Finite Group Databasebeta

← #530458

Subgroups — #530458

Order 512.

2574 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2#14688C211yesmaximal, characteristic, complemented
3#14504C214yesmaximal, complemented
4#14469C212yesmaximal, complemented
5#14444C212yesmaximal
6#14288C212yesmaximal
7#14125C214yesmaximal, complemented
8#13925C212yesmaximal, complemented
9#13879C212yesmaximal, complemented
10#13876C212yesmaximal, complemented
11#13875C212yesmaximal, complemented
12#13871C212yesmaximal
13#13801C212yesmaximal, complemented
14#13781C212yesmaximal, complemented
15C4×#384C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20C4×#124C4116yescomplemented
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#388C2214yes—
24C2×#388C2212yes—
25C2×#388C2212yes—
26C2×#384C2212yes—
27C2×#383C2212yes—
28C2×#382C2212yes—
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C43C2211yescharacteristic
41#1446C2214yes—
42#1445C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2212yes—
71#1434C2214yes—
72#1434C2212yes—
73#1434C2212yes—
74#1434C2214yes—
75C4×#95C2214yes—
76C23×Q8C2311yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C2×#124—24no—
80C2×#124—24no—
81C2×#124C2×C418yes—
82C2×C4×Q8—44no—
83C2×C4×Q8C2×C418yes—
84C2×C4×D4—44no—
85C2×C4×D4C2×C418yes—
86C2×#117—28no—
87C2×#117—44no—
88C2×#117—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#391C2×C418yes—
120#390C2×C418yes—
121#388—48no—
122#388C2×C4116yes—
123#388C2×C4116yes—
124#384—28no—
125#383—28no—
126#382—44no—
127#382C2×C418yes—
128#380—28no—
129#380—44no—
130#380—44no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—44no—
133C4×C4⋊C4—42no—
134C4×C4⋊C4—42no—
135C4×C4⋊C4—44no—
136C4×C22⋊C4C2×C4116yes—
137C4×C22⋊C4C2×C4116yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C43C2×C418yes—
165C25C2412yes—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8C22×C414yes—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4C22×C414yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4C22×C418yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197#124—232no—
198C4×Q8—416no—
199C4×D4—416no—
200#117—432no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C418yes—
225C2×C4⋊C4#4118yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C22×C418yes—
249C2×C22⋊C4C22×C418yes—
250C2×C22⋊C4C22×C418yes—
251C2×C22⋊C4C22×C418yes—
252C2×C22⋊C4C22×C418yes—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42C22×C414yes—
276C2×C42C22×C418yes—
277C2×C42C22×C414yes—
278C2×C42C22×C414yes—
279C2×C42C22×C414yes—
280C2×C42C22×C414yes—
281C2×C42C22×C414yes—
282C2×C42C22×C414yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95C22×C418yes—
308#95C22×C418yes—
309#95C22×C418yes—
310#95C22×C418yes—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24C23×C414yes—
326C24C2511yescharacteristic
327C2×Q8—28no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×D4—28no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4C23×C414yes—
408C22×C4C23×C412yes—
409C22×C4C23×C414yes—
410C22×C4C23×C414yes—
411C22×C4C23×C412yes—
412C22×C4C23×C412yes—
413C22×C4C23×C412yes—
414C22×C4C23×C412yes—
415C22×C4C23×C412yes—
416C22×C4C23×C412yes—
417C22×C4C2×#4118yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4114yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+412yes—
429C22×C42−1+412yes—
430C4⋊C4—216no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C22⋊C4—216no—
440C22⋊C4—216no—
441C22⋊C4—216no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C42—28no—
449C42—28no—
450C42—216no—
451C42—28no—
452C42—28no—
453C42—44no—
454C42—44no—
455C42—416no—
456C42—84no—
457C42—84no—
458C42—88no—
459C42#117116yescomplemented
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23C24×C411yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4111yescharacteristic
498C23C22×#4112yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×2−1+411yescharacteristic
503C23C2×2−1+411yescharacteristic
504C23#58512yes—
505C23#58512yes—
506Q8—416no—
507D4—416no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
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—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—432no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4C2×#11714yes—
587C2×C4C2×#11714yes—
588C2×C4C2×#11714yes—
589C2×C4C4×#4114yes—
590C2×C4#51814yes—
591C2×C4#52014yes—
592C2×C4#52014yes—
593C2×C4#53818yes—
594C2×C4#54218yes—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C22×#11711yescharacteristic
611C22C2×C4×#4111yescharacteristic
612C22C2×#51811yescharacteristic
613C22C2×#52011yescharacteristic
614C22C2×#52011yescharacteristic
615C22#342811yescharacteristic
616C22#342811yescharacteristic
617C22C2×#53812yes—
618C22C2×#54212yes—
619C22#347512yes—
620C22#348012yes—
621C22#350612yes—
622C22#350612yes—
623C22#350811yescharacteristic
624C22#350911yescharacteristic
625C22#351111yescharacteristic
626C22#351111yescharacteristic
627C22#351312yes—
628C22#351612yes—
629C22#354612yes—
630C22#354712yes—
631C22#356212yes—
632C22#356512yes—
633C4—24no—
634C4—24no—
635C4—24no—
636C4—24no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—42no—
641C4—48no—
642C4—44no—
643C4—84no—
644C4—84no—
645C4—84no—
646C4#233718yes—
647C2—84no—
648C2C2×#233711yescharacteristic
649C2#3412111yescharacteristic
650C2#3418611yescharacteristic
651C2#3418811yescharacteristic
652C2#3424111yescharacteristic
653C2#3436711yescharacteristic
654C2#3438511yescharacteristic
655C2#3952512yes—
656C2#3952712yes—
657C2#3978112yes—
658C2#3978312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14688C2semidirect
3. #14504C2semidirect
4. #14469C2semidirect
7. #14125C2semidirect
8. #13925C2semidirect
9. #13879C2semidirect
10. #13876C2semidirect
11. #13875C2semidirect
13. #13801C2semidirect
14. #13781C2semidirect
15. C4×#384C2semidirect
20. C4×#124C4semidirect
459. C42#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
#146882
#145042
#144692
#144442
#142882
#141252
#139252
#138792
#138762
#138752
#138712
#138012
#137812
C4×#3842