Finite Group Databasebeta

← #712702

Subgroups — #712702

Order 512.

2702 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
1C2×#2439C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic
3#16891C212yesmaximal, complemented
4#16889C212yesmaximal, complemented
5#15930C214yesmaximal
6#15922C214yesmaximal
7#15253C212yesmaximal, complemented
8#15191C212yesmaximal, complemented
9#15186C212yesmaximal, complemented
10#15053C212yesmaximal, complemented
11#14343C211yesmaximal, characteristic
12#14142C212yesmaximal, complemented
13#14003C212yesmaximal, complemented
14#13922C211yesmaximal, characteristic
15#13922C211yesmaximal, characteristic
16#13853C211yesmaximal, characteristic
17#13801C211yesmaximal, characteristic
18C24×Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2439—28no—
23C2×#396C2212yes—
24C2×#393C2212yes—
25#2310—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#387C2212yes—
38C2×#382C2211yescharacteristic
39C2×#381C2214yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2214yes—
46#1446C2214yes—
47#1446C2212yes—
48#1446C2212yes—
49#1446C2212yes—
50#1446C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2211yescharacteristic
54#1445C2211yescharacteristic
55#1445C2211yescharacteristic
56#1445C2211yescharacteristic
57#1445C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1440C2214yes—
68#1440C2212yes—
69#1440C2212yes—
70#1437C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76#1434C2214yes—
77#1434C2214yes—
78#1434C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2214yes—
81C22⋊#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8—28no—
86C23×Q8—24no—
87C23×Q8—24no—
88C23×Q8—24no—
89C23×Q8—44no—
90C23×Q8C2312yes—
91C23×Q8C2312yes—
92C23×Q8C2312yes—
93C24×C4C2312yes—
94C24×C4C2311yescharacteristic
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—44no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—44no—
101C2×#117D418yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2312yes—
125#396—28no—
126#393—28no—
127#391—28no—
128#391—28no—
129#391—28no—
130#391—28no—
131#391—28no—
132#391—28no—
133#391—28no—
134#391—44no—
135#391—44no—
136#389—24no—
137#389—24no—
138#387—28no—
139#382—24no—
140#381—216no—
141#380—24no—
142#380—24no—
143C4×C4⋊C4—44no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—24no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C25C2411yescharacteristic
184C22×Q8—232no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—28no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—42no—
200C22×Q8—44no—
201C22×Q8—432no—
202C22×Q8—432no—
203C22×Q8—416no—
204C22×Q8—416no—
205C22×Q8#4118yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C4⋊Q8—416no—
233#117—416no—
234#117—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
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—24no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D418yes—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—28no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C2×D418yes—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24C22×D412yes—
339C24C2×#4112yes—
340C24C2×#4112yes—
341C24C2511yescharacteristic
342C2×Q8—216no—
343C2×Q8—216no—
344C2×Q8—216no—
345C2×Q8—216no—
346C2×Q8—416no—
347C2×Q8—416no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—4128no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—84no—
358C2×Q8—88no—
359C2×Q8—88no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4#124116yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C42−1+414yes—
460C22×C42−1+412yes—
461C22×C42−1+412yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—84no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C42—48no—
480C42—44no—
481C42—48no—
482C42—44no—
483C42—48no—
484C42—88no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C2×#12414yes—
508C23C23×D411yescharacteristic
509C23C22×#4112yes—
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C2×H(5,𝔽2)12yes—
513C23C2×2−1+412yes—
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+411yescharacteristic
516C23#58512yes—
517C23#58511yescharacteristic
518C23#58511yescharacteristic
519Q8—432no—
520Q8—432no—
521Q8—816no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—232no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—432no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4#52514yes—
611C2×C4C2×#12414yes—
612C2×C4C2×#12414yes—
613C2×C4#53414yes—
614C2×C4#55118yes—
615C2×C4#55118yes—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22C2×#52511yescharacteristic
628C22C22×#12411yescharacteristic
629C22C2×#53411yescharacteristic
630C22#346112yes—
631C22C2×#55112yes—
632C22C2×#55112yes—
633C22#348911yescharacteristic
634C22#348911yescharacteristic
635C22#349512yes—
636C22#349911yescharacteristic
637C22#350011yescharacteristic
638C22#350711yescharacteristic
639C22#350711yescharacteristic
640C22#350911yescharacteristic
641C22#351011yescharacteristic
642C22#351812yes—
643C22#351812yes—
644C22#352312yes—
645C22#352412yes—
646C22#355212yes—
647C22#356112yes—
648C22#356112yes—
649C22#356212yes—
650C4—24no—
651C4—48no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—82no—
658C4—84no—
659C4—81no—
660C4—82no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4#256118yes—
666C2—44no—
667C2C2×#256111yescharacteristic
668C2#3531211yescharacteristic
669C2#3562811yescharacteristic
670C2#3600711yescharacteristic
671C2#3607711yescharacteristic
672C2#3825112yes—
673C2#3907112yes—
674C2#4291211yescharacteristic
675C2#4294311yescharacteristic
676C2#4511612yes—
677C2#4529012yes—
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. #16891C2semidirect
4. #16889C2semidirect
7. #15253C2semidirect
8. #15191C2semidirect
9. #15186C2semidirect
10. #15053C2semidirect
12. #14142C2semidirect
13. #14003C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#23102
#168912
#168892
#159302
#159222
#152532
#151912
#151862
#150532
#143432
#141422
#140032
#139222
#138532
#138012