Finite Group Databasebeta

← #395667

Subgroups — #395667

Order 512.

2190 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#2066C212yesmaximal, complemented
3C2×#1853C211yesmaximal, characteristic
4C2×#1850C211yesmaximal, characteristic
5C2×#1787C211yesmaximal, characteristic
6#13922C211yesmaximal, characteristic
7#13907C211yesmaximal, characteristic
8#11122C212yesmaximal, complemented
9#11077C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×#103—28no—
19C22×#103—24no—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#2066—28no—
33#1853—24no—
34#1850—24no—
35#1787—24no—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1446—22no—
42#1441—22no—
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8—21no—
48C23×Q8—22no—
49C23×Q8—24no—
50C23×Q8—24no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—42no—
56C23×Q8—42no—
57C23×Q8Q814yes—
58C23×Q8C2312yes—
59C24×C4—21no—
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C23×C8—22no—
63C23×C8—24no—
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—42no—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2312yes—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4D414yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42C2312yes—
86C2×#106—24no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—42no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—42no—
93C2×#103—232no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#103D418yes—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8D414yes—
109#391—24no—
110#391—24no—
111#391—42no—
112#391—42no—
113#389—24no—
114#389—42no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—22no—
119#380—44no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95D412yes—
132#340—24no—
133#340—22no—
134#340—22no—
135#340—42no—
136#336—24no—
137C25C2×D411yescharacteristic
138C22×Q8—24no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—28no—
143C22×Q8—216no—
144C22×Q8—28no—
145C22×Q8—216no—
146C22×Q8—28no—
147C22×Q8—28no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—44no—
157C22×Q8—44no—
158C22×Q8—48no—
159C22×Q8—48no—
160C22×Q8—48no—
161C22×Q8—48no—
162C22×Q8—48no—
163C22×Q8—48no—
164C22×Q8—48no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—48no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8C2×D414yes—
173C22×Q8C2×Q814yes—
174C22×Q8#4114yes—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—82no—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×Q812yes—
201C23×C4#4112yes—
202C23×C4#4112yes—
203C23×C4#4112yes—
204C23×C4C2411yescharacteristic
205C22×C8—28no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—48no—
221C22×C8—48no—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8C2×D414yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D412yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4SD1614yes—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275#106—44no—
276#106—44no—
277C4⋊C8—48no—
278#103—216no—
279#103—216no—
280#103—48no—
281#103—48no—
282C22⋊C8—28no—
283#95—24no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—42no—
291#95—44no—
292#95—82no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24#12112yes—
298C24#12112yes—
299C24C22⋊Q812yes—
300C24C2×SD1612yes—
301C24#13512yes—
302C24#13712yes—
303C24#13712yes—
304C24C22×D411yescharacteristic
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—232no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—416no—
322C2×Q8—432no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—416no—
326C2×Q8—416no—
327C2×Q8—416no—
328C2×Q8—432no—
329C2×Q8—416no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—48no—
343C2×Q8C22⋊Q818yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—416no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—416no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—48no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—48no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—84no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4#12114yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4C22⋊Q814yes—
421C22×C4C22⋊Q814yes—
422C22×C4C22⋊Q812yes—
423C22×C4#12614yes—
424C22×C4C2×SD1614yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×Q811yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)12yes—
434C22×C4H(5,𝔽2)12yes—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—416no—
443C2×C8—41no—
444C2×C8—42no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—48no—
458C4⋊C4—42no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—88no—
470C4⋊C4#12014yes—
471C22⋊C4—28no—
472C22⋊C4—44no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23#39614yes—
496C23#46114yes—
497C23#46414yes—
498C23#46514yes—
499C23#47014yes—
500C23#47514yes—
501C23#47914yes—
502C23C2×#12111yescharacteristic
503C23C2×#12111yescharacteristic
504C23C2×C22⋊Q811yescharacteristic
505C23D4212yes—
506C23#54612yes—
507C23C22×SD1611yescharacteristic
508C23C2×#13511yescharacteristic
509C23C2×#13711yescharacteristic
510C23C2×#13711yescharacteristic
511C23#57712yes—
512C23#57812yes—
513Q8—28no—
514Q8—28no—
515Q8—432no—
516Q8—44no—
517Q8—44no—
518Q8—44no—
519Q8—44no—
520Q8—44no—
521Q8—44no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—432no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—816no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#39614yes—
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×#12111yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×C22⋊Q812yes—
587C2×C4C2×C22⋊Q811yescharacteristic
588C2×C4#52512yes—
589C2×C4#52511yescharacteristic
590C2×C4C2×#12611yescharacteristic
591C2×C4#53411yescharacteristic
592C2×C4#53511yescharacteristic
593C2×C4#53812yes—
594C2×C4#54312yes—
595C2×C4D4212yes—
596C2×C4#54612yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4#55112yes—
600C2×C4#55112yes—
601C8—42no—
602C8—44no—
603C8—42no—
604C22—28no—
605C22—24no—
606C22—28no—
607C22—24no—
608C22—24no—
609C22—28no—
610C22—28no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22#206618yescomplemented
615C22C2×#39611yescharacteristic
616C22#241011yescharacteristic
617C22#241111yescharacteristic
618C22#242412yes—
619C22#243712yes—
620C22C2×#46111yescharacteristic
621C22C2×#46411yescharacteristic
622C22#304211yescharacteristic
623C22#304411yescharacteristic
624C22C2×#46511yescharacteristic
625C22C2×#47011yescharacteristic
626C22#306312yes—
627C22C2×#47511yescharacteristic
628C22C2×#47911yescharacteristic
629C22#308212yes—
630C22#326612yes—
631C22#327012yes—
632C22D4×SD1612yes—
633C22#328512yes—
634C22#329812yes—
635C22#330112yes—
636C22#331212yes—
637C22#331612yes—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—48no—
642C4—82no—
643C4—82no—
644C4—162no—
645C4—161no—
646C4—161no—
647C4#240811yescharacteristic
648C4#243712yes—
649C4#248012yes—
650C4#253411yescharacteristic
651C4#262611yescharacteristic
652C4#284811yescharacteristic
653C2—28no—
654C2#1472611yescharacteristic
655C2C2×#206611yescharacteristic
656C2#2361511yescharacteristic
657C2#2435811yescharacteristic
658C2#2436111yescharacteristic
659C2#2448011yescharacteristic
660C2#2448311yescharacteristic
661C2#2511412yes—
662C2#2512612yes—
663C2#2595512yes—
664C2#2596512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2066C2semidirect
8. #11122C2semidirect
9. #11077C2semidirect
10. #11067C2semidirect
614. C22#2066semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#20662
C2×#18532
C2×#18502
C2×#17872
#139222
#139072
#111222
#110772
#110672
#107212