Finite Group Databasebeta

← #613843

Subgroups — #613843

Order 512.

2970 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic, complemented
3C2×#2536C211yesmaximal, characteristic
4C2×C42×Q8C211yesmaximal, characteristic, complemented
5#15252C214yesmaximal, complemented
6#14846C214yesmaximal, complemented
7#14780C214yesmaximal
8#14709C211yesmaximal, characteristic, complemented
9#14701C214yesmaximal, complemented
10#13969C212yesmaximal, complemented
11#13969C212yesmaximal, complemented
12#13955C211yesmaximal, characteristic, complemented
13#13876C212yesmaximal, complemented
14#13875C212yesmaximal, complemented
15C4×#390C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2565—28no—
22#2562—28no—
23#2536—28no—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#394C2212yes—
28C2×#394C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#390C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C42×Q8—28no—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C43C2212yes—
51C2×C43C2211yescharacteristic
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2214yes—
62#1437C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2212yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77#1434C2214yes—
78#1434C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C2×#125—24no—
83C2×#125—24no—
84C2×#125—24no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C4×Q8—28no—
89C2×C4×Q8—28no—
90C2×C4×Q8—24no—
91C2×C4×Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8D418yes—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#400—28no—
131#398—216no—
132#398—28no—
133#394—28no—
134#394—28no—
135#391—24no—
136#391—24no—
137#390—28no—
138#388—216no—
139#388—216no—
140#388—216no—
141#387—216no—
142#384—24no—
143#382—24no—
144#382—24no—
145#380—28no—
146#380—28no—
147#380—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—216no—
152C4×C4⋊C4D418yes—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C43—28no—
179C43—24no—
180C25C2412yes—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8C2×D414yes—
185C22×Q8#4118yes—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—42no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212#125—416no—
213C22⋊Q8—432no—
214C4×Q8—264no—
215C4×Q8—232no—
216C4×D4—416no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D418yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D418yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4#4118yes—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42C2×D418yes—
306C2×C42C2×D418yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4118yes—
310C2×C42#4114yes—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24C2×#4114yes—
349C24C2511yescharacteristic
350C2×Q8—28no—
351C2×Q8—216no—
352C2×Q8—216no—
353C2×Q8—28no—
354C2×Q8#124116yes—
355C2×D4—28no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—44no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4C22×D414yes—
432C22×C4C22×D418yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4118yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C42−1+414yes—
451C4⋊C4—232no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C22⋊C4—216no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C42—28no—
474C42—232no—
475C42—216no—
476C42—216no—
477C42—48no—
478C42—48no—
479C42—416no—
480C42—44no—
481C42—44no—
482C42—44no—
483C42#121116yes—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23C23×D411yescharacteristic
520C23C23×D411yescharacteristic
521C23C22×#4111yescharacteristic
522C23C22×#4112yes—
523C23C22×#4112yes—
524C23C22×#4112yes—
525C23C2×2−1+412yes—
526C23#58512yes—
527C23#58512yes—
528Q8—232no—
529D4—416no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—816no—
606C2×C4C2×#12114yes—
607C2×C4C2×#12114yes—
608C2×C4C2×#12114yes—
609C2×C4#52518yes—
610C2×C4C2×#12418yes—
611C2×C4C2×#12414yes—
612C2×C4#52918yes—
613C2×C4#53214yes—
614C2×C4#55218yes—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C22×#12111yescharacteristic
631C22C2×#52512yes—
632C22C22×#12411yescharacteristic
633C22C2×#52912yes—
634C22C2×#53211yescharacteristic
635C22#344411yescharacteristic
636C22#345511yescharacteristic
637C22#346112yes—
638C22D4×#4112yes—
639C22D4×#4112yes—
640C22C2×#55212yes—
641C22#347512yes—
642C22#348012yes—
643C22#348212yes—
644C22#348511yescharacteristic
645C22#348611yescharacteristic
646C22#349712yes—
647C22#350712yes—
648C22#351212yes—
649C22#351612yes—
650C22#352412yes—
651C4—28no—
652C4—24no—
653C4—28no—
654C4—24no—
655C4—48no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4#285018yes—
665C2—84no—
666C2#3474311yescharacteristic
667C2#3489812yes—
668C2#3515311yescharacteristic
669C2#3518312yes—
670C2#3623212yes—
671C2#3627012yes—
672C2#3643612yes—
673C2#3646712yes—
674C2C2×#285011yescharacteristic
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
2. C2×#2565C2semidirect
4. C2×C42×Q8C2semidirect
5. #15252C2semidirect
6. #14846C2semidirect
8. #14709C2semidirect
9. #14701C2semidirect
10. #13969C2semidirect
11. #13969C2semidirect
12. #13955C2semidirect
13. #13876C2semidirect
14. #13875C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#25652
C2×#25362
C2×C42×Q82
#152522
#148462
#147802
#147092
#147012
#139692
#139552
#138762
#138752
C4×#3902