Finite Group Databasebeta

← #1010228

Subgroups — #1010228

Order 512.

3602 conjugacy classes of subgroups.

The subgroup lattice has 799 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2699C212yesmaximal, complemented
2C2×#2428C212yesmaximal, complemented
3#17082C212yesmaximal, complemented
4#16596C212yesmaximal, complemented
5#16268C214yesmaximal, complemented
6#15952C212yesmaximal, complemented
7#15319C214yesmaximal, complemented
8#15151C211yesmaximal, characteristic, complemented
9#15130C212yesmaximal, complemented
10#15123C212yesmaximal, complemented
11#14154C211yesmaximal, characteristic, complemented
12#13974C212yesmaximal, complemented
13#13970C212yesmaximal, complemented
14#13916C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2214yes—
19C22×#121C2212yes—
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22#2699—216no—
23#2428—216no—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2211yescharacteristic
30C2×#392C2214yes—
31C2×#392C2214yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#388C2214yes—
39C2×#386C2214yes—
40C2×#386C2214yes—
41C2×#384C2211yescharacteristic
42C2×#381C2214yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1443C2214yes—
52#1442C2211yescharacteristic
53#1441C2211yescharacteristic
54#1440C2212yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2211yescharacteristic
66#1438C2212yes—
67#1438C2212yes—
68#1438C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2211yescharacteristic
77C23×D4C2312yes—
78C23×D4C2314yes—
79C23×D4C2314yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C23×D4C2312yes—
83C24×C4C2312yes—
84C24×C4C2314yes—
85C2×#127—42no—
86C2×#127—42no—
87C2×#127D418yescomplemented
88C2×#121—216no—
89C2×#121—28no—
90C2×#121—28no—
91C2×#121—28no—
92C2×#121—216no—
93C2×#121—48no—
94C2×#121—44no—
95C2×#121—44no—
96C2×#121—44no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—44no—
100C2×C4×D4—42no—
101C2×C4×D4—44no—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#398—28no—
133#398—44no—
134#394—216no—
135#394—216no—
136#394—48no—
137#394D418yescomplemented
138#392—216no—
139#392—216no—
140#390—28no—
141#390—28no—
142#390—44no—
143#390—44no—
144#390—44no—
145#389—28no—
146#388—216no—
147#386—216no—
148#386—216no—
149#384—42no—
150#381—216no—
151#380—28no—
152#380—44no—
153C4×C4⋊C4—42no—
154C4×C4⋊C4—42no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—44no—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C25C2414yes—
179C25C2412yes—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—41no—
194C22×D4—42no—
195C22×D4—41no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—42no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4C2×D418yes—
215C22×D4C2×D418yes—
216C22×D4C2×D418yes—
217C22×D4C2×D414yes—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—28no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#127—416no—
246#121—464no—
247#121—432no—
248#121—432no—
249C4×D4—432no—
250C4×D4—416no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4#4118yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4C2×D418yes—
324C2×C22⋊C4C2×D418yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42C2×D414yes—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—41no—
347#95—42no—
348#95—44no—
349#95—42no—
350#95—42no—
351#95—41no—
352#95—41no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—42no—
357#95—44no—
358#95C2×D414yes—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—22no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—22no—
386C24—22no—
387C24C22×D418yes—
388C24C22×D414yes—
389C24C2511yescharacteristic
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—216no—
393C2×D4—216no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—416no—
402C2×D4—44no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—416no—
406C2×D4—416no—
407C2×D4—416no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—88no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—84no—
423C2×D4—88no—
424C2×D4—84no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—416no—
494C22×C4—416no—
495C22×C4—416no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—88no—
504C22×C4—88no—
505C22×C4—88no—
506C22×C4—88no—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D414yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C2×#4118yes—
