Finite Group Databasebeta

← #814043

Subgroups — #814043

Order 512.

2866 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2429C212yesmaximal
3C2×#2360C211yesmaximal, characteristic
4#16820C212yesmaximal, complemented
5#15676C214yesmaximal, complemented
6#15590C214yesmaximal
7#15194C211yesmaximal, characteristic, complemented
8#15188C211yesmaximal, characteristic, complemented
9#15178C211yesmaximal, characteristic
10#15165C212yesmaximal
11#15012C212yesmaximal, complemented
12#14229C212yesmaximal
13#14142C212yesmaximal, complemented
14#14139C212yesmaximal
15#14024C212yesmaximal, complemented
16C4×#397C211yesmaximal, characteristic
17Q8×#95C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C4×Q8C2212yes—
23C22×C4×Q8C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25#2546—28no—
26#2429—216no—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#397C2211yescharacteristic
30C2×#396C2214yes—
31#2360—28no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#388C2214yes—
44C2×#387C2212yes—
45C2×#385C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1443C2212yes—
57#1443C2214yes—
58#1442C2211yescharacteristic
59#1442C2212yes—
60#1442C2214yes—
61#1442C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2212yes—
69#1439C2211yescharacteristic
70#1439C2212yes—
71#1439C2212yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80#1434C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—44no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—44no—
100C2×C22⋊Q8D418yes—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×Q8—44no—
104C2×C4×Q8—44no—
105C2×C4×Q8D418yes—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—44no—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2312yes—
138#398—216no—
139#398—216no—
140#397Q818yescomplemented
141#396—216no—
142#391—24no—
143#391—28no—
144#391—28no—
145#391—28no—
146#391—44no—
147#389—28no—
148#389—216no—
149#389—28no—
150#389—24no—
151#389—28no—
152#389—28no—
153#388—216no—
154#387—28no—
155#385—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—28no—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C25C2411yescharacteristic
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8C2×D414yes—
203C22×Q8C2×D414yes—
204C22×Q8#4114yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C4⋊Q8—416no—
232C4⋊Q8—432no—
233C22⋊Q8—232no—
234C22⋊Q8—432no—
235C4×Q8—416no—
236C4×Q8—432no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4#4118yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4C2×D418yes—
297C2×C22⋊C4C2×Q818yes—
298C2×C22⋊C4C2×Q814yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C2×D414yes—
315C2×C42C2×D418yes—
316#95—22no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—22no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95C2×Q818yes—
346#95C2×Q814yes—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24—22no—
355C24C22×D412yes—
356C24C22×Q812yes—
357C242−1+412yes—
358C24C2511yescharacteristic
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—216no—
362C2×Q8—216no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8#121116yes—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—84no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q812yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C42−1+414yes—
472C22×C42−1+412yes—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—216no—
478C4⋊C4—28no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—44no—
489C4⋊C4—84no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—28no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—416no—
496C42—28no—
497C42—48no—
498C42—416no—
499C42—48no—
500C42—44no—
501C42—48no—
502C42—84no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23D4×Q814yes—
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C23×Q811yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4112yes—
531C23C2×H(5,𝔽2)12yes—
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+411yescharacteristic
534C23C2×2−1+412yes—
535C23#58511yescharacteristic
536C23#58512yes—
537Q8—232no—
538Q8—816no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—816no—
626C2×C4C2×#12118yes—
627C2×C4C2×#12114yes—
628C2×C4#53214yes—
629C2×C4#53414yes—
630C2×C4#54618yes—
631C2×C4D4×Q814yes—
632C2×C4#55318yes—
633C22—24no—
634C22—28no—
635C22—42no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C22×#12111yescharacteristic
645C22C2×#53211yescharacteristic
646C22C2×#53411yescharacteristic
647C22C2×#54612yes—
648C22C2×D4×Q811yescharacteristic
649C22C2×D4×Q811yescharacteristic
650C22C2×#55312yes—
651C22Q8×#4111yescharacteristic
652C22Q8×#4112yes—
653C22#348611yescharacteristic
654C22#348911yescharacteristic
655C22#348912yes—
656C22#349711yescharacteristic
657C22#350011yescharacteristic
658C22#350312yes—
659C22#351512yes—
660C22#351512yes—
661C22#351612yes—
662C22#352312yes—
663C22#352311yescharacteristic
664C22#355212yes—
665C22#356112yes—
666C22#356212yes—
667C4—28no—
668C4—24no—
669C4—42no—
670C4—48no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—44no—
678C4—84no—
679C4—81no—
680C4—82no—
681C4—82no—
682C4#254318yes—
683C2—44no—
684C2C2×#254311yescharacteristic
685C2#3573711yescharacteristic
686C2#3594911yescharacteristic
687C2Q8×#12111yescharacteristic
688C2#3656211yescharacteristic
689C2#3959311yescharacteristic
690C2#4294311yescharacteristic
691C2#4517512yes—
692C2#4518412yes—
693C2#5909612yes—
694C2#5969412yes—
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×#2546C2semidirect
4. #16820C2semidirect
5. #15676C2semidirect
7. #15194C2semidirect
8. #15188C2semidirect
11. #15012C2semidirect
13. #14142C2semidirect
15. #14024C2semidirect
17. Q8×#95C2semidirect
140. #397Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#24292
C2×#23602
#168202
#156762
#155902
#151942
#151882
#151782
#151652
#150122
#142292
#141422
#141392
#140242
C4×#3972
Q8×#952