Finite Group Databasebeta

← #1007283

Subgroups — #1007283

Order 512.

3106 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2677C212yesmaximal
2C2×#2546C212yesmaximal, complemented
3C2×#2440C212yesmaximal
4#17114C212yesmaximal, complemented
5#16321C212yesmaximal
6#16293C214yesmaximal, complemented
7#15762C212yesmaximal, complemented
8#15279C214yesmaximal
9#15143C212yesmaximal, complemented
10#15137C212yesmaximal, complemented
11#15135C211yesmaximal, characteristic
12#14133C212yesmaximal, complemented
13#14130C211yesmaximal, characteristic
14#13974C212yesmaximal
15C4×#393C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2214yes—
18C22×C22⋊Q8C2214yes—
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2677—216no—
22#2546—216no—
23#2440—216no—
24C2×#399C2212yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#395C2214yes—
32C2×#393C2211yescharacteristic
33C2×#391C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#388C2214yes—
40C2×#387C2214yes—
41C2×#385C2212yes—
42C2×#385C2214yes—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50#1446C2211yescharacteristic
51#1445C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2212yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2211yescharacteristic
62#1442C2212yes—
63#1442C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2211yescharacteristic
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2211yescharacteristic
77C4×#95C2212yes—
78C23×Q8C2314yes—
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4⋊Q8—216no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—216no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—44no—
87C2×C4⋊Q8—48no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—48no—
91C2×C4×Q8—28no—
92C2×C4×Q8—42no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123#399—28no—
124#398—216no—
125#398—216no—
126#398—216no—
127#398—216no—
128#395—216no—
129#395—216no—
130#395—48no—
131#393D418yes—
132#391—216no—
133#389—28no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389D4116yes—
138#388—216no—
139#387—216no—
140#385—28no—
141#385—216no—
142#384—28no—
143#384—42no—
144#380—28no—
145#380Q818yescomplemented
146C4×C4⋊C4—28no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—42no—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2312yes—
176C25C2411yescharacteristic
177C22×Q8—24no—
178C22×Q8—28no—
179C22×Q8—28no—
180C22×Q8—28no—
181C22×Q8—24no—
182C22×Q8—28no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8C2×D414yes—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—42no—
198C23×C4C2×Q814yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C4⋊Q8—432no—
214C4⋊Q8—464no—
215C22⋊Q8—464no—
216C4×Q8—416no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D4116yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×Q818yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4C2×D418yes—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×Q818yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42C2×D418yes—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—41no—
316#95—41no—
317#95—41no—
318#95—44no—
319#95—44no—
320#95C2×D4116yes—
321#95C2×D418yes—
322#95C2×D414yes—
323#95C2×Q818yes—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—21no—
332C24C22×D412yes—
333C24C22×Q812yes—
334C242−1+412yes—
335C24C2511yescharacteristic
336C2×Q8—216no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—48no—
340C2×Q8—44no—
341C2×Q8—416no—
342C2×Q8—416no—
343C2×Q8—416no—
344C2×Q8—416no—
345C2×Q8—88no—
346C2×Q8—88no—
347C2×Q8—88no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
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—22no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—88no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4C22×D412yes—
427C22×C4C22×D414yes—
428C22×C4C22×D418yes—
429C22×C4C22×D414yes—
430C22×C4C22×D418yes—
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q818yes—
437C22×C4C22×Q812yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C42−1+414yes—
441C22×C42−1+412yes—
442C22×C42−1+412yes—
443C22×C42−1+414yes—
444C4⋊C4—216no—
445C4⋊C4—216no—
446C4⋊C4—232no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—416no—
467C42—216no—
468C42—416no—
469C42—48no—
470C42—44no—
471C42—44no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23D4×Q814yes—
495C23C23×D412yes—
496C23C23×D411yescharacteristic
497C23C23×Q811yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+412yes—
502C23C2×2−1+411yescharacteristic
503C23C2×2−1+412yes—
504C23C2×2−1+411yescharacteristic
505C23#58511yescharacteristic
506Q8—416no—
507Q8—832no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—416no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—816no—
590C2×C4#53618yes—
591C2×C4#54414yes—
592C2×C4D4218yes—
593C2×C4D4214yes—
594C2×C4D4×Q818yes—
595C2×C4#56314yes—
596C22—24no—
597C22—28no—
598C22—42no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22C2×#53612yes—
608C22C2×#54411yescharacteristic
609C22C2×D4211yescharacteristic
610C22C2×D4212yes—
611C22C2×D4×Q812yes—
612C22C2×D4×Q811yescharacteristic
613C22#348612yes—
614C22#348812yes—
615C22#349611yescharacteristic
616C22#349912yes—
617C22#349912yes—
618C22#349911yescharacteristic
619C22#350211yescharacteristic
620C22#350312yes—
621C22C2×#56311yescharacteristic
622C22#353711yescharacteristic
623C22#353811yescharacteristic
624C22#355011yescharacteristic
625C22#355212yes—
626C22#355212yes—
627C22#355212yes—
628C22#356311yescharacteristic
629C22#356312yes—
630C4—28no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—48no—
635C4—42no—
636C4—42no—
637C4—48no—
638C4—44no—
639C4—48no—
640C4—42no—
641C4—81no—
642C4—84no—
643C4#267718yescomplemented
644C2—44no—
645C2#3936012yes—
646C2#3936811yescharacteristic
647C2#4281412yes—
648C2#4293311yescharacteristic
649C2#4527212yes—
650C2#4532111yescharacteristic
651C2#4614711yescharacteristic
652C2C2×#267711yescharacteristic
653C2#5298711yescharacteristic
654C2#5899711yescharacteristic
655C2#5912712yes—
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
2. C2×#2546C2semidirect
4. #17114C2semidirect
6. #16293C2semidirect
7. #15762C2semidirect
9. #15143C2semidirect
10. #15137C2semidirect
12. #14133C2semidirect
145. #380Q8semidirect
643. C4#2677semidirect

Maximal subgroups

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

subgroupindex
C2×#26772
C2×#25462
C2×#24402
#171142
#163212
#162932
#157622
#152792
#151432
#151372
#151352
#141332
#141302
#139742
C4×#3932