515C22×C4C2×#4114yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C4H(5,𝔽2)14yes—
521C4⋊C4—216no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—416no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C22⋊C4—216no—
531C22⋊C4—48no—
532C22⋊C4—416no—
533C22⋊C4—416no—
534C22⋊C4—416no—
535C22⋊C4—416no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—416no—
539C22⋊C4—48no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C22⋊C4—88no—
544C22⋊C4—88no—
545C22⋊C4—88no—
546C42—28no—
547C42—48no—
548C42—44no—
549C42—44no—
550C42—84no—
551C23—28no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—24no—
557C23—24no—
558C23—28no—
559C23—28no—
560C23—24no—
561C23—28no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—44no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—42no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—42no—
585C23—44no—
586C23—42no—
587C23—44no—
588C23—44no—
589C23—42no—
590C23—42no—
591C23—44no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—44no—
602C23—44no—
603C23—42no—
604C23—44no—
605C23—42no—
606C23—44no—
607C23—44no—
608C23—42no—
609C23—44no—
610C23—42no—
611C23—44no—
612C23—42no—
613C23—44no—
614C23—42no—
615C23—44no—
616C23—42no—
617C23—44no—
618C23—44no—
619C23—42no—
620C23—42no—
621C23C23×D411yescharacteristic
622C23C23×D411yescharacteristic
623C23C22×#4112yes—
624C23C2×H(5,𝔽2)11yescharacteristic
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)12yes—
627C23C2×H(5,𝔽2)11yescharacteristic
628C23C2×H(5,𝔽2)11yescharacteristic
629C23#58512yes—
630C23#58512yes—
631C23#58511yescharacteristic
632D4—416no—
633D4—432no—
634D4—816no—
635D4—832no—
636D4—816no—
637D4—832no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—28no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—44no—
698C2×C4—42no—
699C2×C4—41no—
700C2×C4—42no—
701C2×C4—41no—
702C2×C4—44no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—44no—
707C2×C4—42no—
708C2×C4—41no—
709C2×C4—44no—
710C2×C4—44no—
711C2×C4—832no—
712C2×C4—816no—
713C2×C4#53414yes—
714C2×C4#54018yes—
715C2×C4D4214yes—
716C2×C4#54618yes—
717C2×C4#55514yes—
718C2×C4#56014yes—
719C22—48no—
720C22—48no—
721C22—48no—
722C22—48no—
723C22—44no—
724C22—48no—
725C22—44no—
726C22—44no—
727C22—48no—
728C22—48no—
729C22—44no—
730C22—44no—
731C22—44no—
732C22—44no—
733C22—82no—
734C22—82no—
735C22—84no—
736C22—84no—
737C22—82no—
738C22—82no—
739C22—82no—
740C22—84no—
741C22—82no—
742C22—84no—
743C22—82no—
744C22—84no—
745C22—82no—
746C22—84no—
747C22—84no—
748C22—84no—
749C22C2×#53411yescharacteristic
750C22C2×#54012yes—
751C22C2×D4211yescharacteristic
752C22C2×#54612yes—
753C22D4×#4112yes—
754C22C2×#55511yescharacteristic
755C22#348512yes—
756C22#349611yescharacteristic
757C22#349612yes—
758C22#349611yescharacteristic
759C22#349811yescharacteristic
760C22#351412yes—
761C22#351712yes—
762C22#352412yes—
763C22C2×#56011yescharacteristic
764C22#353711yescharacteristic
765C22#353912yes—
766C22#354011yescharacteristic
767C22#354212yes—
768C22#354612yes—
769C22#355111yescharacteristic
770C22#355811yescharacteristic
771C22#356212yes—
772C4—24no—
773C4—44no—
774C4—48no—
775C4—42no—
776C4—42no—
777C4—42no—
778C4—42no—
779C4—82no—
780C4—82no—
781C4—84no—
782C4—81no—
783C4—84no—
784C4#269918yescomplemented
785C2—88no—
786C2—84no—
787C2#3935811yescharacteristic
788C2#3940212yes—
789C2#4267311yescharacteristic
790C2#4275912yes—
791C2#4489512yes—
792C2#4540811yescharacteristic
793C2#4547811yescharacteristic
794C2C2×#269911yescharacteristic
795C2#5296611yescharacteristic
796C2#5903512yes—
797C2#5984111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2699C2semidirect
2. C2×#2428C2semidirect
3. #17082C2semidirect
4. #16596C2semidirect
5. #16268C2semidirect
6. #15952C2semidirect
7. #15319C2semidirect
8. #15151C2semidirect
9. #15130C2semidirect
10. #15123C2semidirect
11. #14154C2semidirect
12. #13974C2semidirect
13. #13970C2semidirect
14. #13916C2semidirect
15. C4×#394C2semidirect
87. C2×#127D4semidirect
137. #394D4semidirect
784. C4#2699semidirect

Maximal subgroups

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

subgroupindex
C2×#26992
C2×#24282
#170822
#165962
#162682
#159522
#153192
#151512
#151302
#151232
#141542
#139742
#139702
#139162
C4×#3942