Finite Group Databasebeta

← #482217

Subgroups — #482217

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60708C211yesmaximal, characteristic
2#33422C211yesmaximal, characteristic, complemented
3#20763C211yesmaximal, characteristic, complemented
4#11860C212yesmaximal, complemented
5#11859C212yesmaximal, complemented
6#11857C212yesmaximal, complemented
7#11855C212yesmaximal, complemented
8#11706C211yesmaximal, characteristic, complemented
9#11704C211yesmaximal, characteristic
10#11703C211yesmaximal, characteristic
11C2×#1348C211yesmaximal, characteristic, complemented
12#3495—21no—
13#3181—22no—
14#3178—22no—
15#3177C412yescomplemented
16#3175C412yescomplemented
17#3170C2212yes—
18#3168C2212yes—
19#3164C411yescharacteristic, complemented
20#3156C411yescharacteristic, complemented
21#3137—21no—
22#3056—21no—
23#3050C2211yescharacteristic
24C2×#467C2211yescharacteristic
25#2951C411yescharacteristic, complemented
26#2950C411yescharacteristic, complemented
27#2947—21no—
28C2×#405C2211yescharacteristic
29#2317—21no—
30#2164—22no—
31#2163C2212yes—
32#2162C2212yes—
33#2151—21no—
34#2112C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36#2108—21no—
37#1836—22no—
38#1835C2212yes—
39#1834C2212yes—
40#1736—21no—
41#1735C2211yescharacteristic
42C2×#340C2211yescharacteristic
43#1348C2212yes—
44#1348C2212yes—
45#1348C2211yescharacteristic
46#1348C2212yes—
47#1348C2212yes—
48#1348C2211yescharacteristic
49#1348C2212yes—
50#1347C2211yescharacteristic
51#1347C2212yes—
52#1347C2211yescharacteristic
53C2×#137—21no—
54C2×#135C2×C411yescharacteristic
55C22×Q16C2×C411yescharacteristic
56C22×M16D411yescharacteristic
57C23×C8C2311yescharacteristic
58#556—41no—
59#554—41no—
60#552—42no—
61#552—41no—
62#552—41no—
63#551—42no—
64#544—21no—
65#542—22no—
66#541—22no—
67#536—21no—
68#533—22no—
69#529—22no—
70#525—21no—
71C2×C22⋊Q8—21no—
72#520—21no—
73#519—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#497—22no—
83#495C2×C412yescomplemented
84#494C2×C412yescomplemented
85#491—22no—
86#489—22no—
87#488—22no—
88#487—22no—
89#470—22no—
90#468—22no—
91#467—22no—
92#467—21no—
93#467—21no—
94#467C2×C412yes—
95#467C2×C411yescharacteristic
96#467C2312yes—
97#467C2311yescharacteristic
98#466C2×C411yescharacteristic
99#466C2311yescharacteristic
100#465—22no—
101#465—21no—
102#465—21no—
103#441—22no—
104#441—22no—
105#440—22no—
106#440—22no—
107C4×Q16—22no—
108C4×Q16C2×C412yescomplemented
109C4×Q16C2×C412yescomplemented
110C4×Q16C2×C412yescomplemented
111C4×SD16—22no—
112C4×D8C2×C412yescomplemented
113#428—21no—
114#427C2×C411yescharacteristic
115C2×#107C2×C411yescharacteristic
116#419—21no—
117#417—21no—
118#416—21no—
119C2×#103—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—
150C22×Q8—21no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153#137—44no—
154#135—42no—
155C2×Q16—21no—
156C2×Q16—22no—
157C2×Q16—42no—
158C2×Q16C22×C412yes—
159C2×Q16C22×C411yescharacteristic
160C2×SD16—21no—
161C2×SD16—22no—
162C2×D8C22×C411yescharacteristic
163C2×M16—42no—
164C2×M16C22⋊C411yescharacteristic
165C2×M16C22⋊C411yescharacteristic
166C2×M16C22⋊C412yes—
167C22×C8—42no—
168C22×C8C22×C411yescharacteristic
169C22×C8C22×C411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D412yes—
173C22×C8C2×D411yescharacteristic
174C22×C8C2411yescharacteristic
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—41no—
178#126—42no—
179#126—42no—
180#126—42no—
181#126—42no—
182#125—22no—
183#125—42no—
184#125—42no—
185#125—41no—
186#125—42no—
187#124—22no—
188#124—42no—
189#123—42no—
190#123—41no—
191#123—42no—
192#123—42no—
193C22⋊Q8—22no—
194C22⋊Q8—21no—
195C22⋊Q8—22no—
196C22⋊Q8—21no—
197C22⋊Q8—41no—
198C22⋊Q8—42no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204#121—21no—
205#121—21no—
206#120—41no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—42no—
213C4×D4—22no—
214C4×D4—41no—
215C4×D4—42no—
216C4×D4—41no—
217C4×D4—42no—
218#117—21no—
219#117—42no—
220#117—41no—
221#117—42no—
222#117—42no—
223#117—42no—
