Finite Group Databasebeta

← #2737218

Subgroups — #2737218

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2497C211yesmaximal, characteristic
3#17129C214yesmaximal
4#17113C214yesmaximal, complemented
5#17065C212yesmaximal, complemented
6#17062C212yesmaximal
7#17043C212yesmaximal, complemented
8#15218C212yesmaximal, complemented
9#15216C212yesmaximal
10#15200C211yesmaximal, characteristic, complemented
11#15153C211yesmaximal, characteristic, complemented
12#15135C211yesmaximal, characteristic
13#15133C212yesmaximal, complemented
14#15132C212yesmaximal
15#14628C212yesmaximal
16#13936C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic
18C22×#125C2212yes—
19#2552—28no—
20#2497—28no—
21C2×#400C2212yes—
22C2×#400C2214yes—
23C2×#399C2212yes—
24C2×#399C2211yescharacteristic
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28C2×#396C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#387C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1445C2211yescharacteristic
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1444C2212yes—
54#1444C2211yescharacteristic
55#1444C2214yes—
56#1444C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1440C2214yes—
70#1439C2212yes—
71#1439C2211yescharacteristic
72#1439C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#125—28no—
87C2×#125—44no—
88C2×#125—44no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#400—28no—
122#400—48no—
123#399—24no—
124#399—28no—
125#398—216no—
126#398—48no—
127#397—48no—
128#396—28no—
129#389—24no—
130#389—28no—
131#389—24no—
132#389—28no—
133#389—44no—
134#388—28no—
135#387—28no—
136#384—24no—
137#384—24no—
138#384—28no—
139#380—24no—
140#380—24no—
141#380—44no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—44no—
144C4×C4⋊C4D418yes—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—44no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2411yescharacteristic
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201#125—432no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
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—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4#4114yes—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42C2×D414yes—
285C2×C42C2×D418yes—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—22no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—22no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95C2×D418yes—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—21no—
327C24—22no—
328C24C2×#4112yes—
329C24C2511yescharacteristic
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4C22×D412yes—
414C22×C4C22×D414yes—
415C22×C4C22×D414yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C42−1+412yes—
423C22×C42−1+412yes—
424C22×C42−1+414yes—
425C22×C42−1+412yes—
426C22×C42−1+412yes—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—416no—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—88no—
447C22⋊C4—28no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—88no—
454C42—28no—
455C42—48no—
456C42—44no—
457C42—48no—
458C42—48no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23C23×D411yescharacteristic
483C23C22×#4111yescharacteristic
484C23C2×H(5,𝔽2)12yes—
485C23C2×H(5,𝔽2)12yes—
486C23C2×2−1+411yescharacteristic
487C23C2×2−1+411yescharacteristic
488C23C2×2−1+412yes—
489C23C2×2−1+411yescharacteristic
490C23C2×2−1+411yescharacteristic
491C23#58512yes—
492C23#58511yescharacteristic
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
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—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—88no—
580C2×C4#53514yes—
581C2×C4#53614yes—
582C2×C4#53614yes—
583C2×C4#54814yes—
584C2×C4#56318yes—
585C2×C4#56318yes—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—82no—
592C22—81no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22C2×#53511yescharacteristic
598C22C2×#53611yescharacteristic
599C22C2×#53611yescharacteristic
600C22C2×#54811yescharacteristic
601C22#348511yescharacteristic
602C22#348911yescharacteristic
603C22#349811yescharacteristic
604C22#350711yescharacteristic
605C22#350911yescharacteristic
606C22#351512yes—
607C22#351912yes—
608C22C2×#56312yes—
609C22C2×#56312yes—
610C22#353912yes—
611C22#354112yes—
612C22#354412yes—
613C22#354712yes—
614C22#354812yes—
615C22#355412yes—
616C22#355511yescharacteristic
617C22#355611yescharacteristic
618C22#355912yes—
619C22#356312yes—
620C4—24no—
621C4—42no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—82no—
628C4—84no—
629C4—84no—
630C4—82no—
631C4—81no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#282118yes—
636C2—82no—
637C2C2×#282111yescharacteristic
638C2#3912211yescharacteristic
639C2#3916911yescharacteristic
640C2#3972411yescharacteristic
641C2#4265111yescharacteristic
642C2#4267911yescharacteristic
643C2#4296811yescharacteristic
644C2#4564012yes—
645C2#4564812yes—
646C2#5306712yes—
647C2#5310712yes—
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×#2552C2semidirect
4. #17113C2semidirect
5. #17065C2semidirect
7. #17043C2semidirect
8. #15218C2semidirect
10. #15200C2semidirect
11. #15153C2semidirect
13. #15133C2semidirect
16. #13936C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24972
#171292
#171132
#170652
#170622
#170432
#152182
#152162
#152002
#151532
#151352
#151332
#151322
#146282
#139362
C4×#3992