Finite Group Databasebeta

← #227378

Subgroups — #227378

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23094C211yesmaximal, characteristic, complemented
2#22968C211yesmaximal, characteristic, complemented
3#22503C211yesmaximal, characteristic
4#22489C211yesmaximal, characteristic
5C2×#1646C211yesmaximal, characteristic
6#9840C211yesmaximal, characteristic, complemented
7#9575C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9227C211yesmaximal, characteristic, complemented
10#9217C211yesmaximal, characteristic, complemented
11#9065C211yesmaximal, characteristic, complemented
12#8894C211yesmaximal, characteristic, complemented
13#8893C211yesmaximal, characteristic, complemented
14#8212C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic, complemented
16C2×#555C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#461C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22#2365C2211yescharacteristic
23#2354C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1987C2211yescharacteristic
29#1987C2211yescharacteristic
30#1968—21no—
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1943—21no—
35#1927—21no—
36#1921—21no—
37#1895—21no—
38#1893—21no—
39#1880—21no—
40#1875—21no—
41#1874—21no—
42#1864—21no—
43#1834C2211yescharacteristic
44#1825—21no—
45#1825—21no—
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1762—21no—
51C4×#103—21no—
52C4×#102C2211yescharacteristic
53C4×#102C2211yescharacteristic
54#1736—21no—
55#1735—21no—
56#1646C418yescomplemented
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×SD16—21no—
69C22×M16—21no—
70C23×C8—21no—
71#555—22no—
72#555—22no—
73#555—41no—
74#555C2×C414yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#124—41no—
78C2×#121—21no—
79C2×#121—21no—
80C2×C4×Q8—21no—
81C2×C4×D4—21no—
82C2×C4×D4C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C2×#117—21no—
87C22×C4⋊C4—21no—
88#487—41no—
89#487C2×C414yescomplemented
90#474—22no—
91#474C2×C414yes—
92#461—22no—
93#461C2×C414yescomplemented
94#436—22no—
95#436—41no—
96C8×D4—22no—
97C8×D4—41no—
98C2×#107—21no—
99C2×#106—21no—
100C2×#106—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—41no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102D411yescharacteristic
113C2×#102D411yescharacteristic
114C2×#102D411yescharacteristic
115C2×#102D411yescharacteristic
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#390—21no—
125#389—21no—
126#389—21no—
127#388—41no—
128#387—41no—
129#385—41no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#380—21no—
133#380—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—41no—
153#340D411yescharacteristic
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#336—21no—
158#336—21no—
159#333C2×C414yescomplemented
160#332—41no—
161D4⋊C8C2×C414yes—
162D4⋊C8C2×C414yescomplemented
163#322—41no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×D4—41no—
167C22×D4C2×D411yescharacteristic
168C22×D4C2×D411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—41no—
172C2×SD16—24no—
173C2×SD16—81no—
174C2×SD16—81no—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—81no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—81no—
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C22×C412yes—
197C4⋊Q8C22×C412yes—
198#124—44no—
199#124—44no—
200#121—24no—
201#121—24no—
202#121—42no—
203#121—42no—
204C4×Q8—42no—
205C4×Q8—81no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—42no—
211C4×D4—81no—
212C4×D4C22×C412yes—
213C4×D4C22×C412yes—
214C4×D4#4112yes—
215C4×D4#4112yes—
216#117—24no—
217#117—24no—
218#117—42no—
219#117—42no—
220#117—42no—
221#117—81no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
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—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42C2411yescharacteristic
255#107—24no—
256#106—81no—
257#106—81no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8C22×C412yes—
265C4⋊C8C22×C412yes—
266#103—24no—
267#103—44no—
268#103—42no—
269#103—81no—
270#102—24no—
271#102—22no—
272#102—41no—
273#102—41no—
274#102—41no—
275#102—81no—
276#102—81no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279#97—81no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—81no—
286C4×C8C22×C412yes—
287#95—21no—
288#95—21no—
289#95—41no—
290#95—41no—
291#95—41no—
292#95—41no—
293C24—21no—
294C24—21no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—81no—
304C2×Q8—81no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—44no—
316C2×D4—161no—
317C2×D4—161no—
318C2×D4C4×D412yes—
319C2×D4C4×D412yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—161no—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348SD16—84no—
349M16—84no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—84no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—81no—
404C4⋊C4—81no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4C4×D412yes—
408C4⋊C4C4×D412yes—
409C4⋊C4#13712yes—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—161no—
419C42—44no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—82no—
425C42—82no—
426C42C23×C411yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C42H(5,𝔽2)11yescharacteristic
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23D4211yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—82no—
448Q8—81no—
449Q8—81no—
450Q8—161no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—22no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—164no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—164no—
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C2×C4×D411yescharacteristic
503C2×C4#51811yescharacteristic
504C2×C4#54611yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4#55111yescharacteristic
507C2×C4C2×#13711yescharacteristic
508C2×C4#57511yescharacteristic
509C2×C4#57511yescharacteristic
510C2×C4#57511yescharacteristic
511C2×C4#57611yescharacteristic
512C2×C4#57711yescharacteristic
513C2×C4#57711yescharacteristic
514C2×C4#57711yescharacteristic
515C8—42no—
516C8—41no—
517C8—42no—
518C8—42no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—81no—
523C8—81no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22#235811yescharacteristic
539C22#326611yescharacteristic
540C22#326711yescharacteristic
541C22#326711yescharacteristic
542C22#326811yescharacteristic
543C22#329011yescharacteristic
544C22#329011yescharacteristic
545C4—21no—
546C4—21no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—82no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—161no—
556C4—161no—
557C4#294311yescharacteristic
558C4C4×#13711yescharacteristic
559C4#294611yescharacteristic
560C4#295211yescharacteristic
561C4#330711yescharacteristic
562C4#331911yescharacteristic
563C4#332311yescharacteristic
564C4#333111yescharacteristic
565C2—82no—
566C2—82no—
567C2#2004111yescharacteristic
568C2#2004911yescharacteristic
569C2#2013011yescharacteristic
570C2#2013311yescharacteristic
571C2#2286011yescharacteristic
572C2#2288211yescharacteristic
573C2#2289611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#555C22
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. #23094C2semidirect
2. #22968C2semidirect
6. #9840C2semidirect
7. #9575C2semidirect
8. #9567C2semidirect
9. #9227C2semidirect
10. #9217C2semidirect
11. #9065C2semidirect
12. #8894C2semidirect
13. #8893C2semidirect
14. #8212C2semidirect
15. #8204C2semidirect
56. #1646C4semidirect
89. #487C2×C4semidirect
93. #461C2×C4semidirect
159. #333C2×C4semidirect
162. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#230942
#229682
#225032
#224892
C2×#16462
#98402
#95752
#95672
#92272
#92172
#90652
#88942
#88932
#82122
#82042