Finite Group Databasebeta

← #482964

Subgroups — #482964

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62249C211yesmaximal, characteristic
2#33487C211yesmaximal, characteristic, complemented
3#22706C211yesmaximal, characteristic, complemented
4#11890C212yesmaximal, complemented
5#11885C212yesmaximal, complemented
6#11859C211yesmaximal, characteristic, complemented
7#11858C211yesmaximal, characteristic, complemented
8#11745C211yesmaximal, characteristic, complemented
9#11744C211yesmaximal, characteristic, complemented
10#11704C212yesmaximal, complemented
11#11703C212yesmaximal, complemented
12#3506—21no—
13#3342C412yescomplemented
14#3341C412yescomplemented
15#3304—22no—
16#3303—22no—
17#3287C412yescomplemented
18#3283C412yescomplemented
19#3246—21no—
20#3191—21no—
21#3191—21no—
22#3170C2211yescharacteristic
23#3169C2211yescharacteristic
24#3050C2212yes—
25#3050C2212yes—
26C2×C8×D4C2211yescharacteristic
27#2352—21no—
28#2168—22no—
29#2168—22no—
30#2163C2211yescharacteristic
31#2163C2211yescharacteristic
32#2112C2212yes—
33C2×C22⋊C16C2212yes—
34#1949—21no—
35#1854—21no—
36#1853—21no—
37#1835C2211yescharacteristic
38#1808C2211yescharacteristic
39#1807C2211yescharacteristic
40C4×#106C2211yescharacteristic
41#1735C2211yescharacteristic
42C2×#340C2211yescharacteristic
43#1348C2212yes—
44#1348C2212yes—
45#1348C2212yes—
46#1348C2212yes—
47#1347C2212yes—
48#1347C2212yes—
49#1347C2212yes—
50#1347C2212yes—
51C2×#135C2×C412yes—
52C2×#135C2×C412yescomplemented
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55#559—42no—
56#556—22no—
57#555—22no—
58#553—42no—
59#550—22no—
60#547—22no—
61#538—41no—
62#538—41no—
63#535—41no—
64#533—21no—
65#532—21no—
66#525—22no—
67#525—22no—
68#520—41no—
69C2×C4×D4D411yescharacteristic
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C2×M32—22no—
73C22×C16—22no—
74#500C2×C411yescharacteristic, complemented
75#500C2×C411yescharacteristic, complemented
76#494C2×C411yescharacteristic, complemented
77#493C2×C411yescharacteristic, complemented
78#487—21no—
79#486—21no—
80#485—22no—
81#485—22no—
82#482—22no—
83#482—22no—
84#476—22no—
85#475—22no—
86#467—21no—
87#467C2×C412yes—
88#467C2×C412yescomplemented
89#467C2312yes—
90#466—21no—
91#466C2×C412yes—
92#466C2×C412yescomplemented
93#466C2312yes—
94#465—22no—
95#465—22no—
96#465—22no—
97#465—22no—
98#461—22no—
99#460—22no—
100#452—22no—
101#452—22no—
102#449—22no—
103#449—22no—
104C4×Q16—21no—
105C4×Q16C2×C412yescomplemented
106C4×D8—21no—
107C4×D8C2×C412yescomplemented
108C8×D4C2×C412yes—
109C8×D4C2×C412yes—
110C8×D4D412yes—
111C8×D4D412yes—
112#427C2×C412yes—
113#427C2×C412yescomplemented
114C2×#107—21no—
115C2×#106—21no—
116C2×#106—21no—
117C2×C4⋊C8D411yescharacteristic
118#416—22no—
119#416—22no—
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#388—41no—
124#387—41no—
125#386—41no—
126#382—21no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C22⋊C4—41no—
131C4⋊C16—21no—
132C4⋊C16—21no—
133C22⋊C16—22no—
134C22⋊C16—22no—
135C22⋊C16C2312yes—
136C22⋊C16C2312yes—
137#346—21no—
138#346—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340C2311yescharacteristic
143#340C2311yescharacteristic
144#340C2311yescharacteristic
145#340C2311yescharacteristic
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—41no—
149C22×D4D811yescharacteristic
150C23×C4—41no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153#135—82no—
154#135—82no—
155C2×Q16—21no—
156C2×Q16C22×C412yes—
157C2×SD16—22no—
158C2×SD16—22no—
159C2×D8—21no—
160C2×D8C22×C412yes—
161C22×C8—81no—
162C22×C8—81no—
163C22×C8C22×C412yes—
164C22×C8C22×C412yes—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8C2411yescharacteristic
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174#127—21no—
175#127—42no—
176#126—42no—
177#126—42no—
178#125—41no—
179#124—42no—
180#124—42no—
181#124—42no—
182#124—41no—
183#123—42no—
184#123—42no—
185#123—42no—
186#123—42no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—41no—
191#121—22no—
192#121—22no—
193#121—22no—
194#121—42no—
195#121—42no—
196#121—41no—
197#121—42no—
198#121—42no—
199#120—42no—
200#120—42no—
201C4×Q8—22no—
202C4×Q8—42no—
203C4×Q8—41no—
204C4×D4—22no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—41no—
211C4×D4C22⋊C412yes—
212C4×D4C22⋊C412yes—
213C4×D4SD1612yes—
214C4×D4SD1612yes—
215#117—22no—
216#117—22no—
217#117—41no—
218#117—42no—
219#117—42no—
