Finite Group Databasebeta

← #674805

Subgroups — #674805

Order 512.

3010 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×#2454C212yesmaximal
3C2×#2349C211yesmaximal, characteristic
4#15406C214yesmaximal, complemented
5#15305C214yesmaximal, complemented
6#15218C212yesmaximal, complemented
7#15137C212yesmaximal, complemented
8#14771C214yesmaximal, complemented
9#14150C211yesmaximal, characteristic, complemented
10#14004C214yesmaximal, complemented
11#13977C212yesmaximal
12#13954C211yesmaximal, characteristic, complemented
13#13941C211yesmaximal, characteristic
14C4×#389C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2859—28no—
20#2454—216no—
21C2×#399C2214yes—
22C2×#398C2214yes—
23C2×#396C2214yes—
24#2349—28no—
25C2×#390C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2214yes—
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2214yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2214yes—
49C22×#95C2212yes—
50C2×C43C2211yescharacteristic
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2212yes—
56#1442C2214yes—
57#1442C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C23×D4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2312yes—
81C2×#125—28no—
82C2×#125—28no—
83C2×#125—44no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—44no—
89C2×C4×D4Q818yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#399—216no—
123#398—48no—
124#396—216no—
125#390—28no—
126#390D418yes—
127#390Q818yes—
128#389—28no—
129#389—28no—
130#389—28no—
131#389—216no—
132#389—216no—
133#389—28no—
134#389—44no—
135#385—216no—
136#385—216no—
137#385—216no—
138#385—48no—
139#384—28no—
140#382—24no—
141#382D418yes—
142#380—216no—
143#380—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C22⋊C4—48no—
149C2×#95—28no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C43—24no—
175C25C2412yes—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C2×D414yes—
181C22×D4C2×Q814yes—
182C22×D4C2×Q814yes—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—28no—
187C23×C4—28no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×Q818yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209#125—432no—
210C4×D4—232no—
211C4×D4—432no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
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—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4C2×Q814yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×D4116yes—
280C2×C22⋊C4C2×Q818yes—
281C2×C22⋊C4C2×Q8116yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×Q814yes—
303C2×C42C2×Q814yes—
304C2×C42#4114yes—
305#95—22no—
306#95—28no—
307#95—24no—
308#95—22no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C2×D418yes—
327#95C2×D418yes—
328#95#4118yes—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24C22×D414yes—
338C24C22×Q814yes—
339C24C22×Q814yes—
340C24C2511yescharacteristic
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4C4⋊Q8116yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—216no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D418yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×Q814yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+412yes—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C22⋊C4—216no—
467C22⋊C4—232no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—88no—
472C42—28no—
473C42—28no—
474C42—48no—
475C42—48no—
476C42—48no—
477C42—48no—
478C42—84no—
479C42C22⋊Q8116yes—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23C2×C4⋊Q818yes—
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C23×Q811yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4112yes—
509C23C2×H(5,𝔽2)11yescharacteristic
510C23C2×H(5,𝔽2)12yes—
511C23C2×2−1+411yescharacteristic
512C23#58512yes—
513C23#58512yes—
514D4—232no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—816no—
594C2×C4—832no—
595C2×C4C2×C22⋊Q814yes—
596C2×C4C2×C22⋊Q814yes—
597C2×C4C2×C22⋊Q814yes—
598C2×C4C2×C4⋊Q814yes—
599C2×C4#54414yes—
600C2×C4D4214yes—
601C2×C4#54714yes—
602C2×C4D4×Q814yes—
603C2×C4#55418yes—
604C2×C4#55414yes—
605C22—28no—
606C22—216no—
607C22—48no—
608C22—44no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—48no—
616C22C22×C22⋊Q811yescharacteristic
617C22C22×C4⋊Q811yescharacteristic
618C22#344411yescharacteristic
619C22#345511yescharacteristic
620C22C2×#54411yescharacteristic
621C22C2×D4211yescharacteristic
622C22C2×#54711yescharacteristic
623C22C2×D4×Q811yescharacteristic
624C22D4×#4112yes—
625C22D4×#4112yes—
626C22C2×#55412yes—
627C22C2×#55411yescharacteristic
628C22Q8×#4112yes—
629C22#348812yes—
630C22#349612yes—
631C22#350112yes—
632C22#350412yes—
633C22#350812yes—
634C22#350811yescharacteristic
635C22#351011yescharacteristic
636C22#351512yes—
637C22#355912yes—
638C22#356212yes—
639C4—28no—
640C4—24no—
641C4—24no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—48no—
648C4—44no—
649C4—82no—
650C4—82no—
651C4—84no—
652C4#244018yes—
653C2—48no—
654C2C2×#244011yescharacteristic
655C2#3474511yescharacteristic
656C2#3551511yescharacteristic
657C2#3613511yescharacteristic
658C2#3626112yes—
659C2#3657311yescharacteristic
660C2#3937211yescharacteristic
661C2#3963011yescharacteristic
662C2#3997612yes—
663C2#5912912yes—
664C2#5915512yes—
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
1. C2×#2859C2semidirect
4. #15406C2semidirect
5. #15305C2semidirect
6. #15218C2semidirect
7. #15137C2semidirect
8. #14771C2semidirect
9. #14150C2semidirect
10. #14004C2semidirect
12. #13954C2semidirect
14. C4×#389C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#24542
C2×#23492
#154062
#153052
#152182
#151372
#147712
#141502
#140042
#139772
#139542
#139412
C4×#3892
C4×#3902