Finite Group Databasebeta

← #392783

Subgroups — #392783

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic, complemented
2C2×#1924C212yesmaximal, complemented
3C2×#1870C211yesmaximal, characteristic
4C2×#1869C211yesmaximal, characteristic
5C2×#1851C211yesmaximal, characteristic, complemented
6#13839C211yesmaximal, characteristic
7#10921C212yesmaximal, complemented
8#10876C212yesmaximal, complemented
9#10798C212yesmaximal, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#405—28no—
23C2×#405—24no—
24C22×#97C2212yes—
25C22×C4×C8C2212yes—
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1924—28no—
29#1870—24no—
30#1869—24no—
31#1851—24no—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1441C411yescharacteristic
41#1441C411yescharacteristic
42#1440C412yes—
43#1439C411yescharacteristic
44#1439C411yescharacteristic
45#1438C412yes—
46C4×#95C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16D414yes—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8D414yes—
60C23×C8C2311yescharacteristic
61C23×C8C2312yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2312yes—
65C23×C8C2311yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—24no—
75C2×#117—24no—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C412yes—
83C22×C22⋊C4C2×C412yes—
84C22×C22⋊C4C2×C412yes—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C22×C42C2312yes—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104#405—232no—
105C2×#97—24no—
106C2×#97—24no—
107C2×#97D418yescomplemented
108C2×C4×C8—24no—
109C2×C4×C8—24no—
110C2×C4×C8—24no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—24no—
114C4×C22⋊C4—24no—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123#336—24no—
124#336—22no—
125#336—22no—
126#336—24no—
127#336—24no—
128#336—24no—
129#336—24no—
130#336—24no—
131C25C22×C411yescharacteristic
132C25C22×C411yescharacteristic
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—41no—
139C22×D4—41no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—216no—
170C2×M16—28no—
171C2×M16—28no—
172C2×M16—28no—
173C2×M16—216no—
174C22×C8—216no—
175C22×C8—216no—
176C22×C8—28no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—28no—
188C22×C8—28no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8C2×D414yes—
207C22×C8C2×D412yes—
208C22×C8C2×D414yes—
209C22×C8C2×D414yes—
210C22×C8C2×D412yes—
211C22×C8#4114yes—
212C22×C8#4114yes—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4—44no—
217#117—216no—
218#117—28no—
219#117—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C2×D414yes—
262C2×C42#4114yes—
263C4⋊C8—48no—
264C22⋊C8—44no—
265C22⋊C8—44no—
266C22⋊C8—44no—
267C22⋊C8—48no—
268C22⋊C8—44no—
269#97—216no—
270C4×C8—216no—
271#95—22no—
272#95—24no—
273#95—24no—
274#95—22no—
275#95—24no—
276#95—24no—
277#95—42no—
278#95—42no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—21no—
294C24—21no—
295C24—22no—
296C24—21no—
297C24#11712yes—
298C24C4×D412yes—
299C24C4×D412yes—
300C24#13112yes—
301C24C23×C411yescharacteristic
302C2×D4—28no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4#11712yes—
385C22×C4C4×D414yes—
386C22×C4C4×D414yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4#12114yes—
390C22×C4#12314yes—
391C22×C4#12414yes—
392C22×C4#12614yes—
393C22×C4#13112yes—
394C22×C4#13114yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4112yes—
405M16—232no—
406C2×C8—232no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—42no—
424C2×C8—41no—
425C2×C8—42no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8#12014yes—
439C2×C8#12118yes—
440C2×C8#12314yes—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—44no—
444C22⋊C4—216no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—42no—
455C42—41no—
456C42—41no—
457C42—42no—
458C42—41no—
459C42—41no—
460C42—42no—
461C42—42no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23#38814yes—
508C23C2×#11711yescharacteristic
509C23C2×C4×D411yescharacteristic
510C23C2×C4×D411yescharacteristic
511C23C2×C4×D412yes—
512C23C4×#4112yes—
513C23C2×#13111yescharacteristic
514C23C2×#13111yescharacteristic
515C23C2×#13112yes—
516C23#56811yescharacteristic
517C23#56811yescharacteristic
518C23#56812yes—
519D4—416no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4#38814yes—
582C2×C4C2×#12011yescharacteristic
583C2×C4C2×#12112yes—
584C2×C4C2×#12111yescharacteristic
585C2×C4C2×#12311yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C2×C4#52512yes—
588C2×C4#52511yescharacteristic
589C2×C4#52512yes—
590C2×C4#52512yes—
591C2×C4#52512yes—
592C2×C4#52511yescharacteristic
593C2×C4C2×#12411yescharacteristic
594C2×C4C2×#12611yescharacteristic
595C2×C4#52912yes—
596C2×C4#52912yes—
597C2×C4#52912yes—
598C2×C4#52912yes—
599C2×C4#53212yes—
600C8—28no—
601C8—24no—
602C8—24no—
603C8—81no—
604C8—82no—
605C8—81no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—82no—
626C22—81no—
627C22—82no—
628C22C2×#38811yescharacteristic
629C22#229012yes—
630C22C4×#12011yescharacteristic
631C22C4×#12112yes—
632C22C4×#12311yescharacteristic
633C22#292411yescharacteristic
634C22#292411yescharacteristic
635C22#292512yes—
636C22#292911yescharacteristic
637C22#293011yescharacteristic
638C22#293111yescharacteristic
639C22#293211yescharacteristic
640C22#293311yescharacteristic
641C22#293311yescharacteristic
642C22#293312yes—
643C22#293312yes—
644C22#293312yes—
645C22#293512yes—
646C22#293512yes—
647C22#296612yes—
648C22#296612yes—
649C22#296712yes—
650C22#296712yes—
651C4—24no—
652C4—42no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—81no—
657C4—81no—
658C4—82no—
659C4#284711yescharacteristic
660C4#284911yescharacteristic
661C4#285012yes—
662C4#285212yes—
663C4#285311yescharacteristic
664C4#285411yescharacteristic
665C2—44no—
666C2—82no—
667C2#1469611yescharacteristic
668C2#2105011yescharacteristic
669C2#2105111yescharacteristic
670C2#2111112yes—
671C2#2111212yes—
672C2#2144311yescharacteristic
673C2#2144511yescharacteristic
674C2#2149612yes—
675C2#2149612yes—
676C2#2156611yescharacteristic
677C2#2156911yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#405C2semidirect
2. C2×#1924C2semidirect
5. C2×#1851C2semidirect
7. #10921C2semidirect
8. #10876C2semidirect
9. #10798C2semidirect
10. #10714C2semidirect
11. #10713C2semidirect
107. C2×#97D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#19242
C2×#18702
C2×#18692
C2×#18512
#138392
#109212
#108762
#107982
#107142
#107132