Finite Group Databasebeta

← #203578

Subgroups — #203578

Order 512.

774 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21356C211yesmaximal, characteristic
2#21146C211yesmaximal, characteristic
3C4×#428C211yesmaximal, characteristic
4C2×#1569C211yesmaximal, characteristic
5C2×#1455C211yesmaximal, characteristic
6#9600C211yesmaximal, characteristic, complemented
7#9599C211yesmaximal, characteristic, complemented
8#9598C211yesmaximal, characteristic, complemented
9#9597C211yesmaximal, characteristic, complemented
10#9596C211yesmaximal, characteristic, complemented
11#9595C211yesmaximal, characteristic, complemented
12#9594C211yesmaximal, characteristic, complemented
13#9593C211yesmaximal, characteristic, complemented
14#7904C211yesmaximal, characteristic
15#7903C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#428C411yescharacteristic
18C2×#428C411yescharacteristic
19C2×#423—21no—
20C2×#423C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1940—21no—
27#1940—21no—
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1851—21no—
37#1851—21no—
38#1776C411yescharacteristic
39#1776C411yescharacteristic
40#1776C411yescharacteristic
41#1776C411yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#106C2211yescharacteristic
45C4×#106C2211yescharacteristic
46#1767—21no—
47#1767—21no—
48#1736C411yescharacteristic
49#1736C411yescharacteristic
50#1728—21no—
51#1569—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56#1455—24no—
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69C22×M16—21no—
70C22×M16—21no—
71C22×M16C2×C411yescharacteristic
72C22×M16C2×C411yescharacteristic
73C22×M16D411yescharacteristic
74C22×M16D411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#432—22no—
82#432—22no—
83#428—24no—
84#428—24no—
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93#423—22no—
94#423—22no—
95#423—44no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C22⋊C8—21no—
110C4×M16—22no—
111C4×M16—22no—
112C4×M16—22no—
113C4×M16D414yes—
114C4×M16Q814yes—
115C4×M16Q814yes—
116C2×#97—21no—
117C2×#97—21no—
118C2×#97C2×C411yescharacteristic
119C2×#97C2×C411yescharacteristic
120C2×#97D411yescharacteristic
121C2×#97D411yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140#376—21no—
141C2×#95—21no—
142C43—21no—
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#335—22no—
160#335—22no—
161#334—22no—
162#334—22no—
163#322—22no—
164#322—22no—
165#322—22no—
166C82—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4111yescharacteristic
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—24no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—41no—
180C2×M16—41no—
181C2×M16—41no—
182C2×M16—41no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16—42no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16C4⋊C414yes—
189C2×M16C4⋊C414yes—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204#117—22no—
205#117—22no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×Q812yes—
240C2×C42C2×Q812yes—
241C2×C42#4112yes—
242C2×C42#4111yescharacteristic
243C2×C42C2411yescharacteristic
244#107—22no—
245#107—22no—
246#107—22no—
247#107M1614yescomplemented
248#106—22no—
249#106—22no—
250#106—22no—
251#106M1614yescomplemented
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269C22⋊C8—42no—
270#97—22no—
271#97—22no—
272#97—24no—
273#97—42no—
274#97C4⋊C414yes—
275#97C4⋊C414yes—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8C2×D412yes—
283C4×C8C2×D412yes—
284C4×C8C2×Q812yes—
285C4×C8C2×Q812yes—
286C4×C8C2×Q812yes—
287C4×C8C2×Q812yes—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296C24C4×D411yescharacteristic
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4C2×C4⋊C412yes—
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×Q812yes—
326C22×C4#12112yes—
327C22×C4C22⋊Q812yes—
328C22×C4#12512yes—
329C22×C4C4⋊Q812yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333M16—28no—
334M16—44no—
335M16—44no—
336M16—44no—
337M16—48no—
338M16—84no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8C2×C4⋊C412yes—
374C2×C8C2×C4⋊C412yes—
375C2×C8C2×C4⋊C412yes—
376C2×C8C2×C4⋊C412yes—
377C2×C8C2×M1612yes—
378C2×C8C2×M1612yes—
379C2×C8AGL(1,ℤ/8)12yes—
380C2×C8#13712yes—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4C2×M1612yes—
396C4⋊C4C2×M1612yes—
397C4⋊C4C2×M1612yes—
398C4⋊C4C2×M1612yes—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—81no—
414C42—81no—
415C42C2×C4⋊C412yes—
416C42C4×Q812yes—
417C42#12112yes—
418C42C22⋊Q812yes—
419C42#12512yes—
420C42C4⋊Q812yes—
421C42C22×D411yescharacteristic
422C42C22×Q811yescharacteristic
423C42C22×Q811yescharacteristic
424C42C2×#4111yescharacteristic
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23#38912yes—
432C23C2×C4×D411yescharacteristic
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4#38912yes—
478C2×C4#38912yes—
479C2×C4#38912yes—
480C2×C4C22×C4⋊C411yescharacteristic
481C2×C4C2×C4×Q811yescharacteristic
482C2×C4C2×#12111yescharacteristic
483C2×C4C2×C22⋊Q811yescharacteristic
484C2×C4C2×#12511yescharacteristic
485C2×C4C2×C4⋊Q811yescharacteristic
486C2×C4C22×M1611yescharacteristic
487C2×C4C22×M1611yescharacteristic
488C2×C4#56811yescharacteristic
489C2×C4#56811yescharacteristic
490C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
491C2×C4C2×#13711yescharacteristic
492C2×C4#57511yescharacteristic
493C2×C4#57511yescharacteristic
494C8—24no—
495C8—22no—
496C8—22no—
497C8—22no—
498C8—22no—
499C8—22no—
500C8—42no—
501C8—41no—
502C8—41no—
503C8—81no—
504C8—81no—
505C8—81no—
506C8—81no—
507C8#42314yescomplemented
508C22—22no—
509C22—22no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22C2×#38911yescharacteristic
516C22D4×M1611yescharacteristic
517C22D4×M1611yescharacteristic
518C22C4×AGL(1,ℤ/8)11yescharacteristic
519C22C4×#13711yescharacteristic
520C22#295611yescharacteristic
521C22#295611yescharacteristic
522C4—22no—
523C4—22no—
524C4—21no—
525C4—21no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4C2×#42311yescharacteristic
536C4#290511yescharacteristic
537C4Q8×M1611yescharacteristic
538C4Q8×M1611yescharacteristic
539C4#305511yescharacteristic
540C4#305611yescharacteristic
541C4#316511yescharacteristic
542C4#316611yescharacteristic
543C2—42no—
544C2#1804611yescharacteristic
545C2#1804711yescharacteristic
546C2#1804811yescharacteristic
547C2#1804911yescharacteristic
548C2#2114611yescharacteristic
549C2#2115911yescharacteristic
550C2#2116511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9600C2semidirect
7. #9599C2semidirect
8. #9598C2semidirect
9. #9597C2semidirect
10. #9596C2semidirect
11. #9595C2semidirect
12. #9594C2semidirect
13. #9593C2semidirect
247. #107M16semidirect
251. #106M16semidirect
507. C8#423semidirect

Maximal subgroups

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

subgroupindex
#213562
#211462
C4×#4282
C2×#15692
C2×#14552
#96002
#95992
#95982
#95972
#95962
#95952
#95942
#95932
#79042
#79032