Finite Group Databasebeta

← #723015

Subgroups — #723015

Order 512.

2612 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2324C211yesmaximal, characteristic
3#17066C212yesmaximal, complemented
4#17007C212yesmaximal, complemented
5#15431C214yesmaximal, complemented
6#15326C214yesmaximal, complemented
7#15244C212yesmaximal, complemented
8#15229C212yesmaximal, complemented
9#15216C212yesmaximal, complemented
10#15047C212yesmaximal, complemented
11#14340C211yesmaximal, characteristic, complemented
12#13998C212yesmaximal
13#13941C211yesmaximal, characteristic, complemented
14#13918C212yesmaximal, complemented
15#13906C211yesmaximal, characteristic, complemented
16#13852C211yesmaximal, characteristic, complemented
17#13804C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2552—28no—
23C2×#398C2214yes—
24#2324—28no—
25C2×#390C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2214yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#386C2212yes—
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×#382C2211yescharacteristic
39C2×#382C2214yes—
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2212yes—
55#1441C2211yescharacteristic
56#1441C2211yescharacteristic
57#1439C2212yes—
58#1439C2212yes—
59#1439C2211yescharacteristic
60#1439C2211yescharacteristic
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#125—28no—
90C2×#125—44no—
91C2×#125—44no—
92C2×#121—24no—
93C2×#121—24no—
94C2×#121Q818yes—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—24no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#398—216no—
138#390—24no—
139#390—28no—
140#390—48no—
141#390—44no—
142#390—44no—
143#389—24no—
144#389—28no—
145#389—28no—
146#388—28no—
147#386—28no—
148#385—28no—
149#384—28no—
150#384—24no—
151#382—24no—
152#382—48no—
153#380—24no—
154#380—48no—
155#380—44no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4D418yes—
160C4×C22⋊C4—44no—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2411yescharacteristic
187C25C2412yes—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4C2×Q814yes—
199C22×D4C2×Q818yes—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×Q814yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#125—432no—
228#121—232no—
229C4×D4—416no—
230#117—416no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
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—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
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⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4#4114yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
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⋊C4—44no—
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×C22⋊C4—44no—
312C2×C22⋊C4C2×Q818yes—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42C2×D418yes—
327C2×C42C2×D414yes—
328C2×C42#4114yes—
329C2×C42#4114yes—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—22no—
335#95—28no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—44no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—42no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C2×D418yes—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—24no—
364C24—22no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—22no—
375C24C22×Q814yes—
376C24C22×Q812yes—
377C24C2511yescharacteristic
378C2×D4—28no—
379C2×D4—216no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—416no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—88no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—416no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q814yes—
481C22×C4C22×Q812yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C4⋊C4—28no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4C22⋊Q8116yes—
509C22⋊C4—216no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C42—48no—
524C42—416no—
525C42—44no—
526C42—48no—
527C42—48no—
528C42#124116yes—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—24no—
538C23—28no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—42no—
570C23—44no—
571C23—42no—
572C23—44no—
573C23—44no—
574C23—42no—
575C23C23×D411yescharacteristic
576C23C23×Q811yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4111yescharacteristic
579C23C22×#4111yescharacteristic
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)12yes—
583C23#58512yes—
584C23#58512yes—
585C23#58512yes—
586D4—416no—
587D4—816no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—416no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—816no—
671C2×C4C2×C22⋊Q818yes—
672C2×C4C2×C22⋊Q814yes—
673C2×C4C2×#12414yes—
674C2×C4C2×#12414yes—
675C2×C4C2×#12414yes—
676C2×C4#53514yes—
677C2×C4#55414yes—
678C2×C4#55414yes—
679C2×C4#55514yes—
680C2×C4#55514yes—
681C22—44no—
682C22—48no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22—48no—
687C22—44no—
688C22—42no—
689C22—42no—
690C22—44no—
691C22—81no—
692C22—82no—
693C22—81no—
694C22—82no—
695C22—84no—
696C22—82no—
697C22—81no—
698C22—82no—
699C22—82no—
700C22—81no—
701C22—82no—
702C22—84no—
703C22C22×C22⋊Q811yescharacteristic
704C22C22×#12411yescharacteristic
705C22C2×#53511yescharacteristic
706C22#346012yes—
707C22C2×#55411yescharacteristic
708C22C2×#55411yescharacteristic
709C22C2×#55511yescharacteristic
710C22C2×#55511yescharacteristic
711C22#348511yescharacteristic
712C22#348711yescharacteristic
713C22#349412yes—
714C22#349811yescharacteristic
715C22#350011yescharacteristic
716C22#350112yes—
717C22#350412yes—
718C22#351212yes—
719C22#351212yes—
720C22#351312yes—
721C22#352412yes—
722C22#353912yes—
723C22#354212yes—
724C22#354612yes—
725C22#356212yes—
726C4—24no—
727C4—24no—
728C4—28no—
729C4—42no—
730C4—44no—
731C4—44no—
732C4—81no—
733C4—82no—
734C4—84no—
735C4—84no—
736C4—82no—
737C4—81no—
738C4—82no—
739C4—82no—
740C4#256318yes—
741C2—82no—
742C2—84no—
743C2C2×#256311yescharacteristic
744C2#3526411yescharacteristic
745C2#3564911yescharacteristic
746C2#3583611yescharacteristic
747C2#3604011yescharacteristic
748C2#3828812yes—
749C2#3969212yes—
750C2#4265211yescharacteristic
751C2#4287611yescharacteristic
752C2#4494212yes—
753C2#4505212yes—
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
1. C2×#2552C2semidirect
3. #17066C2semidirect
4. #17007C2semidirect
5. #15431C2semidirect
6. #15326C2semidirect
7. #15244C2semidirect
8. #15229C2semidirect
9. #15216C2semidirect
10. #15047C2semidirect
11. #14340C2semidirect
13. #13941C2semidirect
14. #13918C2semidirect
15. #13906C2semidirect
16. #13852C2semidirect
17. #13804C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23242
#170662
#170072
#154312
#153262
#152442
#152292
#152162
#150472
#143402
#139982
#139412
#139182
#139062
#138522
#138042