Finite Group Databasebeta

← #358117

Subgroups — #358117

Order 512.

1896 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#433C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3C2×#1749C211yesmaximal, characteristic, complemented
4C2×#1750C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
6#11202C212yesmaximal, complemented
7#11147C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10964C212yesmaximal
10#10694C211yesmaximal, characteristic, complemented
11#10694C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#433—24no—
16C2×#433—24no—
17C2×#433—24no—
18C22×C4⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2290—24no—
28#2290—24no—
29C2×#388C412yes—
30C2×#387C412yes—
31C2×#386C412yes—
32C2×#385C412yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#1750—24no—
38#1749—24no—
39C4×C22⋊C8—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C412yes—
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117D414yes—
64C2×#117D414yes—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42C2×C412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#433—232no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97D414yescomplemented
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8D414yescomplemented
113#388—28no—
114#387—28no—
115#386—28no—
116#385—28no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123#376—22no—
124#376—22no—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130#336—24no—
131#336—24no—
132#336—24no—
133#336—24no—
134#336—24no—
135#336—24no—
136#336—24no—
137C25C22×C411yescharacteristic
138C25C22×C411yescharacteristic
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—24no—
142C22×D4—22no—
143C22×D4—42no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C22×C412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4#4114yes—
179C23×C4C2411yescharacteristic
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C2×D414yes—
205C22×C8C2×D414yes—
206C22×C8#4114yes—
207C22×C8#4114yes—
208C4×D4—216no—
209C4×D4—216no—
210#117—28no—
211#117—28no—
212#117—28no—
213#117—28no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C22⋊C418yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4C22⋊C418yes—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42#4114yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271C4⋊C8—216no—
272C22⋊C8—216no—
273C22⋊C8—216no—
274C22⋊C8—216no—
275C22⋊C8—216no—
276#97—28no—
277#97—44no—
278C4×C8—28no—
279C4×C8—44no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24C2×C22⋊C412yes—
305C24#11712yes—
306C24#13114yes—
307C24C23×C411yescharacteristic
308C2×D4—28no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—44no—
312C2×D4—44no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—216no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
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—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—48no—
395C22×C4C2×C22⋊C414yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C414yes—
398C22×C4#11712yes—
399C22×C4#11712yes—
400C22×C4#11712yes—
401C22×C4C4×D414yes—
402C22×C4C4×D414yes—
403C22×C4#13114yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C4⋊C4—28no—
443C4⋊C4—216no—
444C4⋊C4—48no—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—216no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—24no—
455C42—24no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—42no—
461C42—42no—
462C42—48no—
463C42#12014yes—
464C42#12114yes—
465C42#12114yes—
466C42#12314yes—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23#43314yescomplemented
513C23C22×C22⋊C411yescharacteristic
514C23C2×#11711yescharacteristic
515C23C2×#11711yescharacteristic
516C23C2×C4×D411yescharacteristic
517C23C2×C4×D411yescharacteristic
518C23C4×#4111yescharacteristic
519C23C4×#4111yescharacteristic
520C23C2×#13112yes—
521C23C2×#13112yes—
522C23#56812yes—
523C23#56812yes—
524D4—416no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—432no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4#38618yes—
586C2×C4C2×#12011yescharacteristic
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12311yescharacteristic
590C2×C4#52512yes—
591C2×C4#52512yes—
592C2×C4#52512yes—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4#52912yes—
596C2×C4#52912yes—
597C2×C4#52912yes—
598C2×C4#52912yes—
599C2×C4#53212yes—
600C2×C4#53212yes—
601C2×C4#53212yes—
602C8—44no—
603C8—44no—
604C8—44no—
605C8—44no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—82no—
623C22—82no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22#228011yescharacteristic
629C22C2×#38611yescharacteristic
630C22#229011yescharacteristic
631C22#229011yescharacteristic
632C22#229311yescharacteristic
633C22#229311yescharacteristic
634C22#229711yescharacteristic
635C22#288012yes—
636C22#288012yes—
637C22C2×#43311yescharacteristic
638C22#292111yescharacteristic
639C22#292211yescharacteristic
640C22#292311yescharacteristic
641C22#292412yes—
642C22#292512yes—
643C22#292912yes—
644C22#293012yes—
645C22#293312yes—
646C22#293312yes—
647C22#296612yes—
648C22#296712yes—
649C22#296812yes—
650C22#296912yes—
651C4—28no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—82no—
659C4#238812yes—
660C4#239212yes—
661C4#239212yes—
662C4#239512yes—
663C2—44no—
664C2—82no—
665C2#1383911yescharacteristic
666C2#2071611yescharacteristic
667C2#2072011yescharacteristic
668C2#2089912yes—
669C2#2110311yescharacteristic
670C2#2110411yescharacteristic
671C2#2110511yescharacteristic
672C2#2110611yescharacteristic
673C2#2127212yes—
674C2#2127312yes—
675C2#2138112yes—
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×#433C2semidirect
3. C2×#1749C2semidirect
4. C2×#1750C2semidirect
5. C2×C4×C22⋊C8C2semidirect
6. #11202C2semidirect
7. #11147C2semidirect
8. #11042C2semidirect
10. #10694C2semidirect
11. #10694C2semidirect
106. C2×#97D4semidirect
112. C2×C4×C8D4semidirect
512. C23#433semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#22902
C2×#17492
C2×#17502
C2×C4×C22⋊C82
#112022
#111472
#110422
#109642
#106942