Finite Group Databasebeta

← #1409877

Subgroups — #1409877

Order 512.

2156 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2512C211yesmaximal, characteristic
3#17029C212yesmaximal, complemented
4#16868C214yesmaximal, complemented
5#16673C212yesmaximal, complemented
6#16547C214yesmaximal, complemented
7#15244C211yesmaximal, characteristic
8#15231C212yesmaximal, complemented
9#15222C212yesmaximal
10#15152C211yesmaximal, characteristic, complemented
11#15147C211yesmaximal, characteristic
12#14808C212yesmaximal, complemented
13#14291C212yesmaximal, complemented
14#14150C212yesmaximal, complemented
15#14141C212yesmaximal
16C4×#398C211yesmaximal, characteristic, complemented
17#13821C211yesmaximal, characteristic, complemented
18C22×#126C2212yes—
19#2571—28no—
20#2512—28no—
21C2×#400C2214yes—
22C2×#400C2212yes—
23C2×#398C2211yescharacteristic
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#396C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2214yes—
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#382C2212yes—
41C2×#382C2212yes—
42C2×#382C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49#1445C2211yescharacteristic
50#1445C2212yes—
51#1444C2214yes—
52#1444C2211yescharacteristic
53#1442C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2211yescharacteristic
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×D4C2311yescharacteristic
86C2×#126—28no—
87C2×#126—28no—
88C2×#126—44no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2314yes—
124#400—216no—
125#400—44no—
126#398—24no—
127#397—28no—
128#397—44no—
129#396—28no—
130#390—24no—
131#390—24no—
132#390—24no—
133#390—28no—
134#390—44no—
135#389—28no—
136#389—24no—
137#389—28no—
138#389—28no—
139#388—28no—
140#388—48no—
141#384—24no—
142#384—24no—
143#382—28no—
144#382—28no—
145#382—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—44no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4—44no—
151C4×C4⋊C4D418yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C25C2412yes—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—22no—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203#126—432no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4#4114yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
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⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D418yes—
298C2×C42C2×D414yes—
299C2×C42#4118yes—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—22no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95C2×D418yes—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24C2511yescharacteristic
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
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—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
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—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4114yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C42−1+414yes—
441C22×C42−1+412yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—88no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—48no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C42—28no—
471C42—216no—
472C42—216no—
473C42—44no—
474C42—44no—
475C42—48no—
476C42—88no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23C23×D411yescharacteristic
501C23C22×#4112yes—
502C23C22×#4111yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×2−1+412yes—
507C23C2×2−1+411yescharacteristic
508C23#58512yes—
509C23#58512yes—
510C23#58511yescharacteristic
511D4—88no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
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—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4#53214yes—
603C2×C4#53614yes—
604C2×C4#53714yes—
605C2×C4#53918yes—
606C2×C4#54118yes—
607C2×C4#54614yes—
608C22—84no—
609C22—82no—
610C22—84no—
611C22—82no—
612C22—82no—
613C22—84no—
614C22—84no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#53211yescharacteristic
620C22C2×#53611yescharacteristic
621C22C2×#53711yescharacteristic
622C22C2×#53912yes—
623C22C2×#54112yes—
624C22C2×#54611yescharacteristic
625C22#348012yes—
626C22#348711yescharacteristic
627C22#348811yescharacteristic
628C22#350011yescharacteristic
629C22#350612yes—
630C22#350612yes—
631C22#350611yescharacteristic
632C22#351012yes—
633C22#351111yescharacteristic
634C22#351512yes—
635C22#354012yes—
636C22#354711yescharacteristic
637C22#354911yescharacteristic
638C22#355912yes—
639C22#355912yes—
640C22#356512yes—
641C22#356512yes—
642C4—24no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—44no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—84no—
652C4—82no—
653C4—82no—
654C4—84no—
655C4—82no—
656C4—81no—
657C4#266418yes—
658C2—162no—
659C2C2×#266411yescharacteristic
660C2#3595611yescharacteristic
661C2#3641911yescharacteristic
662C2#3890711yescharacteristic
663C2#3946911yescharacteristic
664C2#4002712yes—
665C2#4003412yes—
666C2#4251712yes—
667C2#4253012yes—
668C2#4288211yescharacteristic
669C2#4294411yescharacteristic
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×#2571C2semidirect
3. #17029C2semidirect
4. #16868C2semidirect
5. #16673C2semidirect
6. #16547C2semidirect
8. #15231C2semidirect
10. #15152C2semidirect
12. #14808C2semidirect
13. #14291C2semidirect
14. #14150C2semidirect
16. C4×#398C2semidirect
17. #13821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#25122
#170292
#168682
#166732
#165472
#152442
#152312
#152222
#151522
#151472
#148082
#142912
#141502
#141412
C4×#3982
#138212