Finite Group Databasebeta

← #482962

Subgroups — #482962

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62247C211yesmaximal, characteristic
2#33487C211yesmaximal, characteristic, complemented
3#22705C211yesmaximal, characteristic, complemented
4#11890C212yesmaximal, complemented
5#11887C211yesmaximal, characteristic, complemented
6#11886C211yesmaximal, characteristic, complemented
7#11860C212yesmaximal, complemented
8#11746C212yesmaximal, complemented
9#11704C212yesmaximal, complemented
10C2×#1348C211yesmaximal, characteristic, complemented
11C2×#1347C211yesmaximal, characteristic, 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
27D4×C4⋊C4—21no—
28#2168—22no—
29#2168—22no—
30#2163C2212yescomplemented
31#2112C2212yescomplemented
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34#1947—21no—
35#1852—21no—
36#1852—21no—
37#1834C2211yescharacteristic, complemented
38#1808C2211yescharacteristic
39#1808C2211yescharacteristic, complemented
40C4×#107C2211yescharacteristic, complemented
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic, complemented
43#1348C2212yescomplemented
44#1348C2212yescomplemented
45#1348C2212yescomplemented
46#1348C2212yescomplemented
47#1347C2212yes—
48#1347C2212yes—
49#1347C2212yes—
50#1347C2212yes—
51C2×#135C2×C412yescomplemented
52C22×Q16C2×C411yescharacteristic, complemented
53C22×D8C2×C411yescharacteristic, complemented
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×D4—41no—
71C2×C4×D4D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C2×M32—22no—
75C22×C16—21no—
76C22×C16—21no—
77#499C2×C412yescomplemented
78#495C2×C412yescomplemented
79#488—22no—
80#485—22no—
81#483—22no—
82#482—22no—
83#481—22no—
84#479—22no—
85#478—22no—
86#467—21no—
87#467—21no—
88#467C2×C412yescomplemented
89#467C2×C412yescomplemented
90#467C2312yes—
91#466—21no—
92#466—21no—
93#466C2×C412yescomplemented
94#466C2×C412yescomplemented
95#466C2312yes—
96#465—22no—
97#465—22no—
98#465—22no—
99#464—22no—
100#463—22no—
101#452—22no—
102#450—22no—
103#449—22no—
104#448—22no—
105C4×Q16C2×C412yescomplemented
106C4×SD16—22no—
107C4×D8C2×C412yescomplemented
108C8×D4C2×C412yes—
109C8×D4C2×C412yes—
110C8×D4D412yes—
111C8×D4D412yes—
112#427C2×C412yescomplemented
113C2×#107—21no—
114C2×#107—21no—
115C2×#107—21no—
116C2×#107C2×C411yescharacteristic, complemented
117C2×#107C2×C411yescharacteristic, complemented
118C2×C4⋊C8D411yescharacteristic
119#416—22no—
120C2×#103—21no—
121C2×#102—21no—
122C2×C22⋊C8D411yescharacteristic
123C2×C22⋊C8D411yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#389—41no—
126#385—41no—
127#385—41no—
128#384—21no—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4⋊C16—22no—
133C22⋊C16—21no—
134C22⋊C16—21no—
135C22⋊C16—22no—
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×C4—41no—
152C23×C4—41no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155#135—82no—
156C2×Q16—21no—
157C2×Q16—81no—
158C2×Q16C22×C412yes—
159C2×SD16—22no—
160C2×SD16—22no—
161C2×D8—21no—
162C2×D8—81no—
163C2×D8C22×C412yes—
164C22×C8—81no—
165C22×C8—81no—
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C22×C412yes—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D412yes—
175C22×C8C2411yescharacteristic
176#126—42no—
177#126—42no—
178#126—42no—
179#125—22no—
180#125—42no—
181#124—22no—
182#124—42no—
183#123—42no—
184#123—42no—
185#123—42no—
186#123—42no—
187#123—42no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—42no—
194#121—22no—
195#121—22no—
196#121—22no—
197#121—42no—
198#121—42no—
199#121—41no—
200#121—41no—
201#121—42no—
202#120—42no—
203#120—42no—
204C4×Q8—22no—
205C4×D4—22no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—84no—
212C4×D4C22⋊C412yes—
213C4×D4C22⋊C412yes—
214C4×D4SD1612yes—
215C4×D4SD1612yes—
216#117—22no—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4—81no—
234C2×C4⋊C4D811yescharacteristic
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4D811yescharacteristic
241C2×C22⋊C4D811yescharacteristic
242C2×C42—41no—
243C2×C42C2×D411yescharacteristic
244M32—82no—
245C2×C16—22no—
246C2×C16—22no—
247C2×C16—81no—
248C2×C16—81no—
249#107—21no—
250#107—21no—
251#107—42no—
252#107—81no—
