Finite Group Databasebeta

← #4388150

Subgroups — #4388150

Order 512.

3846 conjugacy classes of subgroups.

The subgroup lattice has 784 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C211yesmaximal, characteristic, complemented
2C2×#2405C211yesmaximal, characteristic
3C2×#2405C211yesmaximal, characteristic
4C2×#2312C211yesmaximal, characteristic, complemented
5#16257C214yesmaximal, complemented
6#15921C214yesmaximal, complemented
7#15612C214yesmaximal, complemented
8#15298C214yesmaximal, complemented
9#15297C214yesmaximal, complemented
10#14838C214yesmaximal, complemented
11#14673C211yesmaximal, characteristic, complemented
12#14341C211yesmaximal, characteristic, complemented
13#14335C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C22×#123C2212yes—
16C22×#123C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2483—28no—
20#2405—28no—
21#2405—28no—
22C2×#398C2214yescomplemented
23C2×#397C2214yes—
24C2×#396C2214yescomplemented
25C2×#394C2214yes—
26C2×#394C2214yes—
27#2312—28no—
28C2×#388C2214yes—
29C2×#388C2214yescomplemented
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yescomplemented
35C2×#385C2212yescomplemented
36C2×#381C2212yes—
37C2×#381C2212yescomplemented
38C2×#381C2212yescomplemented
39C2×#380C2212yescomplemented
40C2×#379C2211yescharacteristic
41C2×#379C2211yescharacteristic
42C2×#379C2211yescharacteristic
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×#379C2211yescharacteristic
47C2×C4×C22⋊C4C2212yescomplemented
48#1444C2214yescomplemented
49#1443C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1440C2214yes—
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2214yescomplemented
57#1439C2214yes—
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1438C2214yes—
64#1438C2214yes—
65#1437C2214yescomplemented
66#1437C2214yescomplemented
67C4⋊#95C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yescomplemented
70C22⋊#95C2212yescomplemented
71C26C2311yescharacteristic
72C26C2311yescharacteristic
73C23×D4—24no—
74C23×D4—24no—
75C23×D4—24no—
76C23×D4—44no—
77C23×D4—44no—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#123—28no—
86C2×#123—28no—
87C2×#123—44no—
88C2×#123—44no—
89C2×#123—44no—
90C2×#123—44no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4D418yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2314yes—
125#398—48no—
126#397—216no—
127#396—48no—
128#394—216no—
129#394—216no—
130#388—216no—
131#388—216no—
132#386—28no—
133#386—28no—
134#385—28no—
135#385—28no—
136#385—28no—
137#385—28no—
138#381—28no—
139#381—28no—
140#381—44no—
141#380—28no—
142#379—24no—
143#379—24no—
144#379—24no—
145#379—28no—
146#379—28no—
147#379—44no—
148#379D418yes—
149C4×C22⋊C4—28no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25—22no—
167C25—24no—
168C25—22no—
169C25—22no—
170C25—42no—
171C25—42no—
172C25—42no—
173C25—42no—
174C25—42no—
175C25—42no—
176C25—42no—
177C25—42no—
178C25—42no—
179C25—42no—
180C25—42no—
181C25C2×D414yes—
182C25C2×D414yes—
183C25#4114yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—416no—
203C22×D4—432no—
204C22×D4—432no—
205C22×D4—416no—
206C22×D4—416no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241#123—432no—
242#123—432no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C22⋊C4—232no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
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—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—416no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4C2×D418yes—
322C2×C22⋊C4C2×D418yes—
323C2×C22⋊C4C2×D418yes—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×D418yes—
326C2×C42—28no—
327C2×C42—44no—
328C2×C42—44no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—28no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—48no—
375C24—48no—
376C24—44no—
377C24—44no—
378C24—48no—
379C24—48no—
380C24—48no—
381C24—48no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24—44no—
386C24—48no—
387C24—48no—
388C24—44no—
389C24—48no—
390C24—48no—
391C24—44no—
392C24—84no—
393C24—84no—
394C24—82no—
395C24—82no—
396C24—82no—
397C24—82no—
398C24—88no—
