Finite Group Databasebeta

← #779441

Subgroups — #779441

Order 512.

3730 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2445C212yesmaximal, complemented
2C2×#2361C212yesmaximal, complemented
3#16338C212yesmaximal, complemented
4#15949C212yesmaximal, complemented
5#15782C212yesmaximal, complemented
6#15405C214yesmaximal, complemented
7#15219C212yesmaximal, complemented
8#15124C211yesmaximal, characteristic
9#14721C214yesmaximal, complemented
10#14171C212yesmaximal, complemented
11#14150C212yesmaximal, complemented
12#14130C211yesmaximal, characteristic, complemented
13#14018C212yesmaximal, complemented
14#14002C212yesmaximal, complemented
15C4×#399C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#123C2214yes—
18C22×#121C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2214yes—
21C23×C4⋊C4C2211yescharacteristic
22#2445—216no—
23C2×#399C2211yescharacteristic
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#392C2212yes—
27C2×#392C2212yes—
28#2361—216no—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#382C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48#1445C2212yes—
49#1445C2211yescharacteristic
50#1444C2211yescharacteristic
51#1444C2212yes—
52#1444C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2211yescharacteristic
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2212yes—
65#1438C2214yes—
66#1438C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×D4C2312yes—
78C23×D4C2314yes—
79C23×D4C2312yes—
80C23×D4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2314yes—
84C2×#127—28no—
85C2×#127—42no—
86C2×#123—216no—
87C2×#123—48no—
88C2×#123—48no—
89C2×#121—28no—
90C2×#121—28no—
91C2×#121—28no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—216no—
95C2×C4×D4—44no—
96C2×C4×D4—48no—
97C2×C4×D4—48no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—42no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130#399—24no—
131#394—216no—
132#394—216no—
133#392—28no—
134#392—44no—
135#390—28no—
136#390—28no—
137#390—28no—
138#390D4116yescomplemented
139#389—28no—
140#389—28no—
141#388—216no—
142#386—216no—
143#386—216no—
144#385—216no—
145#385—48no—
146#385—44no—
147#382—28no—
148#380—24no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—42no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—42no—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C25C2414yes—
177C25C2411yescharacteristic
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4C2×D418yes—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—42no—
218C23×C4#4114yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#127—416no—
233#123—464no—
234#121—264no—
235C4×D4—464no—
236C4×D4—432no—
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—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4#4118yes—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×D4116yes—
309C2×C22⋊C4C2×D4116yes—
310C2×C22⋊C4#4114yes—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D418yes—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—22no—
336#95—41no—
337#95—42no—
338#95—44no—
339#95—41no—
340#95—41no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—42no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24C22×D418yes—
369C24C2×#4112yes—
370C24C2×#4112yes—
371C24H(5,𝔽2)12yes—
372C24C2511yescharacteristic
373C2×D4—232no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—416no—
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—84no—
396C2×D4—84no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—216no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—48no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—88no—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D418yes—
482C22×C4C22×D414yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4118yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+412yes—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C22⋊C4—216no—
504C22⋊C4—232no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—44no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C42—216no—
519C42—416no—
520C42—48no—
521C42—44no—
522C42—44no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23#55914yes—
581C23C23×D412yes—
582C23C22×#4111yescharacteristic
583C23C22×#4112yes—
584C23C22×#4111yescharacteristic
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)12yes—
588C23C2×H(5,𝔽2)11yescharacteristic
589C23C2×2−1+411yescharacteristic
590C23#58512yes—
591C23#58511yescharacteristic
592D4—464no—
593D4—432no—
594D4—816no—
595D4—816no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—416no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—816no—
672C2×C4—832no—
673C2×C4#52518yes—
674C2×C4#53914yes—
675C2×C4D4214yes—
676C2×C4#54718yes—
677C2×C4#55914yes—
678C22—24no—
679C22—28no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—42no—
689C22—48no—
690C22—48no—
691C22—48no—
692C22—48no—
693C22—48no—
694C22—48no—
695C22—48no—
696C22—84no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22—84no—
703C22—84no—
704C22C2×#52512yes—
705C22C2×#53911yescharacteristic
706C22C2×D4211yescharacteristic
707C22C2×#54712yes—
708C22D4×#4112yes—
709C22C2×#55911yescharacteristic
710C22C2×#55911yescharacteristic
711C22#348112yes—
712C22#348412yes—
713C22#348412yes—
714C22#350012yes—
715C22#350511yescharacteristic
716C22#350612yes—
717C22#350611yescharacteristic
718C22#351111yescharacteristic
719C22#351312yes—
720C22#351712yes—
721C22#351712yes—
722C22#353711yescharacteristic
723C22#353711yescharacteristic
724C22#354712yes—
725C22#355111yescharacteristic
726C22#355911yescharacteristic
727C4—48no—
728C4—44no—
729C4—42no—
730C4—44no—
731C4—48no—
732C4—42no—
733C4—42no—
734C4—44no—
735C4—42no—
736C4—81no—
737C4—84no—
738C4—82no—
739C4#245518yes—
740C2—44no—
741C2—88no—
742C2C2×#245511yescharacteristic
743C2#3570412yes—
744C2#3644112yes—
745C2#3976711yescharacteristic
746C2#4252211yescharacteristic
747C2#4487312yes—
748C2#4531511yescharacteristic
749C2#4535511yescharacteristic
750C2#5911712yes—
751C2#5963211yescharacteristic
752C2#5981811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
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×#2445C2semidirect
2. C2×#2361C2semidirect
3. #16338C2semidirect
4. #15949C2semidirect
5. #15782C2semidirect
6. #15405C2semidirect
7. #15219C2semidirect
9. #14721C2semidirect
10. #14171C2semidirect
11. #14150C2semidirect
12. #14130C2semidirect
13. #14018C2semidirect
14. #14002C2semidirect
15. C4×#399C2semidirect
138. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24452
C2×#23612
#163382
#159492
#157822
#154052
#152192
#151242
#147212
#141712
#141502
#141302
#140182
#140022
C4×#3992