Finite Group Databasebeta

← #226090

Subgroups — #226090

Order 512.

755 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22758C211yesmaximal, characteristic
2#22715C211yesmaximal, characteristic, complemented
3C4⋊C4×D8C211yesmaximal, characteristic
4D4×#106C211yesmaximal, characteristic, complemented
5C2×#1586C211yesmaximal, characteristic
6#9621C211yesmaximal, characteristic, complemented
7#9613C211yesmaximal, characteristic, complemented
8#8894C211yesmaximal, characteristic, complemented
9#8894C211yesmaximal, characteristic, complemented
10#8893C211yesmaximal, characteristic, complemented
11#8890C211yesmaximal, characteristic, complemented
12#8821C211yesmaximal, characteristic, complemented
13#8650C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic, complemented
15#8289C211yesmaximal, characteristic, complemented
16C2×C4×D8—21no—
17C2×C4×D8C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C22×#106—21no—
21#2354C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1948C2211yescharacteristic
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1947C2211yescharacteristic
29#1942—21no—
30#1936—21no—
31#1921—21no—
32#1879—21no—
33#1862—21no—
34#1862—21no—
35#1854—21no—
36#1853—21no—
37#1853—21no—
38#1853—21no—
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1809—21no—
46#1807—21no—
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51C4×#106C2211yescharacteristic
52#1770C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54#1762—21no—
55#1736—21no—
56#1586C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×D8D411yescharacteristic
69C22×M16—21no—
70C23×C8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C4×D8—44no—
81C4×D8C2×C414yes—
82C4×D8Q814yes—
83#436—22no—
84#436C2×C414yescomplemented
85C8×D4—22no—
86C8×D4C2×C414yescomplemented
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107D411yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—44no—
97C2×#106D411yescharacteristic
98C2×#106D411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102D411yescharacteristic
108C2×#102D411yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#390—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#388—41no—
121#385—41no—
122#385—41no—
123#385—41no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135#340—21no—
136#340—21no—
137#340—21no—
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—41no—
148#340—41no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—22no—
154#334C2×C414yescomplemented
155D4⋊C8C2×C414yescomplemented
156D4⋊C8C2×C414yescomplemented
157#322C2×C414yescomplemented
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—41no—
165C2×D8—22no—
166C2×D8—22no—
167C2×D8C4⋊C414yes—
168C2×M16—24no—
169C2×M16—81no—
170C2×M16—81no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—24no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—81no—
191C4×D4—81no—
192C4×D4—81no—
193C4×D4C22×C412yes—
194C4×D4C22×C412yes—
195C4×D4C2×Q812yes—
196C4×D4C2×Q812yes—
197#117—42no—
198#117—81no—
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—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—41no—
236C2×C42C2411yescharacteristic
237#107—42no—
238#107—81no—
239#107—81no—
240#107C4⋊C414yes—
241#106—22no—
242#106—41no—
243#106—42no—
244#106—42no—
245#106—42no—
246#106—81no—
247#106—81no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8C22×C412yes—
255C4⋊C8C22×C412yes—
256#102—42no—
257#102—42no—
258#102—42no—
259#102—81no—
260#102C4⋊C414yes—
261#102C4⋊C414yes—
262C22⋊C8—24no—
263C22⋊C8—24no—
264#97—81no—
265C4×C8—21no—
266C4×C8—42no—
267C4×C8C22×C412yes—
268C4×C8C22×C412yes—
269C4×C8C22×C412yes—
270C4×C8C2×Q812yes—
271C4×C8#4112yes—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278C24—21no—
279C24—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4C2×C4⋊C412yes—
289C2×D4C2×C4⋊C412yes—
290C2×D4AGL(1,ℤ/8)12yes—
291C2×D4#13712yes—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—161no—
321C22×C4—161no—
322C22×C4—161no—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326D8—28no—
327M16—84no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—48no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8C2×C4⋊C412yes—
353C2×C8C2×C4⋊C412yes—
354C2×C8C4×D412yes—
355C2×C8C4×D412yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—81no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4—81no—
387C4⋊C4—82no—
388C4⋊C4C2×C4⋊C412yes—
389C4⋊C4C2×C4⋊C412yes—
390C4⋊C4AGL(1,ℤ/8)12yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—161no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—82no—
405C42C23×C411yescharacteristic
406C42C22×Q811yescharacteristic
407C42C2×#4111yescharacteristic
408C422−1+411yescharacteristic
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23D4211yescharacteristic
424D4—22no—
425D4—22no—
426D4—22no—
427D4—22no—
428D4—22no—
429D4—41no—
430D4—41no—
431D4#42814yes—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—164no—
473C2×C4—164no—
474C2×C4—164no—
475C2×C4C22×C4⋊C411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4#52011yescharacteristic
478C2×C4#54711yescharacteristic
479C2×C4D4×Q811yescharacteristic
480C2×C4#55411yescharacteristic
481C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4C2×#13711yescharacteristic
484C2×C4#57511yescharacteristic
485C2×C4#57611yescharacteristic
486C2×C4#57711yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57711yescharacteristic
489C8—22no—
490C8—22no—
491C8—22no—
492C8—22no—
493C8—42no—
494C8—42no—
495C8—42no—
496C8—81no—
497C8—81no—
498C22—24no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22D4×C4⋊C411yescharacteristic
513C22#326411yescharacteristic
514C22#326511yescharacteristic
515C22#326611yescharacteristic
516C22#326711yescharacteristic
517C22#329011yescharacteristic
518C22#329011yescharacteristic
519C4—21no—
520C4—21no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—161no—
528C4—161no—
529C4—161no—
530C4—161no—
531C4C2×#42811yescharacteristic
532C4#291211yescharacteristic
533C4C4×AGL(1,ℤ/8)11yescharacteristic
534C4#294611yescharacteristic
535C4#334411yescharacteristic
536C4#334511yescharacteristic
537C4#338411yescharacteristic
538C4#338911yescharacteristic
539C2—44no—
540C2—82no—
541C2#2005411yescharacteristic
542C2#2006011yescharacteristic
543C2#2006911yescharacteristic
544C2#2010411yescharacteristic
545C2#2265911yescharacteristic
546C2#2267611yescharacteristic
547C2#2269011yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
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. #22715C2semidirect
4. D4×#106C2semidirect
6. #9621C2semidirect
7. #9613C2semidirect
8. #8894C2semidirect
9. #8894C2semidirect
10. #8893C2semidirect
11. #8890C2semidirect
12. #8821C2semidirect
13. #8650C2semidirect
14. #8648C2semidirect
15. #8289C2semidirect
56. #1586C4semidirect
84. #436C2×C4semidirect
86. C8×D4C2×C4semidirect
154. #334C2×C4semidirect
155. D4⋊C8C2×C4semidirect
156. D4⋊C8C2×C4semidirect
157. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#227582
#227152
C4⋊C4×D82
D4×#1062
C2×#15862
#96212
#96132
#88942
#88932
#88902
#88212
#86502
#86482
#82892