Finite Group Databasebeta

← #482216

Subgroups — #482216

Order 512.

786 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#60707C211yesmaximal, characteristic
2#33422C211yesmaximal, characteristic, complemented
3#20763C211yesmaximal, characteristic, complemented
4#11860C212yesmaximal, complemented
5#11858C212yesmaximal, complemented
6#11856C212yesmaximal, complemented
7#11855C212yesmaximal, complemented
8#11705C211yesmaximal, characteristic, complemented
9#11704C211yesmaximal, characteristic, complemented
10#11703C211yesmaximal, characteristic, complemented
11C2×#1347C211yesmaximal, characteristic, complemented
12#3494—21no—
13#3180—22no—
14#3179—22no—
15#3176C412yescomplemented
16#3175C412yescomplemented
17#3169C2212yes—
18#3168C2212yes—
19#3164C411yescharacteristic, complemented
20#3155C411yescharacteristic, complemented
21#3136—21no—
22#3055—21no—
23#3050C2211yescharacteristic
24C2×#466C2211yescharacteristic
25#2951C411yescharacteristic, complemented
26#2948C411yescharacteristic, complemented
27#2946—21no—
28C2×#405C2211yescharacteristic
29#2317—21no—
30#2164—22no—
31#2163C2212yescomplemented
32#2162C2212yescomplemented
33#2151—21no—
34#2112C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36#2108—21no—
37#1836—22no—
38#1835C2212yescomplemented
39#1834C2212yescomplemented
40#1736—21no—
41#1735C2211yescharacteristic
42C2×#340C2211yescharacteristic
43#1348C2211yescharacteristic, complemented
44#1348C2212yescomplemented
45#1348C2211yescharacteristic, complemented
46#1347C2212yes—
47#1347C2212yes—
48#1347C2212yes—
49#1347C2212yes—
50#1347C2211yescharacteristic, complemented
51#1347C2212yescomplemented
52#1347C2211yescharacteristic, complemented
53C2×AGL(1,ℤ/8)—21no—
54C2×#135C2×C411yescharacteristic
55C22×D8C2×C411yescharacteristic
56C22×M16D411yescharacteristic
57C23×C8C2311yescharacteristic
58#555—41no—
59#553—41no—
60#553—41no—
61#551—42no—
62#547—41no—
63#546—42no—
64#544—21no—
65#542—22no—
66#540—22no—
67#535—21no—
68#532—22no—
69#529—22no—
70#525—21no—
71C2×#121—21no—
72#520—21no—
73#518—21no—
74C2×#117—21no—
75C2×#117D411yescharacteristic
76C22×C4⋊C4—21no—
77C2×M32—21no—
78C22×C16—21no—
79#501—22no—
80#500C2×C412yescomplemented
81#499C2×C412yescomplemented
82#496—22no—
83#495C2×C412yescomplemented
84#493C2×C412yescomplemented
85#490—22no—
86#489—22no—
87#488—22no—
88#486—22no—
89#469—22no—
90#468—22no—
91#467C2×C411yescharacteristic
92#467C2311yescharacteristic
93#466—22no—
94#466—21no—
95#466—21no—
96#466C2×C412yes—
97#466C2×C411yescharacteristic
98#466C2312yes—
99#466C2311yescharacteristic
100#465—22no—
101#465—21no—
102#465—21no—
103#442—22no—
104#442—22no—
105#440—22no—
106#440—22no—
107C4×Q16C2×C412yescomplemented
108C4×SD16—22no—
109C4×D8—22no—
110C4×D8C2×C412yescomplemented
111C4×D8C2×C412yescomplemented
112C4×D8C2×C412yescomplemented
113#428—21no—
114#427C2×C411yescharacteristic
115C2×#107C2×C411yescharacteristic
116#418—21no—
117#417—21no—
118#416—21no—
119C2×#102—21no—
120#405C2×C411yescharacteristic
121#405C2×C412yes—
122#405C2×C411yescharacteristic
123#405D411yescharacteristic
124#405D411yescharacteristic
125#405D412yes—
126C2×#97D412yes—
127C2×C4×C8C2312yes—
128#389—41no—
129#389—41no—
130#385—42no—
131#384—22no—
132#382—22no—
