482—22no—
85#481—22no—
86#478—22no—
87#477—21no—
88#476—21no—
89#467C2×C412yes—
90#467C2311yescharacteristic
91#466—21no—
92#466—22no—
93#466—22no—
94#466C2×C412yes—
95#466C2×C412yes—
96#466C2×C412yes—
97#466C2311yescharacteristic
98#466C2312yes—
99#465—21no—
100#465—22no—
101#465—22no—
102#463—22no—
103#460—21no—
104#459—21no—
105#450—22no—
106#449—22no—
107#449—22no—
108#447—22no—
109C4×Q16C2×C411yescharacteristic
110C4×SD16—21no—
111C4×D8—21no—
112C4×D8C2×C412yes—
113C4×D8C2×C411yescharacteristic
114C8×D4C2×C411yescharacteristic
115C8×D4C2×C412yes—
116C8×D4C2×C411yescharacteristic
117C8×D4D411yescharacteristic
118C8×D4D412yes—
119C8×D4D411yescharacteristic
120#427C2×C412yes—
121C2×#107—22no—
122C2×#107C2×C412yes—
123C2×#106—21no—
124C2×C4⋊C8D411yescharacteristic
125#416—22no—
126C2×#102—22no—
127C2×C22⋊C8D412yes—
128C2×C4×C8C2311yescharacteristic
129#390—41no—
130#388—42no—
131#384—21no—
132#380—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—41no—
135C4⋊C16—21no—
136C4⋊C16—21no—
137C22⋊C16—22no—
138C22⋊C16—22no—
139C22⋊C16C2311yescharacteristic
140C22⋊C16C2312yes—
141C22⋊C16C2311yescharacteristic
142#346—21no—
143C4×C16—21no—
144#340—21no—
145#340—22no—
146#340C2311yescharacteristic
147#340C2312yes—
148#340C2311yescharacteristic
149C2×#41—22no—
150C2×#41—41no—
151C22×D4—22no—
152C22×D4—42no—
153C22×D4—41no—
154C22×D4D811yescharacteristic
155C23×C4C2×D412yes—
156#135—82no—
157C2×Q16C22×C411yescharacteristic
158C2×SD16—22no—
159C2×SD16—21no—
160C2×D8—22no—
161C2×D8—21no—
162C2×D8—82no—
163C2×D8C22×C411yescharacteristic
164C2×D8C22×C412yes—
165C22×C8—82no—
166C22×C8C22×C412yes—
167C22×C8C22×C412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2411yescharacteristic
173C4⋊Q8—21no—
174#127—21no—
175#127—41no—
176#126—42no—
177#126—41no—
178#125—21no—
179#125—41no—
180#124—21no—
181#124—42no—
182#124—41no—
183#123—41no—
184#123—42no—
185#123—42no—
186#123—42no—
187C22⋊Q8—22no—
188C22⋊Q8—21no—
189C22⋊Q8—42no—
190#121—22no—
191#121—22no—
192#121—22no—
193#121—21no—
194#121—22no—
195#121—42no—
196#121—41no—
197#121—42no—
198#121—41no—
199#121—42no—
200#121—41no—
201#121—42no—
202#121—42no—
203#120—42no—
204#120—42no—
205#120—42no—
206C4×Q8—21no—
207C4×D4—21no—
208C4×D4—22no—
209C4×D4—41no—
210C4×D4—42no—
211C4×D4—41no—
212C4×D4—42no—
213C4×D4—41no—
214C4×D4—42no—
215C4×D4—41no—
216C4×D4C22⋊C411yescharacteristic
217C4×D4C22⋊C412yes—
218C4×D4C22⋊C411yescharacteristic
219C4×D4SD1611yescharacteristic
220C4×D4SD1612yes—
221C4×D4SD1611yescharacteristic
222#117—22no—
223#117—41no—
224#117—41no—
225#117—44no—
226#117—82no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4D811yescharacteristic
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4D812yes—
239C2×C42—42no—
240C2×C42—41no—
241C2×C42C2×D411yescharacteristic
242M32—82no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—22no—
246C2×C16—82no—
247#107—22no—
248#107—21no—
249#107—82no—
250#107C22×C411yescharacteristic
251#107C22×C412yes—
252#107C22×C411yescharacteristic
253#106—22no—
254#106—21no—
255#106—42no—
256C4⋊C8C22⋊C411yescharacteristic
257C4⋊C8C22⋊C411yescharacteristic
258C4⋊C8C22⋊C412yes—
259#103—22no—
260#103—21no—
261#103—21no—
262#102—22no—
263#102—22no—
264#102—22no—
265#102—21no—
266#102—22no—
267#102—21no—
268#102—22no—
269C22⋊C8C22⋊C412yes—
270C22⋊C8C22⋊C412yes—
271C22⋊C8C22⋊C412yes—
272C22⋊C8C22⋊C412yes—
273C4×C8C22×C411yescharacteristic
274C4×C8C22×C411yescharacteristic
275C4×C8C2×D411yescharacteristic
276C4×C8C2×D411yescharacteristic
277#95—41no—
278#95—42no—
279C24—42no—
280C24C2×D812yes—
281#41—82no—
282#41—82no—
283#41—161no—
284#41—161no—
285C2×Q8—21no—
286C2×Q8—41no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—84no—
302C2×D4—162no—
303C2×D4—162no—