253#107—81no—
254#107C22×C412yes—
255#107C22×C412yes—
256#106—22no—
257#106—22no—
258C4⋊C8C22⋊C412yes—
259C4⋊C8C22⋊C412yes—
260#103—22no—
261#103—22no—
262#103—22no—
263#103—22no—
264#102—22no—
265#102—22no—
266#102—22no—
267#102—22no—
268C22⋊C8C22⋊C412yes—
269C22⋊C8C22⋊C412yes—
270C22⋊C8C22⋊C412yes—
271C22⋊C8C22⋊C412yes—
272C4×C8C22×C412yes—
273C4×C8C2×D412yes—
274#95—41no—
275#95—41no—
276#95—41no—
277C24—41no—
278C24C2×D811yescharacteristic
279C24C2×D811yescharacteristic
280#41—82no—
281#41—82no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—81no—
285C2×Q8—81no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—81no—
295C2×D4—81no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4#10212yes—
299C2×D4#10212yes—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—161no—
318C22×C4—161no—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4C2×D811yescharacteristic
326C22×C4C2×SD1611yescharacteristic
327C22×C4C2×SD1611yescharacteristic
328C22×C4C2×SD1612yes—
329C22×C4C22×D411yescharacteristic
330Q16—42no—
331Q16—81no—
332SD16—82no—
333SD16—82no—
334D8—42no—
335D8—81no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8C2×C22⋊C412yes—
347C2×C8C2×C22⋊C411yescharacteristic
348C2×C8C2×C22⋊C412yes—
349C2×C8C2×C22⋊C411yescharacteristic
350C2×C8C23×C411yescharacteristic
351C2×C8C22×D411yescharacteristic
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—161no—
374C4⋊C4#10212yes—
375C4⋊C4#10212yes—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4#10212yes—
389C22⋊C4#10212yes—
390C22⋊C4#10212yes—
391C22⋊C4#10212yes—
392C42—42no—
393C42—42no—
394C42—82no—
395C42C2×C22⋊C412yes—
396C42C2×SD1612yes—
397C16—42no—
398C16—42no—
399C23—21no—
400C23—21no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—82no—
407C23—161no—
408C23—161no—
409C23C2×#10212yes—
410C23C2×#10211yescharacteristic
411C23C2×#10211yescharacteristic
412C23#50912yes—
413C23#51012yes—
414C23C22×D811yescharacteristic
415Q8—42no—
416Q8—42no—
417Q8—161no—
418D4—24no—
419D4—24no—
420D4—42no—
421D4—42no—
422D4—82no—
423D4—82no—
424D4—84no—
425D4—161no—
426D4—162no—
427D4—162no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—162no—
461C2×C4—162no—
462C2×C4—161no—
463C2×C4—161no—
464C2×C4—164no—
465C2×C4—164no—
466C2×C4C2×#10212yes—
467C2×C4C2×#10212yes—
468C2×C4C2×#10212yes—
469C2×C4C2×#10211yescharacteristic
470C2×C4C2×#10211yescharacteristic
471C2×C4C22×C22⋊C411yescharacteristic
472C2×C4C22×SD1611yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8—22no—
477C8—21no—
478C8—22no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—322no—
496C22#213912yes—
497C22#213912yes—
498C22C22×#10211yescharacteristic
499C22C2×#50911yescharacteristic
500C22C2×#51011yescharacteristic
501C4—22no—
502C4—22no—
503C4—21no—
504C4—22no—
505C4—21no—
506C4—21no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4#288311yescharacteristic
515C4#290111yescharacteristic
516C2—21no—
517C2—21no—
518C2—22no—
519C2—22no—
520C2—42no—
521C2—162no—
522C2#1761311yescharacteristic
523C2C2×#213911yescharacteristic
524C2#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. #22705C2semidirect
4. #11890C2semidirect
5. #11887C2semidirect
6. #11886C2semidirect
7. #11860C2semidirect
8. #11746C2semidirect
9. #11704C2semidirect
10. C2×#1348C2semidirect
11. C2×#1347C2semidirect
13. #3339C4semidirect
14. #3336C4semidirect
17. #3285C4semidirect
18. #3280C4semidirect
30. #2163C22semidirect
31. #2112C22semidirect
37. #1834C22semidirect
39. #1808C22semidirect
40. C4×#107C22semidirect
42. C2×#340C22semidirect
43. #1348C22semidirect
44. #1348C22semidirect
45. #1348C22semidirect
46. #1348C22semidirect
51. C2×#135C2×C4semidirect
52. C22×Q16C2×C4semidirect
53. C22×D8C2×C4semidirect
77. #499C2×C4semidirect
78. #495C2×C4semidirect
88. #467C2×C4semidirect
89. #467C2×C4semidirect
93. #466C2×C4semidirect
94. #466C2×C4semidirect
105. C4×Q16C2×C4semidirect
107. C4×D8C2×C4semidirect
112. #427C2×C4semidirect
116. C2×#107C2×C4semidirect
117. C2×#107C2×C4semidirect

Maximal subgroups

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

subgroupindex
#622472
#334872
#227052
#118902
#118872
#118862
#118602
#117462
#117042
C2×#13482
C2×#13472