Finite Group Databasebeta

← #1395933

Subgroups — #1395933

Order 512.

2832 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×#2420C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#16698C212yesmaximal, complemented
4#16395C212yesmaximal, complemented
5#16306C214yesmaximal, complemented
6#16271C214yesmaximal, complemented
7#15232C212yesmaximal, complemented
8#15195C211yesmaximal, characteristic, complemented
9#15194C211yesmaximal, characteristic, complemented
10#15145C211yesmaximal, characteristic, complemented
11#15135C212yesmaximal, complemented
12#14777C212yesmaximal, complemented
13#14263C212yesmaximal, complemented
14#13977C212yesmaximal, complemented
15#13970C212yesmaximal
16#13916C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2420—28no—
23C2×#399C2214yes—
24C2×#398C2214yes—
25C2×#397C2211yescharacteristic
26C2×#397C2212yes—
27C2×#394C2212yes—
28D4×C4⋊C4—28no—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#386C2212yes—
35C2×#386C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#382C2211yescharacteristic
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1445C2211yescharacteristic
53#1445C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2212yes—
58#1443C2214yes—
59#1443C2212yes—
60#1443C2214yes—
61#1442C2211yescharacteristic
62#1442C2212yes—
63#1442C2212yes—
64#1442C2214yes—
65#1441C2214yes—
66#1440C2212yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2211yescharacteristic
73#1439C2214yes—
74#1439C2212yes—
75#1438C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×#121—24no—
91C2×#121—28no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—44no—
95C2×C4×D4—44no—
96C2×C4×D4D418yes—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—48no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#399—48no—
131#398—48no—
132#397—28no—
133#397—24no—
134#394—28no—
135#390—24no—
136#390—28no—
137#389—28no—
138#389—24no—
139#389—44no—
140#386—28no—
141#386—48no—
142#385—216no—
143#385—48no—
144#385—48no—
145#384—24no—
146#384—44no—
147#382—24no—
148#381—28no—
149#381—28no—
150#380—28no—
151#380—24no—
152#380—44no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—44no—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C25C2411yescharacteristic
186C25C2412yes—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—42no—
194C22×D4—44no—
195C22×D4C2×D414yes—
196C22×D4#4118yes—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C2×D418yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#121—416no—
230C4×D4—232no—
231C4×D4—432no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—432no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D414yes—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D418yes—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42C2×D414yes—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—22no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—42no—
333#95—42no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95#4114yes—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24C22×D414yes—
362C24C2×#4118yes—
363C242−1+412yes—
364C24C2511yescharacteristic
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—216no—
368C2×D4—216no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—48no—
372C2×D4—84no—
373C2×D4—88no—
374C2×D4#123116yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—48no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4114yes—
473C22×C4H(5,𝔽2)14yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+412yes—
478C22×C42−1+414yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—44no—
497C22⋊C4—416no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—88no—
506C42—28no—
507C42—48no—
508C42—44no—
509C42—48no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—24no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—44no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23C2×#12318yes—
556C23C23×D411yescharacteristic
557C23C22×#4112yes—
558C23C22×#4111yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)12yes—
563C23C2×2−1+412yes—
564C23C2×2−1+411yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23#58511yescharacteristic
567D4—232no—
568D4—816no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—432no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—832no—
652C2×C4—88no—
653C2×C4C2×#12314yes—
654C2×C4#53514yes—
655C2×C4#53614yes—
656C2×C4#53918yes—
657C2×C4#53918yes—
658C2×C4#54714yes—
659C22—28no—
660C22—28no—
661C22—28no—
662C22—44no—
663C22—48no—
664C22—44no—
665C22—48no—
666C22—48no—
667C22—44no—
668C22—48no—
669C22—44no—
670C22—42no—
671C22—44no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—81no—
676C22—81no—
677C22—82no—
678C22—81no—
679C22—81no—
680C22—82no—
681C22C22×#12311yescharacteristic
682C22C2×#53511yescharacteristic
683C22C2×#53611yescharacteristic
684C22C2×#53912yes—
685C22C2×#53912yes—
686C22C2×#54711yescharacteristic
687C22#348112yes—
688C22#348411yescharacteristic
689C22#348911yescharacteristic
690C22#349811yescharacteristic
691C22#351112yes—
692C22#351312yes—
693C22#351711yescharacteristic
694C22#352011yescharacteristic
695C22#352012yes—
696C22#352012yes—
697C22#353712yes—
698C22#354312yes—
699C22#354412yes—
700C22#354711yescharacteristic
701C22#354712yes—
702C22#355211yescharacteristic
703C22#355212yes—
704C4—24no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—42no—
709C4—44no—
710C4—82no—
711C4—81no—
712C4—81no—
713C4—82no—
714C4—82no—
715C4—84no—
716C4—84no—
717C4—81no—
718C4#262318yes—
719C2—48no—
720C2—82no—
721C2C2×#262311yescharacteristic
722C2#3598711yescharacteristic
723C2#3623711yescharacteristic
724C2#3892011yescharacteristic
725C2#3948911yescharacteristic
726C2#4005212yes—
727C2#4017712yes—
728C2#4243011yescharacteristic
729C2#4248511yescharacteristic
730C2#4255912yes—
731C2#4260112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×D4×C4⋊C4C2semidirect
3. #16698C2semidirect
4. #16395C2semidirect
5. #16306C2semidirect
6. #16271C2semidirect
7. #15232C2semidirect
8. #15195C2semidirect
9. #15194C2semidirect
10. #15145C2semidirect
11. #15135C2semidirect
12. #14777C2semidirect
13. #14263C2semidirect
14. #13977C2semidirect
16. #13916C2semidirect
17. C4×#397C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24202
C2×D4×C4⋊C42
#166982
#163952
#163062
#162712
#152322
#151952
#151942
#151452
#151352
#147772
#142632
#139772
#139702
#139162
C4×#3972