133C4⋊C16—22no—
134C4⋊C16—22no—
135C22⋊C16—21no—
136C22⋊C16—21no—
137C22⋊C16C2311yescharacteristic
138C22⋊C16C2311yescharacteristic
139C22⋊C16C2312yes—
140#346—22no—
141C4×C16—22no—
142#340—22no—
143#340—22no—
144#340—21no—
145#340—21no—
146#340C2311yescharacteristic
147#340C2312yes—
148#340C2311yescharacteristic
149C2×#41—21no—
150C2×#41—42no—
151C22×D4—21no—
152C22×D4—41no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155AGL(1,ℤ/8)—44no—
156#135—42no—
157C2×Q16C22×C411yescharacteristic
158C2×SD16—21no—
159C2×SD16—22no—
160C2×D8—21no—
161C2×D8—22no—
162C2×D8—42no—
163C2×D8C22×C412yes—
164C2×D8C22×C411yescharacteristic
165C2×M16—42no—
166C2×M16C22⋊C411yescharacteristic
167C2×M16C22⋊C411yescharacteristic
168C2×M16C22⋊C412yes—
169C22×C8—42no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D412yes—
175C22×C8C2×D411yescharacteristic
176C22×C8C2411yescharacteristic
177C4⋊Q8—22no—
178C4⋊Q8—41no—
179#127—22no—
180#126—42no—
181#126—42no—
182#125—22no—
183#125—41no—
184#124—22no—
185#124—42no—
186#124—42no—
187#123—42no—
188#123—41no—
189#123—42no—
190#123—42no—
191#123—42no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196#121—22no—
197#121—21no—
198#121—22no—
199#121—21no—
200#121—41no—
201#121—42no—
202#121—41no—
203#121—41no—
204#121—41no—
205#121—42no—
206#121—42no—
207#120—41no—
208#120—42no—
209C4×Q8—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—41no—
214C4×D4—42no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—42no—
219C4×D4—42no—
220#117—21no—
221#117—41no—
222#117—42no—
223#117—42no—
224#117—42no—
225#117C22⋊C411yescharacteristic
226#117C22⋊C411yescharacteristic
227#117C22⋊C412yes—
228#117SD1612yes—
229#117SD1611yescharacteristic
230#117SD1611yescharacteristic
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4D811yescharacteristic
242C2×C4⋊C4D811yescharacteristic
243C2×C22⋊C4—41no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4D812yes—
247C2×C42—41no—
248C2×C42C2×D412yes—
249M32—42no—
250C2×C16—21no—
251C2×C16—22no—
252C2×C16—21no—
253C2×C16—42no—
254#107—21no—
255#107—22no—
256#107C22×C412yes—
257#107C22×C411yescharacteristic
258#107C22×C411yescharacteristic
259#106—21no—
260#106—22no—
261#103—21no—
262#103—22no—
263#103—21no—
264#102—22no—
265#102—21no—
266#102—22no—
267#102—22no—
268#102—22no—
269#102—22no—
270#102—21no—
271#97C22⋊C412yes—
272#97C22⋊C412yes—
273#97C22⋊C412yes—
274#97C22⋊C412yes—
275C4×C8C22×C412yes—
276C4×C8C22×C412yes—
277C4×C8C2×D412yes—
278C4×C8C2×D412yes—
279#95—42no—
280#95—42no—
281C24—41no—
282C24C2×D811yescharacteristic
283#41—42no—
284#41—42no—
285#41—162no—
286#41—162no—
287C2×Q8—21no—
288C2×Q8—42no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—161no—
302C2×D4—161no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—82no—
314C22×C4—161no—
315C22×C4—161no—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×D812yes—
319C22×C4C2×D811yescharacteristic
320C22×C4C2×D812yes—
321C22×C4C2×SD1611yescharacteristic
322C22×C4C2×SD1611yescharacteristic
323C22×C4C22×D411yescharacteristic
324Q16—41no—
325SD16—42no—
326SD16—44no—
327D8—42no—
328D8—41no—
329D8—42no—
330D8—44no—
331M16—22no—
332M16—22no—
333M16—42no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8C2×C22⋊C412yes—
342C2×C8C2×C22⋊C412yes—
