Finite Group Databasebeta

← #482963

Subgroups — #482963

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62247C211yesmaximal, characteristic
2#33487C211yesmaximal, characteristic, complemented
3#22708C211yesmaximal, characteristic, complemented
4#11889C211yesmaximal, characteristic, complemented
5#11888C211yesmaximal, characteristic, complemented
6#11885C212yesmaximal, complemented
7#11860C212yesmaximal, complemented
8#11746C212yesmaximal, complemented
9#11706C211yesmaximal, characteristic, complemented
10#11705C211yesmaximal, characteristic, complemented
11#11703C212yesmaximal, complemented
12#3505—21no—
13#3339C412yescomplemented
14#3336C412yescomplemented
15#3301—22no—
16#3296—22no—
17#3285C412yescomplemented
18#3280C412yescomplemented
19#3250—21no—
20#3191—21no—
21#3187—21no—
22#3168C2212yes—
23#3050C2212yes—
24C2×#467C2211yescharacteristic
25C2×#466C2211yescharacteristic
26C2×C8×D4C2211yescharacteristic
27#2354—21no—
28#2168—22no—
29#2168—22no—
30#2163C2212yescomplemented
31#2112C2211yescharacteristic
32#2112C2211yescharacteristic
33C2×C22⋊C16C2212yescomplemented
34#1948—21no—
35#1854—21no—
36#1854—21no—
37#1834C2211yescharacteristic, complemented
38#1807C2211yescharacteristic
39#1807C2211yescharacteristic, complemented
40C4×#107C2211yescharacteristic, complemented
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic, complemented
43#1348C2212yescomplemented
44#1348C2212yescomplemented
45#1348C2212yescomplemented
46#1348C2212yescomplemented
47#1347C2212yes—
48#1347C2212yes—
49#1347C2212yes—
50#1347C2212yes—
51C2×#135C2×C412yes—
52C22×Q16C2×C411yescharacteristic
53C22×D8C2×C411yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56#553—22no—
57#553—42no—
58#552—22no—
59#551—42no—
60#548—22no—
61#546—22no—
62#539—41no—
63#538—41no—
64#534—41no—
65#529—22no—
66#525—22no—
67C2×C22⋊Q8—21no—
68C2×#121—21no—
69#518—41no—
70C2×C4×D4D411yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C2×M32—21no—
74C2×M32—21no—
75C22×C16—22no—
76#499C2×C412yes—
77#495C2×C412yes—
78#488—22no—
79#485—22no—
80#483—22no—
81#482—22no—
82#481—22no—
83#479—22no—
84#478—22no—
85#467—21no—
86#467—21no—
87#467C2×C412yes—
88#467C2×C412yes—
89#467C2312yes—
90#466—21no—
91#466—21no—
92#466C2×C412yes—
93#466C2×C412yes—
94#466C2312yes—
95#465—22no—
96#465—22no—
97#465—22no—
98#464—22no—
99#463—22no—
100#452—22no—
101#450—22no—
102#449—22no—
103#448—22no—
104C4×Q16C2×C412yes—
105C4×SD16—22no—
106C4×D8C2×C412yes—
107C8×D4C2×C412yes—
108C8×D4C2×C412yes—
109C8×D4D412yes—
110C8×D4D412yes—
111#427C2×C412yes—
112C2×#107—21no—
113C2×#107C2×C411yescharacteristic
114C2×#107C2×C411yescharacteristic
115C2×#106—21no—
116C2×#106—21no—
117C2×C4⋊C8D411yescharacteristic
118#416—22no—
119C2×#103—21no—
120C2×#102—21no—
121C2×C22⋊C8D411yescharacteristic
122C2×C22⋊C8D411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#390—41no—
125#388—41no—
126#388—41no—
127#384—21no—
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—41no—
132C4⋊C16—22no—
133C22⋊C16—22no—
134C22⋊C16—21no—
135C22⋊C16—21no—
136C22⋊C16C2312yes—
137C22⋊C16C2312yes—
138#346—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340C2311yescharacteristic
143#340C2311yescharacteristic
144#340C2311yescharacteristic
145#340C2311yescharacteristic
146C2×#41—22no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—41no—
150C22×D4D811yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153#135—82no—
154C2×Q16—21no—
155C2×Q16—81no—
156C2×Q16C22×C412yes—
157C2×SD16—22no—
158C2×SD16—22no—
159C2×D8—21no—
160C2×D8—81no—
161C2×D8C22×C412yes—
162C22×C8—81no—
163C22×C8—81no—
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C22×C412yes—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D412yes—
173C22×C8C2411yescharacteristic
174#126—42no—
175#126—42no—
176#126—42no—
177#125—22no—
178#125—42no—
179#124—22no—
180#124—42no—
181#123—42no—
182#123—42no—
183#123—42no—
184#123—42no—
185#123—42no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—42no—
192#121—22no—
193#121—22no—
194#121—22no—
195#121—42no—
196#121—42no—
197#121—41no—
198#121—41no—
199#121—42no—
200#120—42no—
201#120—42no—
202C4×Q8—22no—
203C4×D4—22no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4C22⋊C412yes—
210C4×D4C22⋊C412yes—
211C4×D4SD1612yes—
212C4×D4SD1612yes—
213#117—22no—
214#117—42no—
