Finite Group Databasebeta

← #202891

Subgroups — #202891

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21835C211yesmaximal, characteristic
2#21385C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20535C211yesmaximal, characteristic
5C2×#1559C211yesmaximal, characteristic
6#9576C211yesmaximal, characteristic, complemented
7#9575C211yesmaximal, characteristic, complemented
8#9574C211yesmaximal, characteristic, complemented
9#9573C211yesmaximal, characteristic, complemented
10#9568C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9566C211yesmaximal, characteristic, complemented
13#9565C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic
15#7880C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#428—21no—
21C2×#423C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1989—21no—
29#1989—21no—
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1981C2211yescharacteristic
33#1981C2211yescharacteristic
34#1949—21no—
35#1949—21no—
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1872—21no—
41#1872—21no—
42#1825—21no—
43#1825—21no—
44#1771—21no—
45#1771—21no—
46#1770C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C4×#103—21no—
51C4×#103—21no—
52#1749—21no—
53C4×C22⋊C8—21no—
54#1736—21no—
55#1559C418yescomplemented
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×M16—21no—
69#533—22no—
70#533—22no—
71#533—22no—
72#533C2×C414yes—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×#125—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×#117—21no—
81C2×#117—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#432—22no—
87#432C2×C414yes—
88#431—22no—
89#431—22no—
90#431—22no—
91#431—22no—
92#428—44no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#106—21no—
96C2×#106—21no—
97#423—22no—
98#423C2×C414yes—
99C2×C4⋊C8—21no—
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8D411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×#97—21no—
117C2×#97—21no—
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C2×#95—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—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#333C2×C414yescomplemented
159#333C2×C414yescomplemented
160#332C2×C414yescomplemented
161#332C2×C414yescomplemented
162C22×Q8—21no—
163C22×Q8—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22×C412yes—
189C4⋊Q8C22×C412yes—
190#125—24no—
191#125—42no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×Q8—42no—
200#117—24no—
201#117—42no—
202#117—42no—
203#117—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—81no—
235C2×C42—81no—
236C2×C42C22×C412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42C2411yescharacteristic
244#107—42no—
245#107—42no—
246#106—42no—
247#106—42no—
248C4⋊C8—24no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8C22×C412yes—
260C4⋊C8C22×C412yes—
261C4⋊C8C22×C412yes—
262C4⋊C8C22×C412yes—
263#103—42no—
264#103—42no—
265#103—42no—
266#103—42no—
267C22⋊C8—24no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—42no—
271#97—24no—
272#97—81no—
273#97—81no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—42no—
279C4×C8—81no—
280C4×C8—81no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—41no—
286#95—41no—
287C24#12111yescharacteristic
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—81no—
297C2×Q8—81no—
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—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4#11712yes—
323C22×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12312yes—
329C22×C4#12412yes—
330C22×C4#12612yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334M16—48no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—81no—
392C4⋊C4—81no—
393C4⋊C4AGL(1,ℤ/8)12yes—
394C4⋊C4#13712yes—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—44no—
402C42—81no—
403C42—81no—
404C42—82no—
405C42#11712yes—
406C42C4×D412yes—
407C42C4×D412yes—
408C42#12312yes—
409C42#12412yes—
410C42#12612yes—
411C42C23×C411yescharacteristic
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23#38812yes—
422C23C2×#12111yescharacteristic
423Q8—42no—
424Q8—42no—
425Q8—82no—
426Q8—82no—
427Q8—81no—
428Q8—81no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—84no—
462C2×C4—84no—
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—81no—
473C2×C4#38812yes—
474C2×C4#38812yes—
475C2×C4#38812yes—
476C2×C4C2×#11711yescharacteristic
477C2×C4C2×C4×D411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4C2×#12411yescharacteristic
481C2×C4C2×#12611yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4C2×#13711yescharacteristic
484C2×C4#57511yescharacteristic
485C2×C4#57511yescharacteristic
486C2×C4#57711yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57811yescharacteristic
489C2×C4#57811yescharacteristic
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—42no—
495C8—42no—
496C8—42no—
497C8—42no—
498C8—81no—
499C8—81no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22C2×#38811yescharacteristic
508C22#304211yescharacteristic
509C22#304411yescharacteristic
510C22#305511yescharacteristic
511C22#305611yescharacteristic
512C22#306711yescharacteristic
513C22#306711yescharacteristic
514C4—22no—
515C4—21no—
516C4—21no—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—81no—
523C4—82no—
524C4—82no—
525C4—161no—
526C4—161no—
527C4C4×AGL(1,ℤ/8)11yescharacteristic
528C4C4×#13711yescharacteristic
529C4#294711yescharacteristic
530C4#294711yescharacteristic
531C4#309711yescharacteristic
532C4#309711yescharacteristic
533C4#314111yescharacteristic
534C4#314211yescharacteristic
535C2—42no—
536C2#1946611yescharacteristic
537C2#1946711yescharacteristic
538C2#1947811yescharacteristic
539C2#1947911yescharacteristic
540C2#2106411yescharacteristic
541C2#2107511yescharacteristic
542C2#2109311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9576C2semidirect
7. #9575C2semidirect
8. #9574C2semidirect
9. #9573C2semidirect
10. #9568C2semidirect
11. #9567C2semidirect
12. #9566C2semidirect
13. #9565C2semidirect
55. #1559C4semidirect
158. #333C2×C4semidirect
159. #333C2×C4semidirect
160. #332C2×C4semidirect
161. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#218352
#213852
#211702
#205352
C2×#15592
#95762
#95752
#95742
#95732
#95682
#95672
#95662
#95652
#80462
#78802