Finite Group Databasebeta

← #227234

Subgroups — #227234

Order 512.

828 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23061C211yesmaximal, characteristic, complemented
2#22969C211yesmaximal, characteristic, complemented
3#22485C211yesmaximal, characteristic
4#22417C211yesmaximal, characteristic
5C2×#1635C211yesmaximal, characteristic
6#9826C211yesmaximal, characteristic, complemented
7#9570C211yesmaximal, characteristic, complemented
8#9566C211yesmaximal, characteristic, complemented
9#9215C211yesmaximal, characteristic, complemented
10#9215C211yesmaximal, characteristic, complemented
11#9060C211yesmaximal, characteristic, complemented
12#8889C211yesmaximal, characteristic, complemented
13#8886C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8200C211yesmaximal, characteristic, complemented
16C2×#547C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#476C2211yescharacteristic
19C2×#476C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#416—21no—
23#2369C2211yescharacteristic
24#2354C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1988C2211yescharacteristic
30#1981C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1944—21no—
35#1934—21no—
36#1926—21no—
37#1920—21no—
38#1920—21no—
39#1893—21no—
40#1892—21no—
41#1874—21no—
42#1873—21no—
43#1862—21no—
44#1844C2211yescharacteristic
45#1823—21no—
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1788—21no—
49C4×#106—21no—
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53C4×#102C2211yescharacteristic
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56#1635C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×#41—21no—
69C23×C8—21no—
70C23×C8—21no—
71#547—22no—
72#547—22no—
73#547—41no—
74#547C2×C414yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#123—41no—
78C2×C22⋊Q8—21no—
79C2×#121—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86#487—41no—
87#487C2×C414yescomplemented
88#476—22no—
89#476—22no—
90#476C2×C414yes—
91#476C2×C414yescomplemented
92C8×D4—22no—
93C8×D4—22no—
94C8×D4—41no—
95C8×D4—41no—
96C2×#107—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—21no—
100C2×#106—41no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105#416—44no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—41no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102D411yescharacteristic
116C2×#102D411yescharacteristic
117C2×#102D411yescharacteristic
118C2×#102D411yescharacteristic
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—41no—
127#390—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#388—41no—
132#385—41no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#380—21no—
136#380—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
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#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#336—21no—
159#336—21no—
160#333—41no—
161#332C2×C414yescomplemented
162D4⋊C8C2×C414yes—
163D4⋊C8C2×C414yescomplemented
164#322—41no—
165C2×#41—24no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41—42no—
169C2×#41—81no—
170C2×#41—81no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×D4—41no—
174C22×D4C2×D411yescharacteristic
175C22×D4C2×D411yescharacteristic
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—41no—
179C23×C4—41no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—81no—
190C22×C8—81no—
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8C22×C412yes—
202C4⋊Q8C22×C412yes—
203#123—44no—
204#123—44no—
205C22⋊Q8—24no—
206C22⋊Q8—42no—
207#121—24no—
208#121—42no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—42no—
214C4×D4—81no—
215C4×D4C22×C412yes—
216C4×D4C22×C412yes—
217C4×D4#4112yes—
218C4×D4#4112yes—
219#117—42no—
220#117—81no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
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—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—81no—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42C2411yescharacteristic
255#107—81no—
256#106—24no—
257#106—24no—
258#106—42no—
259#106—82no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8C22×C412yes—
267C4⋊C8C22×C412yes—
268#103—44no—
269#103—42no—
270#103—42no—
271#103—81no—
272#103—81no—
273#102—24no—
274#102—24no—
275#102—22no—
276#102—41no—
277#102—41no—
278#102—42no—
279#102—81no—
280#102SD1614yes—
281C22⋊C8—42no—
282C22⋊C8—42no—
283#97—81no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—42no—
290C4×C8C22×C412yes—
291#95—21no—
292#95—21no—
293#95—41no—
294#95—41no—
295#95—41no—
296#95—41no—
297C24—21no—
298C24—21no—
299#41—48no—
300#41—84no—
301#41—84no—
302#41—84no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—81no—
312C2×Q8—81no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—44no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4C4×D412yes—
325C2×D4C4×D412yes—
326C2×D4C2×SD1612yes—
327C2×D4#13512yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—44no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—161no—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
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—84no—
386C2×C8—84no—
387C2×C8C2×SD1612yes—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—44no—
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—41no—
413C4⋊C4—41no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—82no—
417C4⋊C4—161no—
418C4⋊C4C4×D412yes—
419C4⋊C4C4×D412yes—
420C4⋊C4C2×SD1612yes—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—161no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—82no—
434C42—161no—
435C42C23×C411yescharacteristic
436C42C2×#4111yescharacteristic
437C42C2×#4111yescharacteristic
438C42H(5,𝔽2)11yescharacteristic
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23D4211yescharacteristic
454Q8—42no—
455Q8—42no—
456Q8—82no—
457Q8—81no—
458Q8—81no—
459Q8—161no—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—22no—
464D4—22no—
465D4—41no—
466D4—41no—
467D4—88no—
468D4C4×SD1614yes—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—88no—
500C2×C4—81no—
501C2×C4—81no—
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—164no—
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×C4×D411yescharacteristic
513C2×C4#51811yescharacteristic
514C2×C4#54611yescharacteristic
515C2×C4#54611yescharacteristic
516C2×C4#55111yescharacteristic
517C2×C4C22×SD1611yescharacteristic
518C2×C4C2×#13511yescharacteristic
519C2×C4#57511yescharacteristic
520C2×C4#57511yescharacteristic
521C2×C4#57611yescharacteristic
522C2×C4#57711yescharacteristic
523C2×C4#57711yescharacteristic
524C2×C4#57811yescharacteristic
525C8—22no—
526C8—42no—
527C8—42no—
528C8—42no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—81no—
533C8—81no—
534C22—24no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22#235811yescharacteristic
549C22#326711yescharacteristic
550C22#326911yescharacteristic
551C22#328011yescharacteristic
552C22D4×SD1611yescharacteristic
553C22#328811yescharacteristic
554C22#328911yescharacteristic
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—82no—
564C4—82no—
565C4—161no—
566C4—161no—
567C4C2×C4×SD1611yescharacteristic
568C4#294311yescharacteristic
569C4#294611yescharacteristic
570C4#294711yescharacteristic
571C4#329511yescharacteristic
572C4#329611yescharacteristic
573C4#331111yescharacteristic
574C4#331511yescharacteristic
575C2—44no—
576C2—82no—
577C2#1976611yescharacteristic
578C2#1976811yescharacteristic
579C2#1977511yescharacteristic
580C2#1977811yescharacteristic
581C2#2285611yescharacteristic
582C2#2287411yescharacteristic
583C2#2289311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23061C2semidirect
2. #22969C2semidirect
6. #9826C2semidirect
7. #9570C2semidirect
8. #9566C2semidirect
9. #9215C2semidirect
10. #9215C2semidirect
11. #9060C2semidirect
12. #8889C2semidirect
13. #8886C2semidirect
14. #8202C2semidirect
15. #8200C2semidirect
56. #1635C4semidirect
87. #487C2×C4semidirect
91. #476C2×C4semidirect
161. #332C2×C4semidirect
163. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#230612
#229692
#224852
#224172
C2×#16352
#98262
#95702
#95662
#92152
#90602
#88892
#88862
#82022
#82002