Finite Group Databasebeta

← #712167

Subgroups — #712167

Order 512.

2326 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
1C2×#2346C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#16964C212yesmaximal, complemented
4#16960C212yesmaximal, complemented
5#15341C214yesmaximal
6#15339C214yesmaximal
7#15139C212yesmaximal, complemented
8#15137C212yesmaximal, complemented
9#15122C212yesmaximal, complemented
10#15121C212yesmaximal, complemented
11#14343C211yesmaximal, characteristic
12#14132C212yesmaximal, complemented
13#14130C211yesmaximal, characteristic
14#14029C212yesmaximal, complemented
15#13939C211yesmaximal, characteristic
16#13880C211yesmaximal, characteristic
17#13853C211yesmaximal, characteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#400C2214yes—
22C2×#398C2214yes—
23#2346—28no—
24#2316—28no—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#387C2214yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#384C2212yes—
38C2×#382C2212yes—
39C2×#380C2214yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×#376C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1445C2211yescharacteristic
50#1445C2211yescharacteristic
51#1445C2211yescharacteristic
52#1445C2211yescharacteristic
53#1445C2214yes—
54#1445C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1444C2212yes—
59#1443C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2212yes—
69#1440C2212yes—
70#1440C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80#1434C2212yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125#400—48no—
126#398—48no—
127#389—28no—
128#389—28no—
129#389—24no—
130#389—24no—
131#389—28no—
132#389—28no—
133#389—28no—
134#388—28no—
135#388—28no—
136#387—48no—
137#385—44no—
138#385—48no—
139#384—28no—
140#382—28no—
141#380—24no—
142#380—24no—
143#380—216no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148#376—28no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2411yescharacteristic
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4C2×D414yes—
191C23×C4#4114yes—
192C23×C4#4114yes—
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209#117—232no—
210#117—416no—
211C2×C4⋊C4—232no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
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—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4#4118yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C2×D418yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C2×D418yes—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—44no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327C24—22no—
328C24—21no—
329C24—22no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24C22×D412yes—
336C24C2×#4112yes—
337C24C2×#4112yes—
338C24C2511yescharacteristic
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
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—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—216no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4#123116yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4118yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4114yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C42−1+414yes—
441C22×C42−1+412yes—
442C22×C42−1+414yes—
443C4⋊C4—216no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—216no—
447C4⋊C4—416no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—84no—
466C42—216no—
467C42—48no—
468C42—48no—
469C42—48no—
470C42—44no—
471C42—44no—
472C42—48no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23C2×#12314yes—
496C23C23×D411yescharacteristic
497C23C22×#4111yescharacteristic
498C23C22×#4111yescharacteristic
499C23C22×#4112yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×2−1+412yes—
503C23C2×2−1+411yescharacteristic
504C23C2×2−1+412yes—
505C23#58511yescharacteristic
506C23#58511yescharacteristic
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—232no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—816no—
597C2×C4C2×#12314yes—
598C2×C4C2×#12314yes—
599C2×C4#52514yes—
600C2×C4#53614yes—
601C2×C4#55218yes—
602C2×C4#55218yes—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22C22×#12311yescharacteristic
615C22C2×#52511yescharacteristic
616C22C2×#53611yescharacteristic
617C22#346011yescharacteristic
618C22#346111yescharacteristic
619C22C2×#55212yes—
620C22C2×#55212yes—
621C22#348811yescharacteristic
622C22#348911yescharacteristic
623C22#349212yes—
624C22#349811yescharacteristic
625C22#349911yescharacteristic
626C22#351111yescharacteristic
627C22#351112yes—
628C22#351211yescharacteristic
629C22#351412yes—
630C22#351612yes—
631C22#351912yes—
632C22#352312yes—
633C22#354412yes—
634C22#354412yes—
635C22#354712yes—
636C22#356312yes—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—42no—
641C4—48no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4#248418yes—
653C2—44no—
654C2C2×#248411yescharacteristic
655C2#3525511yescharacteristic
656C2#3553011yescharacteristic
657C2#3599711yescharacteristic
658C2#3600411yescharacteristic
659C2#3826312yes—
660C2#3907912yes—
661C2#4296411yescharacteristic
662C2#4296811yescharacteristic
663C2#4491712yes—
664C2#4491912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16964C2semidirect
4. #16960C2semidirect
7. #15139C2semidirect
8. #15137C2semidirect
9. #15122C2semidirect
10. #15121C2semidirect
12. #14132C2semidirect
14. #14029C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23462
C2×#23162
#169642
#169602
#153412
#153392
#151392
#151372
#151222
#151212
#143432
#141322
#141302
#140292
#139392
#138802
#138532