Finite Group Databasebeta

← #242515

Subgroups — #242515

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21972C211yesmaximal, characteristic, complemented
2#21805C211yesmaximal, characteristic
3#21393C211yesmaximal, characteristic
4#21362C211yesmaximal, characteristic, complemented
5C2×#1572C211yesmaximal, characteristic, complemented
6C2×#1540C211yesmaximal, characteristic
7#10600C211yesmaximal, characteristic, complemented
8#10599C211yesmaximal, characteristic, complemented
9#10596C211yesmaximal, characteristic, complemented
10#10595C211yesmaximal, characteristic, complemented
11#10592C211yesmaximal, characteristic, complemented
12#10591C211yesmaximal, characteristic, complemented
13#10588C211yesmaximal, characteristic, complemented
14#10587C211yesmaximal, characteristic, complemented
15#8044C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2285C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1988C2211yescharacteristic
24#1988C2211yescharacteristic
25#1983—21no—
26#1983—21no—
27#1982C2211yescharacteristic
28#1982C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1875—21no—
42#1875—21no—
43#1854—21no—
44#1854—21no—
45#1850—21no—
46#1850—21no—
47#1842—21no—
48#1831—21no—
49#1788—21no—
50#1788—21no—
51C4×#107—21no—
52C4×#106—21no—
53#1762—21no—
54#1762—21no—
55#1572—24no—
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1540—24no—
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#331C2211yescharacteristic
64C2×#331C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×#41—21no—
70C23×C8—21no—
71#532—22no—
72#532—22no—
73#532—22no—
74#532Q814yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×#127C2311yescharacteristic
77C2×#124—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×#117—21no—
83C2×#117D411yescharacteristic
84C2×#117D411yescharacteristic
85C22×C42C2311yescharacteristic
86#432—22no—
87#432—22no—
88#431—22no—
89#431—22no—
90#431—22no—
91#431D414yes—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121C2×#102—21no—
122C2×#102—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8D411yescharacteristic
126C2×C22⋊C8D411yescharacteristic
127C2×#97—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#382—21no—
134#380—21no—
135#380—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144#376—21no—
145C43—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#335—22no—
159#335—22no—
160#334—22no—
161#334—22no—
162#332—22no—
163#332—22no—
164#331—22no—
165#331—22no—
166C2×#41—24no—
167C2×#41—42no—
168C2×#41—81no—
169C2×#41—81no—
170C2×#41—81no—
171C2×#41—81no—
172C22×Q8—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—41no—
177C23×C4—41no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4111yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—42no—
188C22×C8—81no—
189C22×C8—81no—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C2×Q812yes—
198#127—21no—
199#127—21no—
200#127—21no—
201#127C2×Q812yes—
202#124—24no—
203#124—42no—
204#121—24no—
205#121—24no—
206#121—42no—
207#121—42no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—42no—
211C4×D4—42no—
212#117—22no—
213#117—22no—
214#117—41no—
215#117—41no—
216#117—82no—
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⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—41no—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42#4111yescharacteristic
248C2×C42C2411yescharacteristic
249#107—42no—
250#107—42no—
251#106—42no—
252#106—42no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—42no—
266C4⋊C8—81no—
267C4⋊C8—81no—
268C4⋊C8—81no—
269C4⋊C8—81no—
270C4⋊C8C2×D412yes—
271C4⋊C8C2×D412yes—
272#103—81no—
273#103—81no—
274#103—81no—
275#103—81no—
276#102—42no—
277#102—42no—
278#102—42no—
279#102—42no—
280#102—81no—
281#102—81no—
282#102—81no—
283#102—81no—
284C22⋊C8—24no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—42no—
288#97—42no—
289C4×C8—24no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—41no—
299#95—41no—
300C24—41no—
301C24#12111yescharacteristic
302#41—84no—
303#41—84no—
304#41—84no—
305#41—84no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—44no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—81no—
328C2×D4—81no—
329C2×D4—82no—
330C2×D4—82no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—44no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4C22⋊Q812yes—
359C22×C4#12612yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—84no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4#13512yes—
415C4⋊C4#13512yes—
416C22⋊C4—24no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—82no—
421C42—24no—
422C42—42no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—81no—
428C42—81no—
429C42—82no—
430C42#12112yes—
431C42#12112yes—
432C42C22⋊Q812yes—
433C42C22⋊Q812yes—
434C42C22⋊Q812yes—
435C42#12612yes—
436C42C22×D411yescharacteristic
437C42C22×Q811yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—42no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23#39612yes—
454C23C2×#12111yescharacteristic
455Q8—42no—
456Q8—42no—
457Q8—81no—
458Q8—81no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—82no—
464D4—82no—
465D4—88no—
466D4—81no—
467D4—81no—
468D4—81no—
469D4—81no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—88no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4#39612yes—
511C2×C4#39612yes—
512C2×C4#39612yes—
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×C22⋊Q811yescharacteristic
516C2×C4C2×C22⋊Q811yescharacteristic
517C2×C4C2×C22⋊Q811yescharacteristic
518C2×C4C2×#12611yescharacteristic
519C2×C4C2×#13511yescharacteristic
520C2×C4C2×#13511yescharacteristic
521C2×C4#57511yescharacteristic
522C2×C4#57511yescharacteristic
523C2×C4#57611yescharacteristic
524C2×C4#57611yescharacteristic
525C2×C4#57711yescharacteristic
526C2×C4#57711yescharacteristic
527C8—42no—
528C8—42no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—81no—
534C8—81no—
535C8—81no—
536C8—81no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—161no—
548C22—161no—
549C22—161no—
550C22—161no—
551C22C2×#39611yescharacteristic
552C22#304511yescharacteristic
553C22#304511yescharacteristic
554C22#305011yescharacteristic
555C22#305411yescharacteristic
556C22#306811yescharacteristic
557C22#306911yescharacteristic
558C4—21no—
559C4—21no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4—81no—
567C4—81no—
568C4—161no—
569C4—161no—
570C4#303511yescharacteristic
571C4#303511yescharacteristic
572C4#305711yescharacteristic
573C4#305811yescharacteristic
574C4#307611yescharacteristic
575C4#307611yescharacteristic
576C4#308111yescharacteristic
577C4#308111yescharacteristic
578C2—42no—
579C2—162no—
580C2#1941211yescharacteristic
581C2#1941311yescharacteristic
582C2#1944811yescharacteristic
583C2#1944911yescharacteristic
584C2#2360411yescharacteristic
585C2#2360911yescharacteristic
586C2#2361211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×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
1. #21972C2semidirect
4. #21362C2semidirect
5. C2×#1572C2semidirect
7. #10600C2semidirect
8. #10599C2semidirect
9. #10596C2semidirect
10. #10595C2semidirect
11. #10592C2semidirect
12. #10591C2semidirect
13. #10588C2semidirect
14. #10587C2semidirect
15. #8044C2semidirect

Maximal subgroups

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

subgroupindex
#219722
#218052
#213932
#213622
C2×#15722
C2×#15402
#106002
#105992
#105962
#105952
#105922
#105912
#105882
#105872
#80442