Finite Group Databasebeta

← #360810

Subgroups — #360810

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic, complemented
3C2×#1804C211yesmaximal, characteristic
4C2×C4×#102C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic, complemented
6#11293C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10834C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2349—24no—
20#2349—24no—
21C2×#389C2212yes—
22C2×#385—22no—
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C22×#95—21no—
29#1835—24no—
30#1804—24no—
31C4×#102C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1442—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—21no—
45#1434—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4C2×C414yes—
62C2×C4×D4D414yes—
63C2×C4×D4Q814yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102C2×C414yes—
87C2×#102C2×C414yes—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8C2×C414yescomplemented
91#389—24no—
92#389—24no—
93#385—216no—
94#382—22no—
95#382—22no—
96#382—22no—
97#382D414yes—
98#380—28no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yescomplemented
103C2×#95—28no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#340C2×C414yescomplemented
122#340C2×C414yescomplemented
123#336—42no—
124#336—42no—
125C25—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C22×C412yes—
134C22×D4C2×D412yes—
135C22×D4C2×Q812yes—
136C22×D4#4112yes—
137C22×D4#4112yes—
138C22×D4#4112yes—
139C22×D4#4112yes—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—28no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4—82no—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C22×C412yes—
182C22×C8C22×C412yes—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—28no—
186C4×D4—48no—
187C4×D4—48no—
188C4×D4—48no—
189C4×D4—48no—
190C4×D4C4⋊C4116yes—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×Q812yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—82no—
241C2×C42C22×C412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C4⋊C8—44no—
248#102—28no—
249#102—28no—
250#102—48no—
251C4×C8—28no—
252#95—28no—
253#95—24no—
254#95—22no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—82no—
263#95—82no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24C22×D411yescharacteristic
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4C2×C4⋊C414yes—
298C2×D4#11714yes—
299C2×D4#11714yes—
300C2×D4#12414yes—
301C2×D4#12414yes—
302C2×D4#12514yes—
303C2×D4#12514yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—216no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—416no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—416no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—88no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—88no—
357C22×C4—88no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4#13514yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×Q811yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C42−1+411yescharacteristic
382C22×C42−1+411yescharacteristic
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
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—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—48no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—88no—
425C4⋊C4C2×C4⋊C414yes—
426C22⋊C4—216no—
427C22⋊C4—416no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—84no—
433C42C2×C4⋊C414yes—
434C42C4×D414yes—
435C42C22⋊Q814yes—
436C42#12314yes—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23C2×C4×D411yescharacteristic
461C23C2×#12011yescharacteristic
462C23C2×C22⋊Q811yescharacteristic
463C23#54611yescharacteristic
464C23#54611yescharacteristic
465C23#54811yescharacteristic
466C23#54811yescharacteristic
467C23C2×#13512yes—
468C23#57512yes—
469C23#57612yes—
470C23#57712yes—
471D4—24no—
472D4—24no—
473D4—24no—
474D4—24no—
475D4—24no—
476D4—24no—
477D4—24no—
478D4#38218yes—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—816no—
516C2×C4—832no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4#38514yes—
533C2×C4#38514yes—
534C2×C4C22×C4⋊C411yescharacteristic
535C2×C4C2×#11711yescharacteristic
536C2×C4C2×#11711yescharacteristic
537C2×C4C2×C4×D411yescharacteristic
538C2×C4#52011yescharacteristic
539C2×C4#52011yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4C2×#12311yescharacteristic
542C2×C4C2×#12411yescharacteristic
543C2×C4C2×#12411yescharacteristic
544C2×C4C2×#12511yescharacteristic
545C2×C4C2×#12511yescharacteristic
546C2×C4#53511yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#54411yescharacteristic
549C2×C4#54411yescharacteristic
550C2×C4#54611yescharacteristic
551C2×C4#54611yescharacteristic
552C2×C4#54811yescharacteristic
553C2×C4#54811yescharacteristic
554C2×C4#55211yescharacteristic
555C2×C4#55211yescharacteristic
556C2×C4#55211yescharacteristic
557C2×C4#55211yescharacteristic
558C2×C4#55311yescharacteristic
559C2×C4#55311yescharacteristic
560C2×C4#55311yescharacteristic
561C2×C4#55311yescharacteristic
562C8—42no—
563C8—42no—
564C8—82no—
565C22—28no—
566C22—48no—
567C22—48no—
568C22—48no—
569C22—48no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22C2×#38511yescharacteristic
577C22#235211yescharacteristic
578C22#235211yescharacteristic
579C22#243511yescharacteristic
580C22#243611yescharacteristic
581C22#244211yescharacteristic
582C22#244211yescharacteristic
583C22C4×#13511yescharacteristic
584C22#294311yescharacteristic
585C22#294811yescharacteristic
586C22#294911yescharacteristic
587C22#300212yes—
588C22#301212yes—
589C22#307612yes—
590C22#308112yes—
591C22#329612yes—
592C22#329712yes—
593C22#332212yes—
594C22#332312yes—
595C22#336012yes—
596C22#336112yes—
597C22#337012yes—
598C22#337612yes—
599C4—24no—
600C4—24no—
601C4—44no—
602C4—44no—
603C4—82no—
604C4—84no—
605C4—82no—
606C4—82no—
607C4—162no—
608C4C2×#38211yescharacteristic
609C4#231711yescharacteristic
610C4#235211yescharacteristic
611C4#235211yescharacteristic
612C4#244211yescharacteristic
613C4#244211yescharacteristic
614C4#250311yescharacteristic
615C4#250411yescharacteristic
616C2—48no—
617C2#1388111yescharacteristic
618C2#2087211yescharacteristic
619C2#2087911yescharacteristic
620C2#2276011yescharacteristic
621C2#2276411yescharacteristic
622C2#2276811yescharacteristic
623C2#2277211yescharacteristic
624C2#2589012yes—
625C2#2593812yes—
626C2#2611512yes—
627C2#2611712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1835C2semidirect
5. #13955C2semidirect
6. #11293C2semidirect
7. #11292C2semidirect
8. #10850C2semidirect
9. #10834C2semidirect
31. C4×#102C4semidirect
90. C2×C4×C8C2×C4semidirect
102. C4×C4⋊C4C2×C4semidirect
121. #340C2×C4semidirect
122. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#18352
C2×#18042
C2×C4×#1022
#139552
#112932
#112922
#108502
#108342
#107882