343C2×C8C2×C22⋊C411yescharacteristic
344C2×C8C2×C22⋊C411yescharacteristic
345C2×C8C23×C411yescharacteristic
346C2×C8C22×D411yescharacteristic
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—82no—
364C4⋊C4—162no—
365C4⋊C4—162no—
366C4⋊C4#10211yescharacteristic
367C4⋊C4#10211yescharacteristic
368C4⋊C4#10211yescharacteristic
369C4⋊C4#10211yescharacteristic
370C4⋊C4#10212yes—
371C4⋊C4#10212yes—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—41no—
384C22⋊C4—82no—
385C22⋊C4#10212yes—
386C22⋊C4#10212yes—
387C22⋊C4#10212yes—
388C22⋊C4#10212yes—
389C42—41no—
390C42—42no—
391C42—82no—
392C42C2×C22⋊C412yes—
393C42C2×C22⋊C412yes—
394C42C2×SD1612yes—
395C42C2×SD1612yes—
396C16—41no—
397C16—41no—
398C16—42no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—82no—
406C23—161no—
407C23—161no—
408C23C2×#10211yescharacteristic
409C23C2×#10211yescharacteristic
410C23C22×D811yescharacteristic
411Q8—41no—
412Q8—41no—
413Q8—162no—
414D4—42no—
415D4—41no—
416D4—41no—
417D4—42no—
418D4—82no—
419D4—81no—
420D4—82no—
421D4—81no—
422D4—82no—
423D4—82no—
424D4—162no—
425D4—162no—
426D4—161no—
427D4—162no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—162no—
453C2×C4—162no—
454C2×C4—162no—
455C2×C4—161no—
456C2×C4—164no—
457C2×C4#41012yes—
458C2×C4C2×#10212yes—
459C2×C4C2×#10212yes—
460C2×C4C2×#10212yes—
461C2×C4C2×#10212yes—
462C2×C4C2×#10211yescharacteristic
463C2×C4C2×#10211yescharacteristic
464C2×C4#42112yes—
465C2×C4C22×C22⋊C411yescharacteristic
466C2×C4C22×SD1611yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8—21no—
471C8—22no—
472C8—22no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—82no—
484C22—82no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22—322no—
489C22#151712yes—
490C22C22×#10211yescharacteristic
491C22#341511yescharacteristic
492C22#341611yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—21no—
498C4—21no—
499C4—82no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—161no—
504C4—161no—
505C4—162no—
506C4#151712yes—
507C4C2×#41011yescharacteristic
508C4C2×#42111yescharacteristic
509C2—21no—
510C2—21no—
511C2—22no—
512C2—41no—
513C2—161no—
514C2—162no—
515C2C2×#151711yescharacteristic
516C2#3338111yescharacteristic
517C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#60707C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33422C2semidirect
3. #20763C2semidirect
4. #11860C2semidirect
5. #11858C2semidirect
6. #11856C2semidirect
7. #11855C2semidirect
8. #11705C2semidirect
9. #11704C2semidirect
10. #11703C2semidirect
11. C2×#1347C2semidirect
15. #3176C4semidirect
16. #3175C4semidirect
19. #3164C4semidirect
20. #3155C4semidirect
25. #2951C4semidirect
26. #2948C4semidirect
31. #2163C22semidirect
32. #2162C22semidirect
38. #1835C22semidirect
39. #1834C22semidirect
43. #1348C22semidirect
44. #1348C22semidirect
45. #1348C22semidirect
50. #1347C22semidirect
51. #1347C22semidirect
52. #1347C22semidirect
80. #500C2×C4semidirect
81. #499C2×C4semidirect
83. #495C2×C4semidirect
84. #493C2×C4semidirect
107. C4×Q16C2×C4semidirect
110. C4×D8C2×C4semidirect
111. C4×D8C2×C4semidirect
112. C4×D8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#607072
#334222
#207632
#118602
#118582
#118562
#118552
#117052
#117042
#117032
C2×#13472