Finite Group Databasebeta

← #399115

Subgroups — #399115

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#106C211yesmaximal, characteristic, complemented
2C2×#2077C212yesmaximal, complemented
3C2×#1869C211yesmaximal, characteristic
4#13872C211yesmaximal, characteristic
5#13872C211yesmaximal, characteristic
6#11378C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106—28no—
17C22×#106—24no—
18C22×#106C2212yes—
19C22×#106C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#386—21no—
24C2×#386—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29#2077—28no—
30#1869—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1441—21no—
40#1441—21no—
41#1439—21no—
42#1439—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—24no—
56C23×C8D414yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4D411yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#106—232no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106D418yescomplemented
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103#386—44no—
104#386—44no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—24no—
110#380—24no—
111#380—42no—
112#380—42no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340—42no—
127#336—22no—
128#336—22no—
129C25C2×D411yescharacteristic
130C25C2×D411yescharacteristic
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4SD1614yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C22×C8—216no—
178C22×C8—216no—
179C22×C8—28no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—28no—
187C22×C8—28no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8C2×D414yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8#4114yes—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
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—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D414yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—82no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267#106—216no—
268#106—216no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—44no—
278#95—82no—
279#95—82no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24#12112yes—
295C24#12312yes—
296C24#12312yes—
297C24C2×SD1612yes—
298C24C2×SD1612yes—
299C24#13512yes—
300C24AGL(1,ℤ/8)12yes—
301C24#13712yes—
302C24C22×D411yescharacteristic
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—28no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4—88no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4#12112yes—
399C22×C4#12314yes—
400C22×C4#12312yes—
401C22×C4#12312yes—
402C22×C4#12414yes—
403C22×C4#12614yes—
404C22×C4#12614yes—
405C22×C4C2×SD1612yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+412yes—
416C2×C8—232no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8#12014yes—
436C2×C8#12314yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23#39714yes—
506C23#46514yes—
507C23#46814yes—
508C23#48114yes—
509C23#48214yes—
510C23#48314yes—
511C23#48514yes—
512C23C2×#12111yescharacteristic
513C23C2×#12311yescharacteristic
514C23C2×#12311yescharacteristic
515C23D4212yes—
516C23#54712yes—
517C23C22×SD1611yescharacteristic
518C23C2×#13511yescharacteristic
519C23C2×AGL(1,ℤ/8)11yescharacteristic
520C23C2×#13711yescharacteristic
521C23#57712yes—
522C23#57712yes—
523D4—416no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—816no—
558C2×C4—816no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#39714yes—
588C2×C4C2×#12011yescharacteristic
589C2×C4C2×#12311yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4#52512yes—
592C2×C4#52512yes—
593C2×C4C2×#12411yescharacteristic
594C2×C4C2×#12611yescharacteristic
595C2×C4C2×#12611yescharacteristic
596C2×C4#53511yescharacteristic
597C2×C4#53611yescharacteristic
598C2×C4#53912yes—
599C2×C4#53912yes—
600C2×C4#54612yes—
601C2×C4#54812yes—
602C2×C4#54812yes—
603C2×C4#55212yes—
604C2×C4#55312yes—
605C2×C4#55312yes—
606C8—24no—
607C8—24no—
608C8—81no—
609C8—81no—
610C22—24no—
611C22—24no—
612C22—28no—
613C22—28no—
614C22—28no—
615C22—28no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—82no—
627C22—82no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#207718yescomplemented
632C22C2×#39711yescharacteristic
633C22#240811yescharacteristic
634C22#244312yes—
635C22#244312yes—
636C22#248611yescharacteristic
637C22C2×#46511yescharacteristic
638C22C2×#46811yescharacteristic
639C22#305711yescharacteristic
640C22#305711yescharacteristic
641C22C2×#48311yescharacteristic
642C22C2×#48211yescharacteristic
643C22C2×#48511yescharacteristic
644C22C2×#48111yescharacteristic
645C22#309912yes—
646C22#309912yes—
647C22#326512yes—
648C22#326612yes—
649C22D4×SD1612yes—
650C22#328212yes—
651C22#333512yes—
652C22#333712yes—
653C22#334412yes—
654C22#334512yes—
655C4—24no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—84no—
660C4—84no—
661C4—162no—
662C4—162no—
663C4#243611yescharacteristic
664C4#248112yes—
665C4#248112yes—
666C4#250411yescharacteristic
667C4#262311yescharacteristic
668C4#284911yescharacteristic
669C2—28no—
670C2—82no—
671C2#1477611yescharacteristic
672C2C2×#207711yescharacteristic
673C2#2364011yescharacteristic
674C2#2426411yescharacteristic
675C2#2426911yescharacteristic
676C2#2612812yes—
677C2#2613012yes—
678C2#2613612yes—
679C2#2613812yes—
680C2#2765811yescharacteristic
681C2#2766311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#106C2semidirect
2. C2×#2077C2semidirect
6. #11378C2semidirect
7. #11257C2semidirect
8. #11095C2semidirect
9. #10730C2semidirect
10. #10730C2semidirect
11. #10721C2semidirect
95. C2×#106D4semidirect
631. C22#2077semidirect

Maximal subgroups

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

subgroupindex
C23×#1062
C2×#20772
C2×#18692
#138722
#113782
#112572
#110952
#107302
#107212