399C24—88no—
400C24—88no—
401C24—88no—
402C24—88no—
403C24—88no—
404C24—88no—
405C24—88no—
406C24#121116yes—
407C24C22×D412yes—
408C24C22×D412yes—
409C24C22×D412yes—
410C24C22×D412yes—
411C24C22×D412yes—
412C24C22×D412yes—
413C24C2×#4112yes—
414C24C2×#4112yes—
415C24C2×#4112yes—
416C24H(5,𝔽2)14yes—
417C24C2511yescharacteristic
418C2×D4—216no—
419C2×D4—216no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—4128no—
431C2×D4—48no—
432C2×D4—48no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—84no—
498C22×C4—84no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—84no—
504C22×C4—84no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4C22×D414yes—
510C22×C4C22×D418yes—
511C22×C4C22×D414yes—
512C22×C4H(5,𝔽2)14yes—
513C4⋊C4—416no—
514C4⋊C4—416no—
515C4⋊C4—88no—
516C4⋊C4—88no—
517C4⋊C4—88no—
518C22⋊C4—216no—
519C22⋊C4—216no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—88no—
533C22⋊C4—88no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C42—88no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—28no—
581C23—28no—
582C23—416no—
583C23—416no—
584C23—416no—
585C23—416no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—42no—
599C23—44no—
600C23—42no—
601C23—42no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—88no—
617C23—88no—
618C23—88no—
619C23—88no—
620C23—88no—
621C23—88no—
622C23—88no—
623C23—832no—
624C23—832no—
625C23—88no—
626C23—832no—
627C23—832no—
628C23—88no—
629C23—88no—
630C23C2×#12114yes—
631C23C2×#12114yes—
632C23C2×#12114yes—
633C23C23×D411yescharacteristic
634C23C23×D411yescharacteristic
635C23C22×#4111yescharacteristic
636C23C2×H(5,𝔽2)12yes—
637C23C2×H(5,𝔽2)12yes—
638C23#58512yes—
639C23#58512yes—
640C23#58512yes—
641C23#58512yes—
642D4—432no—
643D4—432no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—24no—
654C2×C4—416no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
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—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—88no—
701C2×C4—88no—
702C2×C4—88no—
703C2×C4—88no—
704C22—24no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—24no—
709C22—24no—
710C22—24no—
711C22—24no—
712C22—24no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—48no—
723C22—48no—
724C22—48no—
725C22—48no—
726C22—48no—
727C22—48no—
728C22—42no—
729C22—42no—
730C22—42no—
731C22—44no—
732C22—44no—
733C22—48no—
734C22—48no—
735C22—44no—
736C22—832no—
737C22—832no—
738C22—82no—
739C22—82no—
740C22—82no—
741C22—82no—
742C22—84no—
743C22—84no—
744C22C22×#12111yescharacteristic
745C22#344912yes—
746C22#348412yes—
747C22#348412yes—
748C22#348512yes—
749C22#348512yes—
750C22#348512yes—
751C22#349011yescharacteristic
752C22#349011yescharacteristic
753C22#349111yescharacteristic
754C22#349111yescharacteristic
755C22#349111yescharacteristic
756C22#349111yescharacteristic
757C22#354514yes—
758C22#355114yes—
759C22#356014yes—
760C22#356214yes—
761C4—44no—
762C4—44no—
763C4—44no—
764C4—48no—
765C4—82no—
766C4—82no—
767C4—84no—
768C4—84no—
769C4—84no—
770C2—44no—
771C2—44no—
772C2—44no—
773C2—84no—
774C2#3508911yescharacteristic
775C2#3534811yescharacteristic
776C2#3544711yescharacteristic
777C2#4141514yes—
778C2#4638811yescharacteristic
779C2#4639011yescharacteristic
780C2#4683711yescharacteristic
781C2#4719511yescharacteristic
782C2#4877414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2483C2semidirect
4. C2×#2312C2semidirect
5. #16257C2semidirect
6. #15921C2semidirect
7. #15612C2semidirect
8. #15298C2semidirect
9. #15297C2semidirect
10. #14838C2semidirect
11. #14673C2semidirect
12. #14341C2semidirect
22. C2×#398C22semidirect
24. C2×#396C22semidirect
29. C2×#388C22semidirect
34. C2×#385C22semidirect
35. C2×#385C22semidirect
37. C2×#381C22semidirect
38. C2×#381C22semidirect
39. C2×#380C22semidirect
47. C2×C4×C22⋊C4C22semidirect
48. #1444C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. C4⋊#95C22semidirect
68. #1434C22semidirect
69. C22⋊#95C22semidirect
70. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C2×#24052
C2×#23122
#162572
#159212
#156122
#152982
#152972
#148382
#146732
#143412
#143352