Finite Group Databasebeta

← #1006134

Subgroups — #1006134

Order 512.

3158 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2693C212yesmaximal
2C2×#2486C212yesmaximal, complemented
3#17134C212yesmaximal, complemented
4#16462C212yesmaximal, complemented
5#16302C214yesmaximal, complemented
6#15952C212yesmaximal, complemented
7#15463C214yesmaximal, complemented
8#15150C211yesmaximal, characteristic, complemented
9#15139C212yesmaximal, complemented
10#15131C212yesmaximal, complemented
11#14130C211yesmaximal, characteristic, complemented
12#13907C212yesmaximal, complemented
13#13872C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×#123C2214yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21#2693—216no—
22#2486—216no—
23C2×#400C2214yes—
24C2×#398C2214yes—
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#392C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#381C2214yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2214yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1445C2212yes—
47#1445C2212yes—
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1439C2214yes—
61#1439C2211yescharacteristic
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2211yescharacteristic
66#1438C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C4×#95C2211yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2314yes—
82C24×C4C2312yes—
83C2×#123—216no—
84C2×#123—216no—
85C2×#123—48no—
86C2×C4×D4—28no—
87C2×C4×D4—28no—
88C2×C4×D4—28no—
89C2×C4×D4—44no—
90C2×C4×D4—42no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124#400—216no—
125#398—216no—
126#397—28no—
127#397—216no—
128#397—44no—
129#396—216no—
130#392D418yes—
131#389—28no—
132#389—28no—
133#389—44no—
134#386—216no—
135#386—216no—
136#385—28no—
137#384—42no—
138#381—216no—
139#380—28no—
140#380—28no—
141#380—216no—
142#380—44no—
143#380—44no—
144#380D418yescomplemented
145C4×C22⋊C4—42no—
146C4×C22⋊C4—44no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C25C2411yescharacteristic
173C25C2412yes—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4—41no—
183C22×D4—42no—
184C22×D4C2×D418yes—
185C22×D4C2×D414yes—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4#4118yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223#123—464no—
224C4×D4—432no—
225C4×D4—416no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—432no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4C2×D418yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C2×D418yes—
299C2×C22⋊C4C2×D418yes—
300C2×C22⋊C4C2×D414yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—42no—
318#95—41no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—41no—
325#95—41no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95C2×D418yes—
332#95C2×D414yes—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—22no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C22×D414yes—
356C24C2×#4114yes—
357C24H(5,𝔽2)12yes—
358C242−1+414yes—
359C24C2511yescharacteristic
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—44no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—44no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—84no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4—84no—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D418yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C4⋊C4—216no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—88no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C22⋊C4—216no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—48no—
501C42—44no—
502C42—44no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23D4214yes—
560C23#54718yes—
561C23C23×D411yescharacteristic
562C23C23×D411yescharacteristic
563C23C22×#4112yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)12yes—
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×2−1+411yescharacteristic
568C23C2×2−1+412yes—
569C23C2×2−1+411yescharacteristic
570C23#58512yes—
571C23#58511yescharacteristic
572D4—416no—
573D4—432no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—416no—
616C2×C4—432no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—832no—
651C2×C4—816no—
652C2×C4#53714yes—
653C2×C4#53918yes—
654C2×C4#54618yes—
655C2×C4#55114yes—
656C2×C4#56014yes—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—24no—
661C22—28no—
662C22—42no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—42no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—42no—
680C22—44no—
681C22—42no—
682C22—44no—
683C22C2×#53711yescharacteristic
684C22C2×#53912yes—
685C22C2×D4211yescharacteristic
686C22C2×#54612yes—
687C22C2×#54712yes—
688C22C2×#55111yescharacteristic
689C22#348912yes—
690C22#349412yes—
691C22#349512yes—
692C22#349611yescharacteristic
693C22#349612yes—
694C22#349711yescharacteristic
695C22#349911yescharacteristic
696C22#350712yes—
697C22C2×#56011yescharacteristic
698C22#353611yescharacteristic
699C22#353612yes—
700C22#353812yes—
701C22#353811yescharacteristic
702C22#355112yes—
703C22#355111yescharacteristic
704C22#355911yescharacteristic
705C22#356312yes—
706C4—42no—
707C4—42no—
708C4—48no—
709C4—44no—
710C4—42no—
711C4—44no—
712C4—42no—
713C4—82no—
714C4—81no—
715C4—84no—
716C4—84no—
717C4—84no—
718C4#269318yescomplemented
719C2—48no—
720C2—44no—
721C2#3935911yescharacteristic
722C2#3937612yes—
723C2#4295612yes—
724C2#4298011yescharacteristic
725C2#4511312yes—
726C2#4511511yescharacteristic
727C2#4632811yescharacteristic
728C2C2×#269311yescharacteristic
729C2#5299311yescharacteristic
730C2#5901512yes—
731C2#5919811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2486C2semidirect
3. #17134C2semidirect
4. #16462C2semidirect
5. #16302C2semidirect
6. #15952C2semidirect
7. #15463C2semidirect
8. #15150C2semidirect
9. #15139C2semidirect
10. #15131C2semidirect
11. #14130C2semidirect
12. #13907C2semidirect
13. #13872C2semidirect
14. #13872C2semidirect
144. #380D4semidirect
718. C4#2693semidirect

Maximal subgroups

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

subgroupindex
C2×#26932
C2×#24862
#171342
#164622
#163022
#159522
#154632
#151502
#151392
#151312
#141302
#139072
#138722
C4×#3922