215#117—42no—
216#117—42no—
217#117—42no—
218#117—81no—
219#117—81no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4D811yescharacteristic
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4D811yescharacteristic
238C2×C22⋊C4D811yescharacteristic
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42C2×D411yescharacteristic
243M32—81no—
244M32—81no—
245C2×C16—22no—
246C2×C16—22no—
247C2×C16—82no—
248#107—21no—
249#107—21no—
250#107—42no—
251#107C22×C412yes—
252#107C22×C412yes—
253#106—22no—
254#106—22no—
255#106—81no—
256#106—81no—
257C4⋊C8C22⋊C412yes—
258C4⋊C8C22⋊C412yes—
259#103—22no—
260#103—22no—
261#103—22no—
262#103—22no—
263#102—22no—
264#102—22no—
265#102—22no—
266#102—22no—
267C22⋊C8C22⋊C412yes—
268C22⋊C8C22⋊C412yes—
269C22⋊C8C22⋊C412yes—
270C22⋊C8C22⋊C412yes—
271C4×C8C22×C412yes—
272C4×C8C2×D412yes—
273#95—41no—
274#95—41no—
275#95—41no—
276C24—41no—
277C24C2×D811yescharacteristic
278C24C2×D811yescharacteristic
279#41—82no—
280#41—82no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4#10212yes—
298C2×D4#10212yes—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×D811yescharacteristic
316C22×C4C2×D811yescharacteristic
317C22×C4C2×D811yescharacteristic
318C22×C4C2×D811yescharacteristic
319C22×C4C2×SD1611yescharacteristic
320C22×C4C2×SD1611yescharacteristic
321C22×C4C2×SD1612yes—
322C22×C4C22×D411yescharacteristic
323Q16—42no—
324Q16—81no—
325SD16—82no—
326SD16—82no—
327D8—42no—
328D8—81no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8C2×C22⋊C412yes—
340C2×C8C2×C22⋊C411yescharacteristic
341C2×C8C2×C22⋊C412yes—
342C2×C8C2×C22⋊C411yescharacteristic
343C2×C8C23×C411yescharacteristic
344C2×C8C22×D411yescharacteristic
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—82no—
364C4⋊C4—161no—
365C4⋊C4#10212yes—
366C4⋊C4#10212yes—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—161no—
380C22⋊C4—161no—
381C22⋊C4#10212yes—
382C22⋊C4#10212yes—
383C22⋊C4#10212yes—
384C22⋊C4#10212yes—
385C42—42no—
386C42—42no—
387C42—82no—
388C42—82no—
389C42—82no—
390C42C2×C22⋊C412yes—
391C42C2×SD1612yes—
392C16—42no—
393C16—42no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—82no—
404C23—161no—
405C23—161no—
406C23C2×#10211yescharacteristic
407C23C2×#10212yes—
408C23C2×#10211yescharacteristic
409C23C22×D811yescharacteristic
410Q8—42no—
411Q8—42no—
412Q8—161no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—84no—
418D4—82no—
419D4—82no—
420D4—161no—
421D4—162no—
422D4—162no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—162no—
456C2×C4—162no—
457C2×C4—161no—
458C2×C4—161no—
459C2×C4C2×#10212yes—
460C2×C4C2×#10212yes—
461C2×C4C2×#10212yes—
462C2×C4C2×#10211yescharacteristic
463C2×C4C2×#10211yescharacteristic
464C2×C4C22×C22⋊C411yescharacteristic
465C2×C4C22×SD1611yescharacteristic
466C8—21no—
467C8—21no—
468C8—21no—
469C8—22no—
470C8—21no—
471C8—22no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—22no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—21no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—322no—
491C22C22×#10211yescharacteristic
492C22#341411yescharacteristic
493C22#341411yescharacteristic
494C4—22no—
495C4—22no—
496C4—21no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—161no—
505C4—161no—
506C4—161no—
507C4#288311yescharacteristic
508C4#290111yescharacteristic
509C2—21no—
510C2—21no—
511C2—41no—
512C2—42no—
513C2—41no—
514C2—162no—
515C2#1761311yescharacteristic
516C2#3338011yescharacteristic
517C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62247C2
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. #33487C2semidirect
3. #22708C2semidirect
4. #11889C2semidirect
5. #11888C2semidirect
6. #11885C2semidirect
7. #11860C2semidirect
8. #11746C2semidirect
9. #11706C2semidirect
10. #11705C2semidirect
11. #11703C2semidirect
13. #3339C4semidirect
14. #3336C4semidirect
17. #3285C4semidirect
18. #3280C4semidirect
30. #2163C22semidirect
33. C2×C22⋊C16C22semidirect
37. #1834C22semidirect
39. #1807C22semidirect
40. C4×#107C22semidirect
42. #1735C22semidirect
43. #1348C22semidirect
44. #1348C22semidirect
45. #1348C22semidirect
46. #1348C22semidirect

Maximal subgroups

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

subgroupindex
#622472
#334872
#227082
#118892
#118882
#118852
#118602
#117462
#117062
#117052
#117032