Finite Group Databasebeta

← #669705

Subgroups — #669705

Order 512.

2966 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C212yesmaximal
2C2×#2344C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4#15454C214yesmaximal, complemented
5#15414C214yesmaximal, complemented
6#15139C212yesmaximal, complemented
7#15137C212yesmaximal, complemented
8#15073C212yesmaximal
9#14919C212yesmaximal
10#14685C211yesmaximal, characteristic
11#14133C212yesmaximal, complemented
12#14130C211yesmaximal, characteristic
13#14128C212yesmaximal
14#14029C212yesmaximal, complemented
15#13904C211yesmaximal, characteristic
16#13880C211yesmaximal, characteristic
17C4×#381C211yesmaximal, characteristic
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2484—216no—
24C2×#400C2214yes—
25C2×#398C2214yes—
26#2344—28no—
27C2×#389—28no—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#387C2214yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#387C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#384C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C22×#95C2212yes—
55#1445C2214yes—
56#1445C2214yes—
57#1444C2214yes—
58#1444C2212yes—
59#1444C2214yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77#1434C2212yes—
78#1434C2214yes—
79#1434C2212yes—
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
84C4×#95C2212yes—
85C4×#95C2212yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C42—24no—
124C22×C42—24no—
125C22×C42—24no—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133#400—216no—
134#398—216no—
135#389—24no—
136#389—24no—
137#389—28no—
138#389—216no—
139#389—28no—
140#389—28no—
141#387—28no—
142#387—28no—
143#387—216no—
144#387—216no—
145#385—28no—
146#385—28no—
147#385—44no—
148#384—28no—
149#381—28no—
150#381—28no—
151#381—28no—
152#381—28no—
153#381—24no—
154#381D418yes—
155#380—24no—
156#380—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—24no—
161C2×#95—28no—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C25C2411yescharacteristic
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4C2×D414yes—
220C23×C4C2×D418yes—
221C23×C4C2×D414yes—
222C23×C4#4114yes—
223C23×C4#4114yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240#117—432no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—432no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4#4118yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4#4114yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—416no—
324#95—28no—
325#95—28no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—42no—
350#95—42no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C2×D418yes—
355#95C2×D418yes—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C2×#4112yes—
363C24C2×#4112yes—
364C24C2×#4112yes—
365C24H(5,𝔽2)12yes—
366C242−1+412yes—
367C24C2511yescharacteristic
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4#123116yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4118yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4114yes—
491C22×C42−1+414yes—
492C22×C42−1+414yes—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—216no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—88no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C42—48no—
515C42—48no—
516C42—48no—
517C42—48no—
518C42—44no—
519C42—44no—
520C42—48no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23C2×#12314yes—
538C23#53214yes—
539C23#54214yes—
540C23D4214yes—
541C23#54714yes—
542C23#54714yes—
543C23#55314yes—
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4112yes—
549C23C22×#4111yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×2−1+412yes—
552C23C2×2−1+411yescharacteristic
553C23C2×2−1+412yes—
554C23#58512yes—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—232no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—432no—
586C2×C4—416no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—432no—
593C2×C4—416no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4C2×#12314yes—
651C2×C4C2×#12314yes—
652C2×C4#52514yes—
653C2×C4#53614yes—
654C2×C4#55218yes—
655C2×C4#55218yes—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—28no—
660C22—24no—
661C22—28no—
662C22—28no—
663C22—24no—
664C22—28no—
665C22—24no—
666C22#242218yes—
667C22C22×#12311yescharacteristic
668C22C2×#52511yescharacteristic
669C22C2×#53211yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#54211yescharacteristic
672C22C2×D4211yescharacteristic
673C22C2×#54711yescharacteristic
674C22C2×#54711yescharacteristic
675C22D4×#4112yes—
676C22C2×#55212yes—
677C22C2×#55212yes—
678C22C2×#55311yescharacteristic
679C22#348212yes—
680C22#348611yescharacteristic
681C22#348711yescharacteristic
682C22#349512yes—
683C22#349512yes—
684C22#349912yes—
685C22#351312yes—
686C22#351512yes—
687C22#352312yes—
688C22#354412yes—
689C22#356312yes—
690C4—24no—
691C4—42no—
692C4—48no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—44no—
699C4—42no—
700C4—82no—
701C4—81no—
702C4—81no—
703C4—84no—
704C4—82no—
705C4#248418yescomplemented
706C2—28no—
707C2C2×#242211yescharacteristic
708C2C2×#248411yescharacteristic
709C2#3567211yescharacteristic
710C2#3610211yescharacteristic
711C2#3626512yes—
712C2#3642311yescharacteristic
713C2#3806912yes—
714C2#3891911yescharacteristic
715C2#3933711yescharacteristic
716C2#5920912yes—
717C2#5929012yes—
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
4. #15454C2semidirect
5. #15414C2semidirect
6. #15139C2semidirect
7. #15137C2semidirect
11. #14133C2semidirect
14. #14029C2semidirect
705. C4#2484semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#23442
C22×#3892
#154542
#154142
#151392
#151372
#150732
#149192
#146852
#141332
#141302
#141282
#140292
#139042
#138802
C4×#3812