Finite Group Databasebeta

← #793207

Subgroups — #793207

Order 512.

2916 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2495C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#16965C212yesmaximal, complemented
4#15527C214yesmaximal, complemented
5#15514C214yesmaximal, complemented
6#15242C211yesmaximal, characteristic, complemented
7#15181C211yesmaximal, characteristic, complemented
8#15150C211yesmaximal, characteristic, complemented
9#14894C212yesmaximal, complemented
10#14795C212yesmaximal
11#13993C212yesmaximal, complemented
12#13959C212yesmaximal, complemented
13#13905C212yesmaximal, complemented
14#13881C212yesmaximal, complemented
15#13872C212yesmaximal, complemented
16C4×#392C211yesmaximal, characteristic
17#13810C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2495—28no—
23C2×#400C2212yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#392C2211yescharacteristic
28D4×C4⋊C4—28no—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#388C2212yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#384C2211yescharacteristic
36C2×#383C2212yes—
37C2×#382C2211yescharacteristic
38C2×#382C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×#376C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1445C2212yes—
55#1444C2211yescharacteristic
56#1443C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1439C2211yescharacteristic
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2211yescharacteristic
68#1438C2212yes—
69#1438C2212yes—
70#1438C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2212yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—44no—
97C2×C4×D4—44no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132#400—28no—
133#397—28no—
134#397—44no—
135#397—44no—
136#392—24no—
137#389—24no—
138#389—28no—
139#389—24no—
140#388—28no—
141#386—48no—
142#385—216no—
143#384—24no—
144#383—28no—
145#382—24no—
146#382—44no—
147#380—28no—
148#380—28no—
149#380—24no—
150#380—24no—
151#380—44no—
152#380—44no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4D418yescomplemented
155C4×C22⋊C4—24no—
156C4×C22⋊C4—28no—
157#376—44no—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C25C2411yescharacteristic
188C25C2412yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4#4114yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4#4118yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C4×D4—232no—
236C4×D4—432no—
237C4×D4—432no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—432no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
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—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—28no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D414yes—
322C2×C42C2×D418yes—
323C2×C42#4118yes—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—44no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95C2×D418yes—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—21no—
374C24—22no—
375C24C2×#4114yes—
376C24C2×#4114yes—
377C24H(5,𝔽2)12yes—
378C242−1+414yes—
379C24C2511yescharacteristic
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—216no—
385C2×D4—28no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4—88no—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C22⋊C4—216no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—84no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C42—28no—
525C42—216no—
526C42—216no—
527C42—416no—
528C42—44no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—42no—
578C23—42no—
579C23—41no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23—42no—
584C23—41no—
585C23—42no—
586C23#55218yes—
587C23C23×D411yescharacteristic
588C23C22×#4112yes—
589C23C22×#4112yes—
590C23C22×#4111yescharacteristic
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×H(5,𝔽2)11yescharacteristic
593C23C2×H(5,𝔽2)11yescharacteristic
594C23C2×H(5,𝔽2)11yescharacteristic
595C23C2×2−1+412yes—
596C23#58512yes—
597C23#58511yescharacteristic
598D4—232no—
599D4—432no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—432no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—832no—
683C2×C4—88no—
684C2×C4#52514yes—
685C2×C4#53214yes—
686C2×C4#53514yes—
687C2×C4#54614yes—
688C2×C4#55118yes—
689C2×C4#55318yes—
690C22—28no—
691C22—28no—
692C22—28no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—42no—
709C22—42no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—82no—
715C22—82no—
716C22C2×#52511yescharacteristic
717C22C2×#53211yescharacteristic
718C22C2×#53511yescharacteristic
719C22C2×#54611yescharacteristic
720C22C2×#55112yes—
721C22C2×#55212yes—
722C22C2×#55312yes—
723C22#347512yes—
724C22#348611yescharacteristic
725C22#348811yescharacteristic
726C22#349212yes—
727C22#349212yes—
728C22#349412yes—
729C22#349512yes—
730C22#350011yescharacteristic
731C22#350612yes—
732C22#351212yes—
733C22#351411yescharacteristic
734C22#351911yescharacteristic
735C22#354112yes—
736C22#354311yescharacteristic
737C22#354611yescharacteristic
738C22#356112yes—
739C4—24no—
740C4—44no—
741C4—44no—
742C4—44no—
743C4—44no—
744C4—42no—
745C4—44no—
746C4—82no—
747C4—81no—
748C4—81no—
749C4—81no—
750C4—84no—
751C4—82no—
752C4—84no—
753C4#260318yes—
754C2—48no—
755C2—82no—
756C2C2×#260311yescharacteristic
757C2#3592611yescharacteristic
758C2#3608311yescharacteristic
759C2#3616311yescharacteristic
760C2#3641511yescharacteristic
761C2#3891011yescharacteristic
762C2#4278811yescharacteristic
763C2#4492212yes—
764C2#4525012yes—
765C2#5920512yes—
766C2#5930912yes—
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
1. C2×#2495C2semidirect
2. C2×D4×C4⋊C4C2semidirect
3. #16965C2semidirect
4. #15527C2semidirect
5. #15514C2semidirect
6. #15242C2semidirect
7. #15181C2semidirect
8. #15150C2semidirect
9. #14894C2semidirect
11. #13993C2semidirect
12. #13959C2semidirect
13. #13905C2semidirect
14. #13881C2semidirect
15. #13872C2semidirect
17. #13810C2semidirect
154. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24952
C2×D4×C4⋊C42
#169652
#155272
#155142
#152422
#151812
#151502
#148942
#147952
#139932
#139592
#139052
#138812
#138722
C4×#3922
#138102