Finite Group Databasebeta

← #800408

Subgroups — #800408

Order 512.

2474 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic, complemented
2C2×#2345C211yesmaximal, characteristic
3#16991C212yesmaximal, complemented
4#16079C214yesmaximal, complemented
5#15995C214yesmaximal
6#15213C212yesmaximal, complemented
7#15200C211yesmaximal, characteristic, complemented
8#15188C211yesmaximal, characteristic, complemented
9#15178C211yesmaximal, characteristic
10#15045C212yesmaximal
11#14240C212yesmaximal
12#14139C212yesmaximal
13#14021C212yesmaximal, complemented
14#14017C212yesmaximal, complemented
15#13969C212yesmaximal
16#13936C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C2×#399C2211yescharacteristic
23#2365—28no—
24#2345—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2214yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45#1445C2211yescharacteristic
46#1445C2212yes—
47#1445C2212yes—
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1444C2212yes—
51#1444C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2212yes—
65#1439C2211yescharacteristic
66#1439C2212yes—
67#1439C2212yes—
68#1439C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C4×Q8—28no—
94C2×C4×Q8—44no—
95C2×C4×Q8—44no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128#399Q818yescomplemented
129#391—28no—
130#391—28no—
131#391—28no—
132#391—44no—
133#389—24no—
134#389—24no—
135#389—28no—
136#387—28no—
137#387—28no—
138#387—216no—
139#380—24no—
140#380—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—44no—
145C4×C4⋊C4D418yes—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—44no—
149C2×#95C2311yescharacteristic
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C25C2411yescharacteristic
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8#4118yes—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4#4114yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—232no—
217C22⋊Q8—432no—
218C4×Q8—432no—
219C2×C4⋊C4—232no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×Q814yes—
259C2×C4⋊C4C2×Q818yes—
260C2×C4⋊C4#4114yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4C2×Q814yes—
278C2×C22⋊C4C2×Q818yes—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295C2×C42C2×D418yes—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—22no—
310#95—24no—
311#95—44no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95C2×D418yes—
328#95C2×Q814yes—
329C24—21no—
330C24—22no—
331C24—21no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24C22×Q812yes—
338C24C2×#4112yes—
339C242−1+412yes—
340C24C2511yescharacteristic
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—216no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4C22×D414yes—
434C22×C4C22×D412yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q812yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q812yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4118yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C42−1+412yes—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C22⋊C4—28no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C42—28no—
473C42—48no—
474C42—416no—
475C42—44no—
476C42—48no—
477C42—48no—
478C42—84no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—24no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23#55714yes—
502C23C23×D411yescharacteristic
503C23C23×Q811yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4112yes—
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×2−1+411yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23#58512yes—
511C23#58511yescharacteristic
512C23#58512yes—
513Q8—432no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—416no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—816no—
602C2×C4#52514yes—
603C2×C4#52514yes—
604C2×C4#53514yes—
605C2×C4D4×Q814yes—
606C2×C4#55118yes—
607C2×C4#55318yes—
608C22—24no—
609C22—28no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22C2×#52511yescharacteristic
620C22C2×#52511yescharacteristic
621C22C2×#53511yescharacteristic
622C22C2×D4×Q811yescharacteristic
623C22C2×#55112yes—
624C22C2×#55312yes—
625C22C2×#55711yescharacteristic
626C22Q8×#4111yescharacteristic
627C22Q8×#4112yes—
628C22#348511yescharacteristic
629C22#348911yescharacteristic
630C22#350011yescharacteristic
631C22#350112yes—
632C22#350711yescharacteristic
633C22#351012yes—
634C22#351612yes—
635C22#351612yes—
636C22#352112yes—
637C22#352311yescharacteristic
638C22#352412yes—
639C22#353912yes—
640C22#354112yes—
641C22#354712yes—
642C4—24no—
643C4—44no—
644C4—42no—
645C4—48no—
646C4—42no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—44no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4#259818yes—
658C2—44no—
659C2C2×#259811yescharacteristic
660C2#3598511yescharacteristic
661C2#3603311yescharacteristic
662C2#3654511yescharacteristic
663C2#3654711yescharacteristic
664C2#3961811yescharacteristic
665C2#4267911yescharacteristic
666C2#4523512yes—
667C2#4539112yes—
668C2#5939312yes—
669C2#5986712yes—
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
1. C2×#2365C2semidirect
3. #16991C2semidirect
4. #16079C2semidirect
6. #15213C2semidirect
7. #15200C2semidirect
8. #15188C2semidirect
13. #14021C2semidirect
14. #14017C2semidirect
16. #13936C2semidirect
128. #399Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#23452
#169912
#160792
#159952
#152132
#152002
#151882
#151782
#150452
#142402
#141392
#140212
#140172
#139692
#139362
C4×#3992