Finite Group Databasebeta

← #4456926

Subgroups — #4456926

Order 512.

2382 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2319C211yesmaximal, characteristic
3#16984C212yesmaximal, complemented
4#16980C212yesmaximal, complemented
5#16604C212yesmaximal, complemented
6#16604C212yesmaximal, complemented
7#16305C214yesmaximal, complemented
8#16302C214yesmaximal, complemented
9#15194C212yesmaximal, complemented
10#15141C212yesmaximal
11#15137C212yesmaximal, complemented
12#15134C212yesmaximal
13#14277C211yesmaximal, characteristic, complemented
14#13954C211yesmaximal, characteristic, complemented
15#13941C211yesmaximal, characteristic
16#13906C211yesmaximal, characteristic
17#13848C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2552—28no—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#396C2214yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26#2319—28no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#388C2212yes—
40C2×#384C2212yes—
41C2×#384C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#381C2212yes—
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×#376C2211yescharacteristic
51#1445C2214yes—
52#1444C2211yescharacteristic
53#1444C2211yescharacteristic
54#1444C2214yes—
55#1444C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2212yes—
59#1442C2211yescharacteristic
60#1442C2211yescharacteristic
61#1442C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2212yes—
77#1439C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#125—28no—
88C2×#125—48no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4Q818yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#398—216no—
128#398—48no—
129#396—48no—
130#396—48no—
131#396—48no—
132#390—24no—
133#390—24no—
134#390—44no—
135#390—44no—
136#389—24no—
137#389—24no—
138#389—28no—
139#389—28no—
140#389—44no—
141#389—44no—
142#389—44no—
143#388—28no—
144#388—28no—
145#384—28no—
146#384—24no—
147#382—24no—
148#381—28no—
149#380—28no—
150#380—48no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—44no—
154C4×C4⋊C4D418yes—
155#376—24no—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2412yes—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4C2×Q814yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×Q818yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213#125—432no—
214C4×D4—232no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×Q814yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×Q818yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—44no—
305C2×C42C2×D418yes—
306C2×C42C2×D414yes—
307C2×C42C2×Q814yes—
308C2×C42#4114yes—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—42no—
326#95—44no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95C2×D418yes—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24C22×Q814yes—
348C24C2511yescharacteristic
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—216no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—416no—
433C22×C4—88no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×Q814yes—
442C22×C4C22×Q812yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q812yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+414yes—
453C22×C42−1+412yes—
454C22×C42−1+412yes—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C4⋊C4—88no—
475C22⋊C4—216no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C42—48no—
485C42—48no—
486C42—44no—
487C42—44no—
488C42—48no—
489C42C22⋊Q8116yes—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23C23×D411yescharacteristic
514C23C23×Q811yescharacteristic
515C23C22×#4111yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)12yes—
519C23C2×H(5,𝔽2)12yes—
520C23C2×2−1+411yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+412yes—
523C23#58512yes—
524D4—416no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—816no—
611C2×C4C2×C22⋊Q814yes—
612C2×C4C2×C22⋊Q814yes—
613C2×C4C2×C22⋊Q814yes—
614C2×C4#53714yes—
615C2×C4#53714yes—
616C2×C4#54014yes—
617C2×C4#54114yes—
618C2×C4#54414yes—
619C2×C4#54414yes—
620C22—48no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22C22×C22⋊Q811yescharacteristic
632C22C2×#53711yescharacteristic
633C22C2×#53711yescharacteristic
634C22C2×#54011yescharacteristic
635C22C2×#54111yescharacteristic
636C22C2×#54411yescharacteristic
637C22C2×#54411yescharacteristic
638C22#348411yescharacteristic
639C22#348611yescharacteristic
640C22#349412yes—
641C22#349811yescharacteristic
642C22#350011yescharacteristic
643C22#350112yes—
644C22#350312yes—
645C22#351112yes—
646C22#353612yes—
647C22#354212yes—
648C22#354312yes—
649C22#354812yes—
650C22#355212yes—
651C22#355412yes—
652C22#355512yes—
653C22#355912yes—
654C4—24no—
655C4—24no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—84no—
667C4—82no—
668C4—84no—
669C4#262918yes—
670C2—84no—
671C2C2×#262911yescharacteristic
672C2#3552911yescharacteristic
673C2#3557411yescharacteristic
674C2#3734812yes—
675C2#3735312yes—
676C2#4242411yescharacteristic
677C2#4242511yescharacteristic
678C2#4249412yes—
679C2#4249912yes—
680C2#4279411yescharacteristic
681C2#4279511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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. #16984C2semidirect
4. #16980C2semidirect
5. #16604C2semidirect
6. #16604C2semidirect
7. #16305C2semidirect
8. #16302C2semidirect
9. #15194C2semidirect
11. #15137C2semidirect
13. #14277C2semidirect
14. #13954C2semidirect
17. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23192
#169842
#169802
#166042
#163052
#163022
#151942
#151412
#151372
#151342
#142772
#139542
#139412
#139062
#138482