Finite Group Databasebeta

← #697410

Subgroups — #697410

Order 512.

2486 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#16990C212yesmaximal, complemented
4#16988C212yesmaximal, complemented
5#15421C214yesmaximal
6#15406C214yesmaximal
7#15228C212yesmaximal, complemented
8#15221C212yesmaximal, complemented
9#15211C212yesmaximal, complemented
10#15012C212yesmaximal, complemented
11#14264C211yesmaximal, characteristic
12#14130C211yesmaximal, characteristic
13#14005C212yesmaximal, complemented
14#13910C212yesmaximal, complemented
15#13880C211yesmaximal, characteristic
16#13844C211yesmaximal, characteristic
17#13801C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2441—28no—
23C2×#399C2212yes—
24C2×#396C2212yes—
25#2317—28no—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#387C2212yes—
37C2×#385C2214yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#384C2211yescharacteristic
42C2×#384C2212yes—
43C2×#382C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#1445C2214yes—
51#1444C2211yescharacteristic
52#1444C2211yescharacteristic
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2212yes—
63#1442C2211yescharacteristic
64#1442C2211yescharacteristic
65#1442C2214yes—
66#1442C2214yes—
67#1442C2212yes—
68#1442C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#125—24no—
88C2×#125—44no—
89C2×#117—24no—
90C2×#117—28no—
91C2×#117—44no—
92C2×#117D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131#399—28no—
132#396—28no—
133#389—24no—
134#389—24no—
135#389—28no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—48no—
140#389—48no—
141#389—44no—
142#389—48no—
143#387—28no—
144#385—216no—
145#385—44no—
146#385—48no—
147#384—28no—
148#384—24no—
149#384—44no—
150#382—28no—
151#380—24no—
152#380Q818yes—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—44no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4Q818yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C25C2411yescharacteristic
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4C2×D414yes—
193C23×C4C2×Q814yes—
194C23×C4C2×Q814yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211#125—416no—
212#117—232no—
213#117—416no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
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—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—416no—
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⋊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×Q818yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×Q818yes—
291C2×C22⋊C4C2×Q818yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C2×D418yes—
305C2×C42C2×Q818yes—
306C2×C42#4118yes—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95C2×Q814yes—
332#95C2×Q814yes—
333#95C2×Q818yes—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24C22×D412yes—
343C24C22×Q812yes—
344C24C22×Q812yes—
345C24C2511yescharacteristic
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—48no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4C4⋊Q8116yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q818yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q814yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q812yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C42−1+412yes—
447C22×C42−1+414yes—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—416no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—84no—
478C42—216no—
479C42—216no—
480C42—48no—
481C42—44no—
482C42—44no—
483C42—88no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23C2×C4⋊Q814yes—
507C23C23×D411yescharacteristic
508C23C23×Q811yescharacteristic
509C23C23×Q811yescharacteristic
510C23C22×#4112yes—
511C23C2×H(5,𝔽2)12yes—
512C23C2×H(5,𝔽2)12yes—
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×2−1+412yes—
515C23C2×2−1+411yescharacteristic
516C23#58511yescharacteristic
517C23#58511yescharacteristic
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—232no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—816no—
605C2×C4#52514yes—
606C2×C4C2×C4⋊Q814yes—
607C2×C4C2×C4⋊Q814yes—
608C2×C4#53414yes—
609C2×C4#54418yes—
610C2×C4#55418yes—
611C2×C4#55418yes—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22C2×#52511yescharacteristic
624C22C22×C4⋊Q811yescharacteristic
625C22C2×#53411yescharacteristic
626C22C2×#54412yes—
627C22C2×#55412yes—
628C22C2×#55412yes—
629C22#348811yescharacteristic
630C22#348911yescharacteristic
631C22#349212yes—
632C22#349212yes—
633C22#349811yescharacteristic
634C22#349911yescharacteristic
635C22#350111yescharacteristic
636C22#350211yescharacteristic
637C22#350312yes—
638C22#350311yescharacteristic
639C22#350411yescharacteristic
640C22#350612yes—
641C22#350712yes—
642C22#354312yes—
643C22#354612yes—
644C22#355212yes—
645C22#355412yes—
646C4—24no—
647C4—28no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—48no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—84no—
661C4#258318yes—
662C2—44no—
663C2C2×#258311yescharacteristic
664C2#3527511yescharacteristic
665C2#3556611yescharacteristic
666C2#3568611yescharacteristic
667C2#3568911yescharacteristic
668C2#3787412yes—
669C2#3949812yes—
670C2#4291711yescharacteristic
671C2#4296911yescharacteristic
672C2#4490812yes—
673C2#4492312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16990C2semidirect
4. #16988C2semidirect
7. #15228C2semidirect
8. #15221C2semidirect
9. #15211C2semidirect
10. #15012C2semidirect
13. #14005C2semidirect
14. #13910C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#23172
#169902
#169882
#154212
#154062
#152282
#152212
#152112
#150122
#142642
#141302
#140052
#139102
#138802
#138442
#138012