Finite Group Databasebeta

← #242605

Subgroups — #242605

Order 512.

812 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21964C211yesmaximal, characteristic, complemented
2#21809C211yesmaximal, characteristic
3#21410C211yesmaximal, characteristic
4#21145C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6C2×#1547C211yesmaximal, characteristic
7#10602C211yesmaximal, characteristic, complemented
8#10599C211yesmaximal, characteristic, complemented
9#10598C211yesmaximal, characteristic, complemented
10#10595C211yesmaximal, characteristic, complemented
11#10594C211yesmaximal, characteristic, complemented
12#10591C211yesmaximal, characteristic, complemented
13#10590C211yesmaximal, characteristic, complemented
14#10587C211yesmaximal, characteristic, complemented
15#8044C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#424—21no—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1982C2211yescharacteristic
27#1982C2211yescharacteristic
28#1981C2211yescharacteristic
29#1981C2211yescharacteristic
30#1959—21no—
31#1959—21no—
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1948—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947—21no—
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1940—21no—
41#1940—21no—
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1939C2211yescharacteristic
45#1939C2211yescharacteristic
46#1895—21no—
47#1895—21no—
48#1809—21no—
49#1809—21no—
50#1788—21no—
51#1788—21no—
52#1762—21no—
53#1762—21no—
54#1729—21no—
55#1569—24no—
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1547—24no—
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#331C2211yescharacteristic
64C2×#331C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×#41—21no—
70C22×M16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73C23×C8—21no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532Q814yes—
78C2×C4⋊Q8C2311yescharacteristic
79C2×#127C2311yescharacteristic
80C2×#124—21no—
81C2×#121—21no—
82C2×#121—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×#117—21no—
86C2×#117—21no—
87C22×C42C2311yescharacteristic
88#432—22no—
89#432—22no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#106—21no—
93C2×#106—21no—
94#424—44no—
95#423—22no—
96#423D414yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
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×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—
124C4×M16—22no—
125C4×M16—22no—
126C2×#97—21no—
127C2×#97D411yescharacteristic
128C2×#97D411yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#384D411yescharacteristic
132#384D411yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#382—21no—
136#382—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380D411yescharacteristic
141#380D411yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2311yescharacteristic
146C4×C4⋊C4C2311yescharacteristic
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—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#332—22no—
164#332—22no—
165#331—22no—
166#331—22no—
167C2×#41—24no—
168C2×#41—42no—
169C2×#41—81no—
170C2×#41—81no—
171C2×#41—81no—
172C2×#41—81no—
173C22×Q8—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—41no—
178C23×C4—41no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—24no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—41no—
188C2×M16—41no—
189C2×M16—81no—
190C2×M16—81no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8C2×Q812yes—
208#127—21no—
209#127—21no—
210#127—21no—
211#127C2×Q812yes—
212#124—24no—
213#124—42no—
214#121—24no—
215#121—24no—
216#121—42no—
217#121—42no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—42no—
221C4×D4—42no—
222#117—42no—
223#117—42no—
224#117—82no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—41no—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257#107—81no—
258#107—81no—
259#106—81no—
260#106—81no—
261C4⋊C8—24no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—42no—
273C4⋊C8—42no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8C2×D412yes—
278C4⋊C8C2×D412yes—
279#103—81no—
280#103—81no—
281#103—81no—
282#103—81no—
283#102—42no—
284#102—42no—
285#102—42no—
286#102—42no—
287#102—81no—
288#102—81no—
289#102—81no—
290#102—81no—
291C22⋊C8—42no—
292#97—22no—
293#97—22no—
294#97—42no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—41no—
304#95—41no—
305C24—41no—
306C24#12111yescharacteristic
307#41—84no—
308#41—84no—
309#41—84no—
310#41—84no—
311C2×Q8—22no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—44no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—81no—
333C2×D4—81no—
334C2×D4—82no—
335C2×D4—82no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—44no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4C22⋊Q812yes—
362C22×C4C22⋊Q812yes—
363C22×C4C22⋊Q812yes—
364C22×C4#12612yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368M16—44no—
369M16—44no—
370M16—84no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
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—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—84no—
400C2×C8AGL(1,ℤ/8)12yes—
401C2×C8#13712yes—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—82no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—82no—
438C42#12112yes—
439C42#12112yes—
440C42C22⋊Q812yes—
441C42C22⋊Q812yes—
442C42C22⋊Q812yes—
443C42#12612yes—
444C42C22×D411yescharacteristic
445C42C22×Q811yescharacteristic
446C42C2×#4111yescharacteristic
447C42C2×#4111yescharacteristic
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—42no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23#39612yes—
462C23C2×#12111yescharacteristic
463Q8—42no—
464Q8—42no—
465Q8—81no—
466Q8—81no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—82no—
472D4—82no—
473D4—88no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—81no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—88no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4#39612yes—
519C2×C4#39612yes—
520C2×C4#39612yes—
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×#12611yescharacteristic
527C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
528C2×C4C2×#13711yescharacteristic
529C2×C4#57511yescharacteristic
530C2×C4#57511yescharacteristic
531C2×C4#57611yescharacteristic
532C2×C4#57611yescharacteristic
533C2×C4#57711yescharacteristic
534C2×C4#57711yescharacteristic
535C8—22no—
536C8—22no—
537C8—42no—
538C8—42no—
539C8—41no—
540C8—41no—
541C8—81no—
542C8—81no—
543C8—81no—
544C8—81no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—161no—
556C22—161no—
557C22—161no—
558C22—161no—
559C22C2×#39611yescharacteristic
560C22#303611yescharacteristic
561C22#303711yescharacteristic
562C22#306211yescharacteristic
563C22#306211yescharacteristic
564C22#306811yescharacteristic
565C22#306911yescharacteristic
566C4—21no—
567C4—21no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—82no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—161no—
577C4—161no—
578C4#304111yescharacteristic
579C4#304311yescharacteristic
580C4#305511yescharacteristic
581C4#305611yescharacteristic
582C4#307911yescharacteristic
583C4#308011yescharacteristic
584C4#308111yescharacteristic
585C4#308111yescharacteristic
586C2—42no—
587C2—162no—
588C2#1936811yescharacteristic
589C2#1937011yescharacteristic
590C2#1937211yescharacteristic
591C2#1937411yescharacteristic
592C2#2360411yescharacteristic
593C2#2361411yescharacteristic
594C2#2361711yescharacteristic
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. #21964C2semidirect
4. #21145C2semidirect
5. C2×#1569C2semidirect
7. #10602C2semidirect
8. #10599C2semidirect
9. #10598C2semidirect
10. #10595C2semidirect
11. #10594C2semidirect
12. #10591C2semidirect
13. #10590C2semidirect
14. #10587C2semidirect
15. #8044C2semidirect

Maximal subgroups

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

subgroupindex
#219642
#218092
#214102
#211452
C2×#15692
C2×#15472
#106022
#105992
#105982
#105952
#105942
#105912
#105902
#105872
#80442