220#117—81no—
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—81no—
229C2×C4⋊C4D811yescharacteristic
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4D811yescharacteristic
234C2×C22⋊C4D811yescharacteristic
235C2×C42—41no—
236C2×C42—41no—
237C2×C42C2×D411yescharacteristic
238M32—82no—
239C2×C16—22no—
240C2×C16—22no—
241C2×C16—82no—
242#107—21no—
243#107—21no—
244#107—81no—
245#107C22×C412yes—
246#107C22×C412yes—
247#106—22no—
248#106—22no—
249#106—42no—
250#106—81no—
251C4⋊C8C22⋊C412yes—
252C4⋊C8C22⋊C412yes—
253#103—22no—
254#103—22no—
255#103—22no—
256#103—22no—
257#102—22no—
258#102—22no—
259#102—22no—
260#102—22no—
261C22⋊C8C22⋊C412yes—
262C22⋊C8C22⋊C412yes—
263C22⋊C8C22⋊C412yes—
264C22⋊C8C22⋊C412yes—
265C4×C8C22×C411yescharacteristic
266C4×C8C22×C411yescharacteristic
267C4×C8C2×D411yescharacteristic
268C4×C8C2×D411yescharacteristic
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24C2×D811yescharacteristic
275C24C2×D811yescharacteristic
276#41—82no—
277#41—82no—
278#41—82no—
279#41—82no—
280#41—82no—
281#41—82no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4#10212yes—
296C2×D4#10212yes—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—161no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×D811yescharacteristic
314C22×C4C2×D811yescharacteristic
315C22×C4C2×D811yescharacteristic
316C22×C4C2×D811yescharacteristic
317C22×C4C2×SD1612yes—
318C22×C4C2×SD1612yes—
319C22×C4C22×D411yescharacteristic
320Q16—42no—
321Q16—81no—
322SD16—82no—
323SD16—82no—
324D8—42no—
325D8—81no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8C2×C22⋊C411yescharacteristic
335C2×C8C2×C22⋊C411yescharacteristic
336C2×C8C2×C22⋊C412yes—
337C2×C8C2×C22⋊C412yes—
338C2×C8C23×C411yescharacteristic
339C2×C8C22×D411yescharacteristic
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—82no—
359C4⋊C4#10212yes—
360C4⋊C4#10212yes—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—161no—
373C22⋊C4#10212yes—
374C22⋊C4#10212yes—
375C22⋊C4#10212yes—
376C22⋊C4#10212yes—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—82no—
381C42—161no—
382C42C2×C22⋊C411yescharacteristic
383C42C2×C22⋊C411yescharacteristic
384C42C2×SD1611yescharacteristic
385C42C2×SD1611yescharacteristic
386C16—42no—
387C16—42no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23C2×#10212yes—
398C23C2×#10212yes—
399C23C22×D811yescharacteristic
400Q8—42no—
401Q8—42no—
402Q8—82no—
403Q8—161no—
404D4—42no—
405D4—42no—
406D4—42no—
407D4—42no—
408D4—82no—
409D4—82no—
410D4—82no—
411D4—161no—
412D4—162no—
413D4—162no—
414D4—162no—
415D4—162no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—84no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—162no—
446C2×C4—162no—
447C2×C4—164no—
448C2×C4C2×#10211yescharacteristic
449C2×C4C2×#10212yes—
450C2×C4C2×#10212yes—
451C2×C4C2×#10211yescharacteristic
452C2×C4C2×#10211yescharacteristic
453C2×C4C2×#10211yescharacteristic
454C2×C4C22×C22⋊C411yescharacteristic
455C2×C4C22×SD1611yescharacteristic
456C8—21no—
457C8—21no—
458C8—22no—
459C8—22no—
460C8—21no—
461C8—21no—
462C22—21no—
463C22—21no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—42no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22C22×#10211yescharacteristic
478C22#341411yescharacteristic
479C22#341411yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—22no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#288311yescharacteristic
495C4#290111yescharacteristic
496C2—21no—
497C2—21no—
498C2—42no—
499C2—42no—
500C2—162no—
501C2#1761311yescharacteristic
502C2#3338011yescharacteristic
503C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62249C2
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. #22706C2semidirect
4. #11890C2semidirect
5. #11885C2semidirect
6. #11859C2semidirect
7. #11858C2semidirect
8. #11745C2semidirect
9. #11744C2semidirect
10. #11704C2semidirect
11. #11703C2semidirect
13. #3342C4semidirect
14. #3341C4semidirect
17. #3287C4semidirect
18. #3283C4semidirect
52. C2×#135C2×C4semidirect
74. #500C2×C4semidirect
75. #500C2×C4semidirect
76. #494C2×C4semidirect
77. #493C2×C4semidirect
88. #467C2×C4semidirect
92. #466C2×C4semidirect
105. C4×Q16C2×C4semidirect
107. C4×D8C2×C4semidirect
113. #427C2×C4semidirect

Maximal subgroups

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

subgroupindex
#622492
#334872
#227062
#118902
#118852
#118592
#118582
#117452
#117442
#117042
#117032