Finite Group Databasebeta

← #876779

Subgroups — #876779

Order 512.

3230 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2677C212yesmaximal
2C2×#2857C212yesmaximal
3C2×#2551C212yesmaximal, complemented
4#16303C212yesmaximal
5#16254C212yesmaximal
6#15360C214yesmaximal, complemented
7#15318C214yesmaximal, complemented
8#15143C212yesmaximal, complemented
9#15135C211yesmaximal, characteristic
10#15133C212yesmaximal, complemented
11#15125C211yesmaximal, characteristic
12#14402C212yesmaximal
13#13981C212yesmaximal, complemented
14#13876C212yesmaximal
15C4×#385C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2214yes—
19C22×C22⋊Q8C2214yes—
20C23×C42C2211yescharacteristic
21#2857—216no—
22#2677—216no—
23#2551—216no—
24C2×#399C2212yes—
25C2×#398C2214yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#393C2212yes—
31C2×#391C2214yes—
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#387C2214yes—
39C2×#385C2211yescharacteristic
40C2×#385C2211yescharacteristic
41C2×#384C2212yes—
42C2×#384C2212yes—
43C2×#384C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51#1446C2212yes—
52#1445C2212yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C4×#95C2212yes—
78C23×Q8C2314yes—
79C23×Q8C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—216no—
85C2×C4⋊Q8—216no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8—48no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—48no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C42—28no—
118C22×C42—42no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125#399—28no—
126#398—216no—
127#395—216no—
128#395—216no—
129#395—216no—
130#395—48no—
131#393—28no—
132#391—216no—
133#389—216no—
134#389—28no—
135#389—28no—
136#389—216no—
137#389D4116yes—
138#388—216no—
139#387—216no—
140#385D418yes—
141#385Q818yescomplemented
142#384—28no—
143#384—28no—
144#384—42no—
145#380—28no—
146#380—28no—
147#380—216no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C22⋊C4—42no—
151C4×C22⋊C4Q818yescomplemented
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25C2411yescharacteristic
178C22×Q8—28no—
179C22×Q8—28no—
180C22×Q8—24no—
181C22×Q8—28no—
182C22×Q8—28no—
183C22×Q8—28no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—42no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—42no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4C2×D414yes—
203C23×C4C2×Q814yes—
204C23×C4C2×Q814yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C4⋊Q8—432no—
220C4⋊Q8—464no—
221C4⋊Q8—416no—
222C22⋊Q8—464no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D4116yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×Q818yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4C2×D414yes—
285C2×C22⋊C4C2×D414yes—
286C2×C22⋊C4C2×Q814yes—
287C2×C22⋊C4C2×Q814yes—
288C2×C22⋊C4C2×Q814yes—
289C2×C22⋊C4C2×Q814yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—416no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42C2×D418yes—
306C2×C42C2×Q818yes—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—42no—
322#95—44no—
323#95—42no—
324#95—41no—
325#95—41no—
326#95—44no—
327#95—44no—
328#95C2×D4116yes—
329#95C2×D418yes—
330#95C2×Q818yes—
331#95C2×Q818yes—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24C22×D412yes—
341C24C22×Q812yes—
342C24C22×Q812yes—
343C24C2511yescharacteristic
344C2×Q8—416no—
345C2×Q8—416no—
346C2×Q8—416no—
347C2×Q8—48no—
348C2×Q8—416no—
349C2×Q8—416no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×Q8—88no—
353C2×Q8—88no—
354C2×Q8—84no—
355C2×Q8—84no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4C22×D412yes—
436C22×C4C22×D418yes—
437C22×C4C22×D412yes—
438C22×C4C22×D418yes—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q812yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+414yes—
455C22×C42−1+412yes—
456C22×C42−1+414yes—
457C4⋊C4—216no—
458C4⋊C4—232no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—416no—
470C4⋊C4—48no—
471C4⋊C4—88no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4C4⋊Q8116yescomplemented
480C42—216no—
481C42—216no—
482C42—48no—
483C42—416no—
484C42—44no—
485C42—44no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C2×C4⋊Q814yes—
509C23C23×D411yescharacteristic
510C23C23×D412yes—
511C23C23×Q811yescharacteristic
512C23C23×Q811yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)12yes—
515C23C2×2−1+411yescharacteristic
516C23C2×2−1+412yes—
517C23C2×2−1+412yes—
518C23#58511yescharacteristic
519C23#58511yescharacteristic
520Q8—832no—
521Q8—816no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—416no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4C2×C4⋊Q818yes—
604C2×C4#53618yes—
605C2×C4#54414yes—
606C2×C4D4218yes—
607C2×C4D4214yes—
608C2×C4D4×Q818yes—
609C2×C4D4×Q818yes—
610C2×C4#56314yes—
611C22—28no—
612C22—24no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22C22×C4⋊Q811yescharacteristic
623C22C2×#53612yes—
624C22C2×#54411yescharacteristic
625C22#345811yescharacteristic
626C22C2×D4212yes—
627C22C2×D4211yescharacteristic
628C22C2×D4×Q812yes—
629C22C2×D4×Q812yes—
630C22#348812yes—
631C22#348912yes—
632C22#349611yescharacteristic
633C22#349711yescharacteristic
634C22#349912yes—
635C22#350111yescharacteristic
636C22#350211yescharacteristic
637C22#350312yes—
638C22#350312yes—
639C22C2×#56311yescharacteristic
640C22#353711yescharacteristic
641C22#353811yescharacteristic
642C22#354712yes—
643C22#355012yes—
644C22#355512yes—
645C4—28no—
646C4—28no—
647C4—28no—
648C4—44no—
649C4—48no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—48no—
654C4—42no—
655C4—42no—
656C4—84no—
657C4—84no—
658C4#267718yescomplemented
659C2—44no—
660C2#3516711yescharacteristic
661C2D4×C4⋊Q812yes—
662C2#3730811yescharacteristic
663C2#3769012yes—
664C2#3934312yes—
665C2#3959712yes—
666C2#4548811yescharacteristic
667C2#4550211yescharacteristic
668C2C2×#267711yescharacteristic
669C2#5899711yescharacteristic
670C2#5899811yescharacteristic
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. C2×#2551C2semidirect
6. #15360C2semidirect
7. #15318C2semidirect
8. #15143C2semidirect
10. #15133C2semidirect
13. #13981C2semidirect
141. #385Q8semidirect
151. C4×C22⋊C4Q8semidirect
479. C22⋊C4C4⋊Q8semidirect
658. 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×#28572
C2×#25512
#163032
#162542
#153602
#153182
#151432
#151352
#151332
#151252
#144022
#139812
#138762
C4×#3852