Finite Group Databasebeta

← #202808

Subgroups — #202808

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21964C211yesmaximal, characteristic
2#21839C211yesmaximal, characteristic
3#21409C211yesmaximal, characteristic
4#20441C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#9576C211yesmaximal, characteristic, complemented
7#9575C211yesmaximal, characteristic, complemented
8#9570C211yesmaximal, characteristic, complemented
9#9569C211yesmaximal, characteristic, complemented
10#9568C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9562C211yesmaximal, characteristic, complemented
13#9561C211yesmaximal, characteristic, complemented
14#8044C211yesmaximal, characteristic
15#7876C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#405—21no—
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24#1992—21no—
25#1992—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1988C2211yescharacteristic
29#1988C2211yescharacteristic
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947—21no—
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1894—21no—
41#1894—21no—
42#1825—21no—
43#1825—21no—
44#1809—21no—
45#1809—21no—
46#1771—21no—
47#1771—21no—
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1763—21no—
53#1763—21no—
54#1729—21no—
55#1550C418yescomplemented
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—
69C22×M16D411yescharacteristic
70C22×M16D411yescharacteristic
71C23×C8—21no—
72#533—22no—
73#533—22no—
74#533—22no—
75#533C2×C414yes—
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×#125—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117—21no—
86C22×C42C2311yescharacteristic
87#432—22no—
88#432—22no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#106—21no—
92C2×#106—21no—
93#423—22no—
94#423C2×C414yes—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8C2×C414yes—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×C22⋊C8—21no—
110#405—44no—
111C4×M16—22no—
112C4×M16—22no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97D411yescharacteristic
117C2×#97D411yescharacteristic
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#384D411yescharacteristic
123#384D411yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#382—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380D411yescharacteristic
132#380D411yescharacteristic
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—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×C414yes—
159#333C2×C414yes—
160#332C2×C414yes—
161#332C2×C414yes—
162C22×Q8—21no—
163C22×Q8—21no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4#4111yescharacteristic
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—24no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—41no—
173C2×M16—41no—
174C2×M16—81no—
175C2×M16—81no—
176C22×C8—24no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8C22×C412yes—
196C4⋊Q8C22×C412yes—
197#125—24no—
198#125—42no—
199C22⋊Q8—24no—
200C22⋊Q8—24no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—42no—
206C4×Q8—42no—
207#117—24no—
208#117—42no—
209#117—42no—
210#117—42no—
211#117—82no—
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—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42C22×C412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42C2411yescharacteristic
248#107—81no—
249#107—81no—
250#106—81no—
251#106—81no—
252C4⋊C8—24no—
253C4⋊C8—24no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8C22×C412yes—
263C4⋊C8C22×C412yes—
264C4⋊C8C22×C412yes—
265C4⋊C8C22×C412yes—
266#103—42no—
267#103—42no—
268#103—42no—
269#103—42no—
270C22⋊C8—42no—
271#97—22no—
272#97—22no—
273#97—42no—
274#97—42no—
275#97—42no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—42no—
281C4×C8—42no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—41no—
287#95—41no—
288C24#12111yescharacteristic
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—81no—
298C2×Q8—81no—
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—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4#11712yes—
318C22×C4C4×D412yes—
319C22×C4C4×D412yes—
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12312yes—
324C22×C4#12412yes—
325C22×C4#12612yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329M16—28no—
330M16—44no—
331M16—84no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—48no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
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×C8AGL(1,ℤ/8)12yes—
362C2×C8#13712yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
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⋊C4—81no—
394C4⋊C4—81no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—82no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—44no—
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—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
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#38812yes—
472C2×C4#38812yes—
473C2×C4#38812yes—
474C2×C4C2×#11711yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×#12311yescharacteristic
478C2×C4C2×#12411yescharacteristic
479C2×C4C2×#12611yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
481C2×C4C2×#13711yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57811yescharacteristic
487C2×C4#57811yescharacteristic
488C8—22no—
489C8—22no—
490C8—41no—
491C8—41no—
492C8—42no—
493C8—42no—
494C8—42no—
495C8—42no—
496C8—81no—
497C8—81no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22C2×#38811yescharacteristic
506C22#303611yescharacteristic
507C22#303711yescharacteristic
508C22#306311yescharacteristic
509C22#306311yescharacteristic
510C22#306811yescharacteristic
511C22#306911yescharacteristic
512C4—22no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—22no—
517C4—82no—
518C4—82no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—161no—
524C4—161no—
525C4C4×AGL(1,ℤ/8)11yescharacteristic
526C4C4×#13711yescharacteristic
527C4#294911yescharacteristic
528C4#295011yescharacteristic
529C4#309211yescharacteristic
530C4#309311yescharacteristic
531C4#313711yescharacteristic
532C4#313711yescharacteristic
533C2—42no—
534C2#1936911yescharacteristic
535C2#1937111yescharacteristic
536C2#1937311yescharacteristic
537C2#1937511yescharacteristic
538C2#2105911yescharacteristic
539C2#2107611yescharacteristic
540C2#2108811yescharacteristic
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. #9570C2semidirect
9. #9569C2semidirect
10. #9568C2semidirect
11. #9567C2semidirect
12. #9562C2semidirect
13. #9561C2semidirect
55. #1550C4semidirect

Maximal subgroups

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

subgroupindex
#219642
#218392
#214092
#204412
C2×#15502
#95762
#95752
#95702
#95692
#95682
#95672
#95622
#95612
#80442
#78762