224#117—42no—
225#117C22⋊C411yescharacteristic
226#117C22⋊C411yescharacteristic
227#117C22⋊C412yes—
228#117SD1611yescharacteristic
229#117SD1612yes—
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—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4D811yescharacteristic
243C2×C4⋊C4D811yescharacteristic
244C2×C22⋊C4—41no—
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—22no—
262#103—21no—
263#103—22no—
264#103—22no—
265#103—22no—
266#103—22no—
267#103—21no—
268#102—21no—
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—
281C24C2×D811yescharacteristic
282#41—42no—
283#41—42no—
284C2×Q8—22no—
285C2×Q8—21no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—161no—
305C22×C4—161no—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×D812yes—
309C22×C4C2×D811yescharacteristic
310C22×C4C2×D812yes—
311C22×C4C2×SD1611yescharacteristic
312C22×C4C2×SD1611yescharacteristic
313C22×C4C22×D411yescharacteristic
314Q16—42no—
315Q16—41no—
316Q16—42no—
317Q16—44no—
318SD16—42no—
319SD16—44no—
320D8—41no—
321M16—22no—
322M16—22no—
323M16—42no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8C2×C22⋊C412yes—
332C2×C8C2×C22⋊C412yes—
333C2×C8C2×C22⋊C411yescharacteristic
334C2×C8C2×C22⋊C411yescharacteristic
335C2×C8C23×C411yescharacteristic
336C2×C8C22×D411yescharacteristic
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—82no—
360C4⋊C4—162no—
361C4⋊C4—162no—
362C4⋊C4#10211yescharacteristic
363C4⋊C4#10211yescharacteristic
364C4⋊C4#10212yes—
365C4⋊C4#10211yescharacteristic
366C4⋊C4#10211yescharacteristic
367C4⋊C4#10212yes—
368C22⋊C4—42no—
369C22⋊C4—41no—
370C22⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—42no—
378C22⋊C4—82no—
379C22⋊C4#10212yes—
380C22⋊C4#10212yes—
381C22⋊C4#10212yes—
382C22⋊C4#10212yes—
383C42—41no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—82no—
388C42C2×C22⋊C412yes—
389C42C2×C22⋊C412yes—
390C42C2×SD1612yes—
391C42C2×SD1612yes—
392C16—41no—
393C16—41no—
394C16—42no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—41no—
399C23—41no—
400C23C2×#10211yescharacteristic
401C23C2×#10211yescharacteristic
402C23C22×D811yescharacteristic
403Q8—41no—
404Q8—42no—
405Q8—42no—
406Q8—41no—
407Q8—82no—
408Q8—162no—
409D4—41no—
410D4—41no—
411D4—81no—
412D4—81no—
413D4—82no—
414D4—161no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—162no—
442C2×C4—161no—
443C2×C4—164no—
444C2×C4#41012yes—
445C2×C4C2×#10212yes—
446C2×C4C2×#10212yes—
447C2×C4C2×#10212yes—
448C2×C4C2×#10212yes—
449C2×C4C2×#10211yescharacteristic
450C2×C4C2×#10211yescharacteristic
451C2×C4#42112yes—
452C2×C4C22×C22⋊C411yescharacteristic
453C2×C4C22×SD1611yescharacteristic
454C8—21no—
455C8—21no—
456C8—21no—
457C8—21no—
458C8—22no—
459C8—22no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—42no—
468C22—42no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#151712yes—
473C22C22×#10211yescharacteristic
474C22#341611yescharacteristic
475C22#341711yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—21no—
481C4—21no—
482C4—81no—
483C4—82no—
484C4—81no—
485C4—81no—
486C4—82no—
487C4—161no—
488C4—161no—
489C4—162no—
490C4#151712yes—
491C4C2×#41011yescharacteristic
492C4C2×#42111yescharacteristic
493C2—21no—
494C2—21no—
495C2—22no—
496C2—41no—
497C2—161no—
498C2C2×#151711yescharacteristic
499C2#3338211yescharacteristic
500C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#3050C22
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. #11859C2semidirect
6. #11857C2semidirect
7. #11855C2semidirect
8. #11706C2semidirect
11. C2×#1348C2semidirect
15. #3177C4semidirect
16. #3175C4semidirect
19. #3164C4semidirect
20. #3156C4semidirect
25. #2951C4semidirect
26. #2950C4semidirect
80. #500C2×C4semidirect
81. #499C2×C4semidirect
83. #495C2×C4semidirect
84. #494C2×C4semidirect
108. C4×Q16C2×C4semidirect
109. C4×Q16C2×C4semidirect
110. C4×Q16C2×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
#607082
#334222
#207632
#118602
#118592
#118572
#118552
#117062
#117042
#117032
C2×#13482