Finite Group Databasebeta

← #402431

Subgroups — #402431

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊C8C211yesmaximal, characteristic
3#14702C211yesmaximal, characteristic
4#14682C211yesmaximal, characteristic
5#11358C211yesmaximal, characteristic, complemented
6#11268C211yesmaximal, characteristic, complemented
7#11199C212yesmaximal
8#11052C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10926C212yesmaximal, complemented
11#10820C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24#2074—24no—
25C4×C22⋊C8—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1436—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38#1434—22no—
39C22⋊#95—22no—
40C22⋊#95—22no—
41C23×Q8C2311yescharacteristic
42C24×C4D412yes—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—42no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C42—22no—
60C22×C42—22no—
61C22×C42—24no—
62C22×C42—42no—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103—42no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C4×C8—24no—
76C2×C4×C8—24no—
77C2×C4×C8—42no—
78#391—22no—
79#391—22no—
80#391—24no—
81#382—22no—
82#382—22no—
83#382—42no—
84#380—24no—
85#380—42no—
86#380—42no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#340—42no—
102#340—42no—
103#336—22no—
104#336D414yes—
105C25C2×D411yescharacteristic
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8#4112yes—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4#4114yes—
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8C2×D414yes—
156C22×C8#4114yes—
157C22⋊Q8—48no—
158C22⋊Q8—48no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4#4112yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—48no—
202C2×C42—48no—
203C2×C42—42no—
204C2×C42C2×D412yes—
205C2×C42#4112yes—
206C2×C42#4114yes—
207#103—48no—
208#103—48no—
209C22⋊C8—216no—
210C4×C8—48no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—42no—
218#95—44no—
219#95—44no—
220#95—44no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24#12412yes—
231C24#13514yes—
232C24C22×D411yescharacteristic
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—216no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—48no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4#12114yes—
325C22×C4#12314yes—
326C22×C4#12412yes—
327C22×C4#13514yes—
328C22×C4#13714yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C4H(5,𝔽2)11yescharacteristic
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C22⋊C4—48no—
369C22⋊C4—88no—
370C22⋊C4—84no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—48no—
376C23—22no—
377C23—24no—
378C23—22no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23—22no—
387C23—24no—
388C23—42no—
389C23—42no—
390C23—41no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23#48814yes—
399C23C2×#12111yescharacteristic
400C23C2×#12311yescharacteristic
401C23#52511yescharacteristic
402C23C2×#12411yescharacteristic
403C23#53211yescharacteristic
404C23#53411yescharacteristic
405C23#53511yescharacteristic
406C23C2×#13512yes—
407C23C2×#13512yes—
408C23C2×#13712yes—
409C23#57512yes—
410Q8—44no—
411Q8—44no—
412Q8—82no—
413Q8—82no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—432no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4#48518yes—
476C2×C4#52512yes—
477C2×C4#52912yes—
478C2×C4#52912yes—
479C2×C4#52912yes—
480C2×C4#53812yes—
481C2×C4#54012yes—
482C2×C4#54611yescharacteristic
483C2×C4#54612yes—
484C2×C4#55011yescharacteristic
485C2×C4#55112yes—
486C2×C4#55112yes—
487C2×C4#55112yes—
488C2×C4#55311yescharacteristic
489C2×C4#55312yes—
490C2×C4#55312yes—
491C2×C4#55511yescharacteristic
492C8—44no—
493C8—44no—
494C22—24no—
495C22—24no—
496C22—24no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—42no—
502C22—44no—
503C22—44no—
504C22—42no—
505C22#239511yescharacteristic
506C22#241211yescharacteristic
507C22#241411yescharacteristic
508C22#250411yescharacteristic
509C22#256111yescharacteristic
510C22#266111yescharacteristic
511C22#285011yescharacteristic
512C22#305012yes—
513C22#305612yes—
514C22C2×#48512yes—
515C22#309112yes—
516C22#310212yes—
517C22#310812yes—
518C22C2×#48811yescharacteristic
519C22#313111yescharacteristic
520C22#313911yescharacteristic
521C22#314211yescharacteristic
522C22#316812yes—
523C22#317212yes—
524C22#319012yes—
525C22#319112yes—
526C22#321012yes—
527C22#321112yes—
528C4—28no—
529C4—44no—
530C4—42no—
531C4—82no—
532C4—82no—
533C4—161no—
534C4—161no—
535C4—162no—
536C4—162no—
537C4#245712yes—
538C4#246712yes—
539C4#253112yes—
540C4#257812yes—
541C2—44no—
542C2#1484211yescharacteristic
543C2#2375912yes—
544C2#2455412yes—
545C2#2465512yes—
546C2#2797511yescharacteristic
547C2#2797711yescharacteristic
548C2#2910611yescharacteristic
549C2#2910811yescharacteristic
550C2#3032112yes—
551C2#3304411yescharacteristic
552C2#3305911yescharacteristic
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×#2074C2semidirect
5. #11358C2semidirect
6. #11268C2semidirect
8. #11052C2semidirect
10. #10926C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20742
C2×C4×C22⋊C82
#147022
#146822
#113582
#112682
#111992
#110522
#109702
#109262
#108202