Finite Group Databasebeta

← #402179

Subgroups — #402179

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2096C211yesmaximal, characteristic, complemented
2C2×#1892C212yesmaximal
3C2×#1807C212yesmaximal
4C22×#336C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6#14686C211yesmaximal, characteristic
7#11378C212yesmaximal, complemented
8#11373C211yesmaximal, characteristic, complemented
9#11323C211yesmaximal, characteristic, complemented
10#11122C212yesmaximal, complemented
11#11050C211yesmaximal, characteristic, complemented
12C24×C8C2212yes—
13C22×C22⋊Q8C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#106C2212yes—
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#380—22no—
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26#2096—24no—
27#1892—28no—
28#1807—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336—24no—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1436—22no—
38C4⋊#95—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C23×Q8C2311yescharacteristic
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C23×C8—24no—
47C23×C8—24no—
48C23×C8—24no—
49C23×C8—24no—
50C23×C8—42no—
51C23×C8—42no—
52C23×C8D418yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8Q814yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C42—24no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—42no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×#106—42no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—42no—
88C2×#103—42no—
89#391—22no—
90#391—22no—
91#391—42no—
92#384—22no—
93#384—22no—
94#384—42no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—48no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—24no—
105C2×#95—21no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336—24no—
111#336—22no—
112#336D414yes—
113C25C2×D411yescharacteristic
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8C2×Q812yes—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4SD1618yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D414yes—
147C23×C4C2×Q812yes—
148C23×C4#4114yes—
149C23×C4#4112yes—
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—216no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—216no—
163C22×C8—48no—
164C22×C8—48no—
165C22×C8—48no—
166C22×C8—48no—
167C22×C8—48no—
168C22×C8C2×D414yes—
169C22×C8C2×D414yes—
170C22×C8C2×D414yes—
171C22×C8#4114yes—
172C22⋊Q8—216no—
173C22⋊Q8—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4C2×Q812yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—44no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—48no—
223C2×C42—48no—
224C2×C42—42no—
225C2×C42C2×D412yes—
226C2×C42#4112yes—
227#106—48no—
228#106—48no—
229#103—48no—
230#103—48no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—48no—
237#95—44no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24C22⋊Q812yes—
245C24#12312yes—
246C24#12712yes—
247C24C2×SD1614yes—
248C24#13714yes—
249C24C22×D411yescharacteristic
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—216no—
295C22×C4—216no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4#12114yes—
334C22×C4C22⋊Q818yes—
335C22×C4C22⋊Q812yes—
336C22×C4#12318yes—
337C22×C4#12312yes—
338C22×C4#12712yes—
339C22×C4C2×SD1614yes—
340C22×C4C2×SD1614yes—
341C22×C4#13514yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D412yes—
344C22×C4C22×Q811yescharacteristic
345C22×C4C2×#4112yes—
346C22×C4C2×#4112yes—
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+411yescharacteristic
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—432no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8#12118yes—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C22⋊C4—48no—
385C22⋊C4—88no—
386C22⋊C4—84no—
387C42—24no—
388C42—24no—
389C42—48no—
390C42—42no—
391C42—42no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—22no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23#39914yes—
411C23#47518yes—
412C23#48318yes—
413C23#49214yes—
414C23#49714yes—
415C23C2×#12111yescharacteristic
416C23C2×C22⋊Q811yescharacteristic
417C23C2×#12311yescharacteristic
418C23C2×#12711yescharacteristic
419C23#53211yescharacteristic
420C23#53411yescharacteristic
421C23#53711yescharacteristic
422C23C22×SD1612yes—
423C23C2×#13512yes—
424C23C2×#13712yes—
425C23#57512yes—
426Q8—44no—
427Q8—44no—
428Q8—82no—
429Q8—82no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—432no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—84no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—82no—
484C2×C4—84no—
485C2×C4—82no—
486C2×C4#39914yes—
487C2×C4#46518yes—
488C2×C4C2×#12112yes—
489C2×C4C2×C22⋊Q812yes—
490C2×C4C2×#12312yes—
491C2×C4#53212yes—
492C2×C4#53412yes—
493C2×C4#53712yes—
494C2×C4#54611yescharacteristic
495C2×C4#54612yes—
496C2×C4D4×Q811yescharacteristic
497C2×C4#55112yes—
498C2×C4#55112yes—
499C2×C4#55112yes—
500C2×C4#55212yes—
501C2×C4#55211yescharacteristic
502C2×C4#55411yescharacteristic
503C2×C4#55412yes—
504C8—28no—
505C8—44no—
506C22—28no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—28no—
513C22—28no—
514C22—28no—
515C22—24no—
516C22#209618yescomplemented
517C22C2×#39911yescharacteristic
518C22#241211yescharacteristic
519C22#241711yescharacteristic
520C22#249811yescharacteristic
521C22#256411yescharacteristic
522C22#265611yescharacteristic
523C22#285611yescharacteristic
524C22C2×#46512yes—
525C22#305612yes—
526C22C2×#47512yes—
527C22#307612yes—
528C22C2×#48312yes—
529C22#309112yes—
530C22C2×#49211yescharacteristic
531C22#314711yescharacteristic
532C22C2×#49711yescharacteristic
533C22#315011yescharacteristic
534C22#316712yes—
535C22#318112yes—
536C22#318712yes—
537C22#319012yes—
538C22#324612yes—
539C22#324812yes—
540C4—28no—
541C4—24no—
542C4—44no—
543C4—82no—
544C4—82no—
545C4—162no—
546C4—161no—
547C4—162no—
548C4—161no—
549C4#241212yes—
550C4#241712yes—
551C4#249812yes—
552C4#256412yes—
553C2—28no—
554C2#1483611yescharacteristic
555C2C2×#209611yescharacteristic
556C2#2373011yescharacteristic
557C2#2455112yes—
558C2#2483812yes—
559C2#2785512yes—
560C2#2915912yes—
561C2#3024011yescharacteristic
562C2#3024411yescharacteristic
563C2#3317011yescharacteristic
564C2#3318411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2096C2semidirect
7. #11378C2semidirect
8. #11373C2semidirect
9. #11323C2semidirect
10. #11122C2semidirect
11. #11050C2semidirect
516. C22#2096semidirect

Maximal subgroups

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

subgroupindex
C2×#20962
C2×#18922
C2×#18072
C22×#3362
#147022
#146862
#113782
#113732
#113232
#111222
#110502