304C2×D4#10211yescharacteristic
305C2×D4#10211yescharacteristic
306C2×D4#10212yes—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×D811yescharacteristic
319C22×C4C2×D811yescharacteristic
320C22×C4C2×D812yes—
321C22×C4C2×SD1612yes—
322C22×C4C2×SD1612yes—
323C22×C4C22×D411yescharacteristic
324Q16—41no—
325SD16—81no—
326SD16—82no—
327D8—41no—
328D8—42no—
329D8—81no—
330D8—82no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8C2×C22⋊C411yescharacteristic
339C2×C8C2×C22⋊C411yescharacteristic
340C2×C8C2×C22⋊C412yes—
341C2×C8C2×C22⋊C412yes—
342C2×C8C23×C411yescharacteristic
343C2×C8C22×D411yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—82no—
361C4⋊C4—161no—
362C4⋊C4#10211yescharacteristic
363C4⋊C4#10211yescharacteristic
364C4⋊C4#10212yes—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—41no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4—42no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4—41no—
376C22⋊C4—42no—
377C22⋊C4—84no—
378C22⋊C4—162no—
379C22⋊C4#10212yes—
380C22⋊C4#10212yes—
381C22⋊C4#10212yes—
382C22⋊C4#10212yes—
383C42—42no—
384C42—41no—
385C42—82no—
386C42—84no—
387C42C2×C22⋊C411yescharacteristic
388C42C2×C22⋊C411yescharacteristic
389C42C2×SD1611yescharacteristic
390C42C2×SD1611yescharacteristic
391C16—41no—
392C16—42no—
393C16—41no—
394C23—22no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—84no—
401C23—162no—
402C23—162no—
403C23C2×#10212yes—
404C23C2×#10212yes—
405C23C22×D811yescharacteristic
406Q8—41no—
407Q8—41no—
408Q8—161no—
409D4—22no—
410D4—22no—
411D4—42no—
412D4—41no—
413D4—42no—
414D4—42no—
415D4—41no—
416D4—84no—
417D4—81no—
418D4—82no—
419D4—81no—
420D4—82no—
421D4—82no—
422D4—162no—
423D4—161no—
424D4—162no—
425D4—162no—
426D4—162no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—162no—
454C2×C4—162no—
455C2×C4—162no—
456C2×C4C2×#10212yes—
457C2×C4C2×#10211yescharacteristic
458C2×C4C2×#10211yescharacteristic
459C2×C4C2×#10211yescharacteristic
460C2×C4C2×#10212yes—
461C2×C4C2×#10211yescharacteristic
462C2×C4C22×C22⋊C411yescharacteristic
463C2×C4C22×SD1611yescharacteristic
464C8—21no—
465C8—21no—
466C8—22no—
467C8—22no—
468C8—21no—
469C8—21no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—81no—
482C22—81no—
483C22—82no—
484C22—82no—
485C22—81no—
486C22—324no—
487C22C22×#10211yescharacteristic
488C22#341511yescharacteristic
489C22#341611yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—22no—
495C4—22no—
496C4—82no—
497C4—81no—
498C4—82no—
499C4—81no—
500C4—81no—
501C4—162no—
502C4—161no—
503C4#214512yes—
504C4#288311yescharacteristic
505C4#290111yescharacteristic
506C2—21no—
507C2—21no—
508C2—42no—
509C2—42no—
510C2—162no—
511C2—161no—
512C2#1761311yescharacteristic
513C2#3338111yescharacteristic
514C2C2×#214511yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62251C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33488C2semidirect
3. #22707C2semidirect
4. #11890C2semidirect
5. #11886C2semidirect
6. #11860C2semidirect
7. #11856C2semidirect
8. #11744C2semidirect
9. #11741C2semidirect
10. #11705C2semidirect
11. #11703C2semidirect
13. #3341C4semidirect
14. #3339C4semidirect
15. #3338C4semidirect
19. #3287C4semidirect
20. #3280C4semidirect
21. D4×D8C4semidirect
33. #2163C22semidirect
34. #2162C22semidirect
35. #2112C22semidirect
36. C2×C22⋊C16C22semidirect
39. #1834C22semidirect
40. #1808C22semidirect
41. C4×#106C22semidirect
42. #1735C22semidirect
43. #1348C22semidirect
44. #1348C22semidirect
45. #1348C22semidirect
49. #1347C22semidirect
50. #1347C22semidirect
51. #1347C22semidirect
52. #1347C22semidirect

Maximal subgroups

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

subgroupindex
#622512
#334882
#227072
#118902
#118862
#118602
#118562
#117442
#117412
#117052
#117032