Finite Group Databasebeta

← #4930407

Subgroups — #4930407

Order 512.

2722 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic
3#15063C212yesmaximal, complemented
4#15015C212yesmaximal, complemented
5#15001C212yesmaximal, complemented
6#14934C212yesmaximal, complemented
7#14831C214yesmaximal, complemented
8#14773C214yesmaximal, complemented
9#14277C211yesmaximal, characteristic, complemented
10#14133C212yesmaximal, complemented
11#14030C212yesmaximal, complemented
12#13979C212yesmaximal
13#13941C211yesmaximal, characteristic
14#13925C211yesmaximal, characteristic, complemented
15#13916C212yesmaximal
16#13911C211yesmaximal, characteristic, complemented
17#13850C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2538—28no—
24C2×#396C2214yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387C2214yes—
30C2×#384C2211yescharacteristic
31C2×#382C2212yes—
32C2×#382C2211yescharacteristic
33C2×#381C2214yes—
34C2×#380C2212yes—
35#2275—28no—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×#376C2211yescharacteristic
44C22×#95C2212yes—
45#1446C2211yescharacteristic
46#1446C2211yescharacteristic
47#1446C2212yes—
48#1445C2211yescharacteristic
49#1445C2211yescharacteristic
50#1445C2212yes—
51#1444C2212yes—
52#1444C2212yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77#1434C2214yes—
78C22⋊#95C2212yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4D418yes—
103C2×C4×D4Q818yescomplemented
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#396—216no—
132#390—28no—
133#390—28no—
134#389—28no—
135#389—28no—
136#387—48no—
137#384—24no—
138#382—28no—
139#382—24no—
140#381—48no—
141#380—28no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149#376—24no—
150C2×#95—28no—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2412yes—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4C2×D414yes—
195C22×D4C2×Q814yes—
196C22×D4#4114yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D418yes—
211C23×C4C2×Q818yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C22⋊Q8—432no—
229C4×Q8—416no—
230C4×D4—232no—
231C4×D4—232no—
232C4×D4—232no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×Q814yes—
266C2×C4⋊C4#4114yes—
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—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×D418yes—
287C2×C22⋊C4C2×Q818yes—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C2×D414yes—
305C2×C42C2×Q814yes—
306C2×C42#4114yes—
307C2×C42#4114yes—
308C2×C42#4114yes—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—28no—
317#95—22no—
318#95—22no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—42no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—42no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24C22×D414yes—
351C24C22×Q814yes—
352C24C2×#4114yes—
353C24C2511yescharacteristic
354C2×Q8—28no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4C22⋊Q8116yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—88no—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q814yes—
461C22×C4C22×Q812yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)14yes—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—416no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C22⋊C4—216no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—416no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—28no—
495C42—28no—
496C42—48no—
497C42—48no—
498C42—84no—
499C42—84no—
500C42—84no—
501C42#123116yes—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C2×C22⋊Q818yes—
525C23C23×D411yescharacteristic
526C23C23×Q811yescharacteristic
527C23C22×#4111yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4111yescharacteristic
532C23C2×H(5,𝔽2)12yes—
533C23#58512yes—
534C23#58512yes—
535C23#58512yes—
536Q8—416no—
537D4—232no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—816no—
624C2×C4—816no—
625C2×C4—816no—
626C2×C4C2×C22⋊Q814yes—
627C2×C4C2×#12314yes—
628C2×C4C2×#12314yes—
629C2×C4C2×#12314yes—
630C2×C4#52514yes—
631C2×C4#52914yes—
632C2×C4#52914yes—
633C2×C4#53314yes—
634C2×C4#53314yes—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—48no—
639C22—48no—
640C22—44no—
641C22—48no—
642C22—44no—
643C22—44no—
644C22—48no—
645C22—44no—
646C22C22×C22⋊Q811yescharacteristic
647C22C22×#12311yescharacteristic
648C22C2×#52511yescharacteristic
649C22C2×#52911yescharacteristic
650C22C2×#52911yescharacteristic
651C22C2×#53311yescharacteristic
652C22C2×#53311yescharacteristic
653C22#348411yescharacteristic
654C22#348511yescharacteristic
655C22#349312yes—
656C22#349412yes—
657C22#349412yes—
658C22#350011yescharacteristic
659C22#350011yescharacteristic
660C22#350112yes—
661C22#350412yes—
662C22#350612yes—
663C22#351012yes—
664C22#351212yes—
665C22#351412yes—
666C22#351712yes—
667C22#352112yes—
668C22#352112yes—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—84no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—82no—
684C4#285518yes—
685C2—48no—
686C2#3554011yescharacteristic
687C2#3561212yes—
688C2#3561512yes—
689C2#3561811yescharacteristic
690C2#3570211yescharacteristic
691C2#3570411yescharacteristic
692C2#3595212yes—
693C2#3603111yescharacteristic
694C2#3603311yescharacteristic
695C2#3604412yes—
696C2C2×#285511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15063C2semidirect
4. #15015C2semidirect
5. #15001C2semidirect
6. #14934C2semidirect
7. #14831C2semidirect
8. #14773C2semidirect
9. #14277C2semidirect
10. #14133C2semidirect
11. #14030C2semidirect
14. #13925C2semidirect
16. #13911C2semidirect
17. #13850C2semidirect
103. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#22752
#150632
#150152
#150012
#149342
#148312
#147732
#142772
#141332
#140302
#139792
#139412
#139252
#139162
#139112
#138502