Finite Group Databasebeta

← #4457315

Subgroups — #4457315

Order 512.

2990 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic
3#17005C212yesmaximal, complemented
4#16986C212yesmaximal, complemented
5#16892C212yesmaximal, complemented
6#16815C212yesmaximal, complemented
7#16426C214yesmaximal, complemented
8#16308C214yesmaximal, complemented
9#15226C212yesmaximal, complemented
10#15186C212yesmaximal, complemented
11#15180C212yesmaximal, complemented
12#15147C212yesmaximal, complemented
13#14258C211yesmaximal, characteristic, complemented
14#14139C211yesmaximal, characteristic, complemented
15#14130C211yesmaximal, characteristic, complemented
16#13937C211yesmaximal, characteristic
17#13844C211yesmaximal, characteristic, complemented
18C23×#41C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2432—28no—
25C2×#397C2214yes—
26C2×#394C2212yes—
27C2×#394C2212yes—
28#2311—28no—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#386C2214yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#1445C2214yes—
49#1444C2211yescharacteristic
50#1444C2211yescharacteristic
51#1444C2212yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2211yescharacteristic
55#1442C2211yescharacteristic
56#1442C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1438C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C22×#41—24no—
86C22×#41—24no—
87C22×#41—44no—
88C22×#41—44no—
89C22×#41D418yes—
90C23×Q8C2311yescharacteristic
91C23×Q8C2311yescharacteristic
92C23×D4C2311yescharacteristic
93C23×D4C2312yes—
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C24×C4C2312yes—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—44no—
99C2×#124—24no—
100C2×#124—44no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×#117—24no—
105C2×#117—44no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—44no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#397—48no—
139#394—28no—
140#394—28no—
141#391—28no—
142#391—24no—
143#391—44no—
144#391—44no—
145#391—44no—
146#390—28no—
147#390—24no—
148#390—44no—
149#389—44no—
150#386—216no—
151#385—28no—
152#385—28no—
153#385—44no—
154#380—28no—
155#380—28no—
156#380—24no—
157C4×C4⋊C4—44no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—24no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C2×#41—232no—
191C2×#41—416no—
192C2×#41—416no—
193C2×#41—432no—
194C2×#41—432no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8C2×D414yes—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4C2×D414yes—
214C22×D4C2×D418yes—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—42no—
224C23×C4—44no—
225C23×C4—42no—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4#4114yes—
229C23×C4#4114yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C4⋊Q8—416no—
246#124—416no—
247C22⋊Q8—416no—
248#117—416no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—416no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331#95—22no—
332#95—22no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—44no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—44no—
347#95—44no—
348#95—42no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24C22×D414yes—
378C24C22×D412yes—
379C24C2×#4112yes—
380C24C2×#4112yes—
381C24C2511yescharacteristic
382#41—4128no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—28no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×D4—216no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—48no—
402C2×D4—48no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—216no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—416no—
488C22×C4—416no—
489C22×C4—48no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4—88no—
493C22×C4—88no—
494C22×C4#123116yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)14yes—
504C22×C4H(5,𝔽2)14yes—
505C22×C42−1+414yes—
506C22×C42−1+412yes—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—88no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C22⋊C4—88no—
533C42—48no—
534C42—44no—
535C42—44no—
536C42—48no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23C2×#12314yes—
583C23C23×D411yescharacteristic
584C23C22×#4111yescharacteristic
585C23C22×#4111yescharacteristic
586C23C2×H(5,𝔽2)12yes—
587C23C2×H(5,𝔽2)12yes—
588C23C2×H(5,𝔽2)11yescharacteristic
589C23C2×2−1+411yescharacteristic
590C23C2×2−1+412yes—
591C23#58511yescharacteristic
592C23#58511yescharacteristic
593C23#58512yes—
594Q8—416no—
595Q8—88no—
596D4—432no—
597D4—416no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—232no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—432no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
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—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—816no—
684C2×C4C2×#12314yes—
685C2×C4C2×#12314yes—
686C2×C4#53414yes—
687C2×C4#53414yes—
688C2×C4#53818yes—
689C2×C4#53818yes—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—44no—
694C22—44no—
695C22—42no—
696C22—42no—
697C22—44no—
698C22—42no—
699C22—44no—
700C22—42no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—84no—
709C22—84no—
710C22—84no—
711C22—84no—
712C22C22×#12311yescharacteristic
713C22C2×#53411yescharacteristic
714C22C2×#53411yescharacteristic
715C22C2×#53812yes—
716C22C2×#53812yes—
717C22#348811yescharacteristic
718C22#348911yescharacteristic
719C22#349512yes—
720C22#349911yescharacteristic
721C22#350011yescharacteristic
722C22#350611yescharacteristic
723C22#350712yes—
724C22#350711yescharacteristic
725C22#351111yescharacteristic
726C22#351211yescharacteristic
727C22#353612yes—
728C22#354112yes—
729C22#354612yes—
730C22#354712yes—
731C22#355112yes—
732C22#355212yes—
733C22#355412yes—
734C22#356212yes—
735C4—24no—
736C4—42no—
737C4—44no—
738C4—42no—
739C4—44no—
740C4—84no—
741C4—82no—
742C4—82no—
743C4—81no—
744C4—82no—
745C4—81no—
746C4—84no—
747C4—82no—
748C4—82no—
749C4#262218yes—
750C2—44no—
751C2—84no—
752C2C2×#262211yescharacteristic
753C2#3551711yescharacteristic
754C2#3561911yescharacteristic
755C2#3733812yes—
756C2#3740212yes—
757C2#4252012yes—
758C2#4269612yes—
759C2#4290811yescharacteristic
760C2#4291711yescharacteristic
761C2#4296711yescharacteristic
762C2#4298411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #17005C2semidirect
4. #16986C2semidirect
5. #16892C2semidirect
6. #16815C2semidirect
7. #16426C2semidirect
8. #16308C2semidirect
9. #15226C2semidirect
10. #15186C2semidirect
11. #15180C2semidirect
12. #15147C2semidirect
13. #14258C2semidirect
14. #14139C2semidirect
15. #14130C2semidirect
17. #13844C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
C2×#23112
#170052
#169862
#168922
#168152
#164262
#163082
#152262
#151862
#151802
#151472
#142582
#141392
#141302
#139372
#138442