Finite Group Databasebeta

← #719807

Subgroups — #719807

Order 512.

3314 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×#2445C212yesmaximal
2C2×#2349C211yesmaximal, characteristic, complemented
3C2×#2328C211yesmaximal, characteristic
4#16401C214yesmaximal, complemented
5#15383C214yesmaximal, complemented
6#15269C214yesmaximal, complemented
7#15221C212yesmaximal, complemented
8#15138C212yesmaximal, complemented
9#14927C214yesmaximal, complemented
10#14276C211yesmaximal, characteristic, complemented
11#13941C211yesmaximal, characteristic, complemented
12#13916C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic, complemented
15#13848C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#123C2214yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2445—216no—
22C2×#397C2214yes—
23C2×#394C2214yes—
24C2×#394C2214yes—
25#2349—28no—
26#2328—28no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#381C2214yes—
42C2×#380C2212yes—
43C2×#380C2214yes—
44C2×#380C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49C22×#95C2212yes—
50#1445C2214yes—
51#1444C2214yes—
52#1444C2212yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C4×#95C2212yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2314yes—
81C24×C4C2314yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×#127—24no—
85C2×#127—24no—
86C2×#127D418yes—
87C2×#123—216no—
88C2×#123—48no—
89C2×#123—48no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#397—48no—
128#394—216no—
129#394—216no—
130#390—24no—
131#390—28no—
132#390—28no—
133#390—28no—
134#390—24no—
135#389—28no—
136#389—28no—
137#386—216no—
138#386—216no—
139#386—216no—
140#385—216no—
141#384—24no—
142#382—24no—
143#382—24no—
144#381—48no—
145#380—28no—
146#380—48no—
147#380—48no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151#376—24no—
152C2×#95—28no—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C25C2412yes—
179C25C2412yes—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4C2×D414yes—
197C22×D4C2×D414yes—
198C22×D4C2×D414yes—
199C22×D4C2×D4116yes—
200C22×D4#4114yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C2×D418yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#127—232no—
232#123—464no—
233C4×D4—232no—
234C4×D4—432no—
235C4×D4—416no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—44no—
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—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D418yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42#4114yes—
317C2×C42#4114yes—
318#95—22no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—22no—
330#95—22no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—44no—
336#95—42no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24C22×D414yes—
362C24C22×D414yes—
363C24C22×D414yes—
364C24C2×#4114yes—
365C24C2511yescharacteristic
366C2×D4—232no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4#121116yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
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—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—216no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C4⋊C4—216no—
488C4⋊C4—28no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—28no—
511C42—48no—
512C42—44no—
513C42—48no—
514C42#123116yescomplemented
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—44no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23C2×#12118yes—
561C23C23×D411yescharacteristic
562C23C23×D411yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4111yescharacteristic
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×2−1+411yescharacteristic
569C23C2×2−1+411yescharacteristic
570C23#58512yes—
571C23#58512yes—
572D4—232no—
573D4—416no—
574D4—832no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—816no—
650C2×C4—816no—
651C2×C4—832no—
652C2×C4C2×#12114yes—
653C2×C4C2×#12314yes—
654C2×C4C2×#12314yes—
655C2×C4C2×#12314yes—
656C2×C4#53614yes—
657C2×C4#54714yes—
658C2×C4#54714yes—
659C2×C4#55214yes—
660C2×C4#55214yes—
661C22—216no—
662C22—28no—
663C22—44no—
664C22—44no—
665C22—48no—
666C22—44no—
667C22—48no—
668C22—44no—
669C22—44no—
670C22—48no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—48no—
675C22—48no—
676C22—44no—
677C22—82no—
678C22—84no—
679C22—82no—
680C22—82no—
681C22—84no—
682C22—82no—
683C22C22×#12111yescharacteristic
684C22C22×#12311yescharacteristic
685C22C2×#53611yescharacteristic
686C22#346012yes—
687C22C2×#54711yescharacteristic
688C22C2×#54711yescharacteristic
689C22C2×#55211yescharacteristic
690C22C2×#55211yescharacteristic
691C22#348412yes—
692C22#348512yes—
693C22#348812yes—
694C22#349412yes—
695C22#349612yes—
696C22#350811yescharacteristic
697C22#350811yescharacteristic
698C22#351011yescharacteristic
699C22#351011yescharacteristic
700C22#351312yes—
701C22#351412yes—
702C22#354712yes—
703C22#354812yes—
704C22#355712yes—
705C22#355912yes—
706C4—24no—
707C4—24no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—84no—
713C4—82no—
714C4—82no—
715C4—84no—
716C4—82no—
717C4—82no—
718C4#248618yes—
719C2—48no—
720C2—84no—
721C2C2×#248611yescharacteristic
722C2#3525411yescharacteristic
723C2#3554111yescharacteristic
724C2#3576112yes—
725C2#3776412yes—
726C2#3822812yes—
727C2#4243912yes—
728C2#4493111yescharacteristic
729C2#4505611yescharacteristic
730C2#4513711yescharacteristic
731C2#4521111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2349C2semidirect
4. #16401C2semidirect
5. #15383C2semidirect
6. #15269C2semidirect
7. #15221C2semidirect
8. #15138C2semidirect
9. #14927C2semidirect
10. #14276C2semidirect
11. #13941C2semidirect
12. #13916C2semidirect
13. #13906C2semidirect
14. #13906C2semidirect
15. #13848C2semidirect
514. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24452
C2×#23492
C2×#23282
#164012
#153832
#152692
#152212
#151382
#149272
#142762
#139412
#139162
#139062
#138482