Finite Group Databasebeta

← #226138

Subgroups — #226138

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22770C211yesmaximal, characteristic
2#22728C211yesmaximal, characteristic, complemented
3C4⋊C4×SD16C211yesmaximal, characteristic
4D4×#106C211yesmaximal, characteristic
5C2×#1582C211yesmaximal, characteristic
6#9613C211yesmaximal, characteristic, complemented
7#9612C211yesmaximal, characteristic, complemented
8#8891C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic
10#8888C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic
12#8819C211yesmaximal, characteristic, complemented
13#8651C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic
15#8285C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C4×SD16—21no—
18C2×C4×SD16C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C22×#106—21no—
21#2355C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1949—21no—
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947—21no—
30#1947C2211yescharacteristic
31#1941—21no—
32#1933—21no—
33#1919—21no—
34#1877—21no—
35#1865—21no—
36#1864—21no—
37#1853—21no—
38#1853—21no—
39#1847—21no—
40#1845C2211yescharacteristic
41#1844C2211yescharacteristic
42#1844C2211yescharacteristic
43#1844C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1807—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1776—21no—
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102—21no—
56#1582C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16D411yescharacteristic
69C23×C8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77#446—22no—
78#446C2×C414yescomplemented
79C4×SD16—44no—
80C4×SD16C2×C414yes—
81C4×SD16Q814yes—
82C8×D4—22no—
83C8×D4C2×C414yes—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—44no—
95C2×#106—41no—
96C2×#106D411yescharacteristic
97C2×#106D411yescharacteristic
98C2×#106D411yescharacteristic
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103—21no—
105C2×#103—21no—
106C2×#103D411yescharacteristic
107C2×#102—21no—
108C2×#102—21no—
109C2×#102D411yescharacteristic
110C2×C22⋊C8—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—41no—
122#387—41no—
123#385—41no—
124#385—41no—
125#384—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#382—21no—
131#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—41no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—22no—
154#334C2×C414yescomplemented
155Q8⋊C8C2×C414yescomplemented
156D4⋊C8C2×C414yes—
157#322C2×C414yescomplemented
158C22×Q8C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—41no—
162C23×C4—41no—
163C2×SD16—22no—
164C2×SD16—22no—
165C2×SD16C4⋊C414yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—24no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—42no—
181C4×Q8—81no—
182C4×Q8C22×C412yes—
183C4×Q8C2×Q812yes—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—81no—
189C4×D4—81no—
190C4×D4C22×C412yes—
191C4×D4C2×Q812yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C2411yescharacteristic
228#107—42no—
229#107—81no—
230#107—81no—
231#106—22no—
232#106—42no—
233#106—42no—
234#106—42no—
235#106—42no—
236#106—81no—
237#106—81no—
238#106—81no—
239#106—82no—
240#106C4⋊C414yes—
241#106SD1614yescomplemented
242C4⋊C8—24no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8C22×C412yes—
250C4⋊C8C22×C412yes—
251#103—42no—
252#103—81no—
253#103C4⋊C414yes—
254#102—42no—
255#102—42no—
256#102C4⋊C414yes—
257C22⋊C8—24no—
258#97—24no—
259C4×C8—21no—
260C4×C8—42no—
261C4×C8—81no—
262C4×C8C22×C412yes—
263C4×C8C22×C412yes—
264C4×C8C22×C412yes—
265C4×C8C2×Q812yes—
266C4×C8#4112yes—
267#95—21no—
268#95—21no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C2×C4⋊C412yes—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C2×C4⋊C412yes—
286C2×D4AGL(1,ℤ/8)12yes—
287C2×D4#13712yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—161no—
310C22×C4—161no—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314SD16—28no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—48no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8C2×C4⋊C412yes—
338C2×C8C2×C4⋊C412yes—
339C2×C8C4×D412yes—
340C2×C8C4×D412yes—
341C2×C8C2×SD1612yes—
342C2×C8#13512yes—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—81no—
375C4⋊C4—82no—
376C4⋊C4—161no—
377C4⋊C4C2×C4⋊C412yes—
378C4⋊C4C2×C4⋊C412yes—
379C4⋊C4C2×SD1612yes—
380C4⋊C4C2×SD1612yes—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C42—22no—
386C42—22no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—82no—
392C42—161no—
393C42C23×C411yescharacteristic
394C42C22×Q811yescharacteristic
395C42C2×#4111yescharacteristic
396C422−1+411yescharacteristic
397C23—22no—
398C23—22no—
399C23—22no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23D4211yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—41no—
408Q8—41no—
409D4—22no—
410D4—22no—
411D4—22no—
412D4#42814yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—164no—
453C2×C4—164no—
454C2×C4C22×C4⋊C411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4#52011yescharacteristic
457C2×C4#54711yescharacteristic
458C2×C4D4×Q811yescharacteristic
459C2×C4#55411yescharacteristic
460C2×C4C22×SD1611yescharacteristic
461C2×C4C2×#13511yescharacteristic
462C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
463C2×C4C2×#13711yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57711yescharacteristic
468C8—22no—
469C8—22no—
470C8—22no—
471C8—42no—
472C8—42no—
473C8—42no—
474C8—81no—
475C8—81no—
476C8C4×SD1614yescomplemented
477C22—24no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22D4×C4⋊C411yescharacteristic
485C22#326511yescharacteristic
486C22#326611yescharacteristic
487C22D4×SD1611yescharacteristic
488C22#328411yescharacteristic
489C22#328811yescharacteristic
490C22#328911yescharacteristic
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4—161no—
501C4—161no—
502C4—161no—
503C4—161no—
504C4C2×#42811yescharacteristic
505C4#291211yescharacteristic
506C4C2×C4×SD1611yescharacteristic
507C4#295211yescharacteristic
508C4#334411yescharacteristic
509C4#334511yescharacteristic
510C4Q8×SD1611yescharacteristic
511C4#338111yescharacteristic
512C2—44no—
513C2#1976411yescharacteristic
514C2#1976511yescharacteristic
515C2#1985211yescharacteristic
516C2#1985311yescharacteristic
517C2#2265911yescharacteristic
518C2C4⋊C4×SD1611yescharacteristic
519C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22728C2semidirect
6. #9613C2semidirect
7. #9612C2semidirect
8. #8891C2semidirect
10. #8888C2semidirect
12. #8819C2semidirect
13. #8651C2semidirect
15. #8285C2semidirect
56. #1582C4semidirect
78. #446C2×C4semidirect
154. #334C2×C4semidirect
155. Q8⋊C8C2×C4semidirect
157. #322C2×C4semidirect
241. #106SD16semidirect
476. C8C4×SD16semidirect

Maximal subgroups

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

subgroupindex
#227702
#227282
C4⋊C4×SD162
D4×#1062
C2×#15822
#96132
#96122
#88912
#88902
#88882
#88862
#88192
#86512
#86442
#82852