Finite Group Databasebeta

← #1409541

Subgroups — #1409541

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2494C211yesmaximal, characteristic
3#16899C212yesmaximal
4#16870C214yesmaximal, complemented
5#16868C214yesmaximal
6#16705C212yesmaximal, complemented
7#15225C212yesmaximal, complemented
8#15220C212yesmaximal
9#15153C211yesmaximal, characteristic, complemented
10#15149C211yesmaximal, characteristic, complemented
11#15135C211yesmaximal, characteristic
12#15020C212yesmaximal
13#14326C212yesmaximal, complemented
14#14006C212yesmaximal, complemented
15#13993C212yesmaximal
16C4×#397C211yesmaximal, characteristic
17#13818C211yesmaximal, characteristic, complemented
18#2571—28no—
19#2494—28no—
20C2×#401C2212yes—
21C2×#400C2212yes—
22C2×#400C2214yes—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#397C2211yescharacteristic
27C2×#397C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#384C2211yescharacteristic
38C2×#382C2212yes—
39C2×#382C2212yes—
40C2×#382C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1445C2212yes—
50#1445C2211yescharacteristic
51#1445C2214yes—
52#1445C2214yes—
53#1444C2212yes—
54#1443C2212yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2212yes—
67#1439C2211yescharacteristic
68#1439C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2314yes—
114C22×C42C2312yes—
115#401—28no—
116#400—216no—
117#400—28no—
118#400—44no—
119#399—44no—
120#398—44no—
121#397—28no—
122#397—24no—
123#389—28no—
124#389—24no—
125#389—28no—
126#389—28no—
127#389—28no—
128#389—24no—
129#389—44no—
130#388—28no—
131#387—28no—
132#384—24no—
133#382—28no—
134#382—24no—
135#382—28no—
136#380—24no—
137#380—24no—
138#380—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4D418yes—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—44no—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C25C2411yescharacteristic
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4#4118yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42C2×D414yes—
278C2×C42C2×D418yes—
279#95—22no—
280#95—22no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—22no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95C2×D418yes—
313#95#4114yes—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—21no—
322C24—22no—
323C24—22no—
324C242−1+412yes—
325C24C2511yescharacteristic
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×D414yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4118yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4114yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+414yes—
424C22×C42−1+412yes—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—216no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—44no—
446C22⋊C4—84no—
447C22⋊C4—88no—
448C42—28no—
449C42—48no—
450C42—44no—
451C42—48no—
452C42—48no—
453C42—88no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23C23×D411yescharacteristic
478C23C22×#4112yes—
479C23C22×#4111yescharacteristic
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×2−1+412yes—
483C23C2×2−1+411yescharacteristic
484C23C2×2−1+411yescharacteristic
485C23#58512yes—
486C23#58512yes—
487C23#58511yescharacteristic
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—44no—
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—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—88no—
577C2×C4#53214yes—
578C2×C4#53614yes—
579C2×C4#53714yes—
580C2×C4#53918yes—
581C2×C4#54118yes—
582C2×C4#55014yes—
583C22—44no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—82no—
589C22—81no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C2×#53211yescharacteristic
595C22C2×#53611yescharacteristic
596C22C2×#53711yescharacteristic
597C22C2×#53912yes—
598C22C2×#54112yes—
599C22C2×#55011yescharacteristic
600C22#348012yes—
601C22#348511yescharacteristic
602C22#348911yescharacteristic
603C22#350011yescharacteristic
604C22#351412yes—
605C22#351512yes—
606C22#351612yes—
607C22#351811yescharacteristic
608C22#352011yescharacteristic
609C22#352112yes—
610C22#354012yes—
611C22#354611yescharacteristic
612C22#355211yescharacteristic
613C22#355912yes—
614C22#356312yes—
615C22#356512yes—
616C22#356512yes—
617C4—24no—
618C4—48no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—82no—
624C4—81no—
625C4—81no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4—84no—
632C4#266418yes—
633C2—82no—
634C2C2×#266411yescharacteristic
635C2#3585611yescharacteristic
636C2#3659711yescharacteristic
637C2#3913411yescharacteristic
638C2#3913611yescharacteristic
639C2#3921412yes—
640C2#3922512yes—
641C2#4265011yescharacteristic
642C2#4275312yes—
643C2#4275712yes—
644C2#4297811yescharacteristic
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
1. C2×#2571C2semidirect
4. #16870C2semidirect
6. #16705C2semidirect
7. #15225C2semidirect
9. #15153C2semidirect
10. #15149C2semidirect
13. #14326C2semidirect
14. #14006C2semidirect
17. #13818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#24942
#168992
#168702
#168682
#167052
#152252
#152202
#151532
#151492
#151352
#150202
#143262
#140062
#139932
C4×#3972
#138182