Finite Group Databasebeta

← #877073

Subgroups — #877073

Order 512.

2686 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2748C212yesmaximal
2#17057C212yesmaximal, complemented
3#16305C212yesmaximal, complemented
4#15746C214yesmaximal, complemented
5#15545C214yesmaximal, complemented
6#15145C212yesmaximal, complemented
7#15135C212yesmaximal, complemented
8#15131C212yesmaximal, complemented
9#14686C212yesmaximal, complemented
10#14401C212yesmaximal
11#14029C212yesmaximal, complemented
12#13906C211yesmaximal, characteristic
13#13880C212yesmaximal, complemented
14#13872C211yesmaximal, characteristic
15C4×#386C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2748—216no—
20C2×#399C2214yes—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#396C2214yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#386C2212yes—
29C2×#386C2211yescharacteristic
30C2×#386C2211yescharacteristic
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#384C2212yes—
34C2×#384C2212yes—
35C2×#384C2211yescharacteristic
36C2×#384C2212yes—
37C2×#380C2214yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1445C2214yes—
47#1445C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2212yes—
62#1437C2212yes—
63#1436C2214yes—
64#1436C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4Q818yes—
85C22×C4⋊C4—44no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C42—28no—
114C22×C42—42no—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120#399—216no—
121#398—216no—
122#398—48no—
123#396—216no—
124#390—28no—
125#389—44no—
126#389—44no—
127#387—216no—
128#386—28no—
129#386D418yes—
130#386Q818yes—
131#385—216no—
132#385—48no—
133#384—28no—
134#384—42no—
135#384—44no—
136#384—44no—
137#380—216no—
138#380—48no—
139#380—44no—
140#380—44no—
141#380—44no—
142#380—42no—
143C4×C4⋊C4—42no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—42no—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4C2×D414yes—
175C22×D4C2×Q814yes—
176C22×D4C2×Q814yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4C2×D414yes—
199C23×C4C2×Q814yes—
200C23×C4C2×Q814yes—
201C23×C4C2×Q814yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C4×D4—416no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
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—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
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⋊C4C2×Q814yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D418yes—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×Q818yes—
290C2×C22⋊C4C2×Q814yes—
291C2×C22⋊C4C2×Q814yes—
292C2×C22⋊C4C2×Q814yes—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—416no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42C2×Q814yes—
308C2×C42#4118yes—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C2×D418yes—
330#95C2×Q818yes—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24C22×D412yes—
348C24C22×D412yes—
349C24C22×Q812yes—
350C24C22×Q812yes—
351C24C22×Q812yes—
352C24C22×Q812yes—
353C24C2511yescharacteristic
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4C4⋊Q8116yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—84no—
447C22×C4—88no—
448C22×C4—84no—
449C22×C4—88no—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D414yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q812yes—
456C22×C4C22×Q812yes—
457C22×C4C22×Q812yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q814yes—
460C22×C4C22×Q812yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4114yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+414yes—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C4⋊C4—84no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C22⋊C4—216no—
486C22⋊C4—216no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C42—28no—
496C42—216no—
497C42—48no—
498C42—48no—
499C42—84no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23C2×C4⋊Q814yes—
545C23C2×C4⋊Q814yes—
546C23C23×D411yescharacteristic
547C23C23×Q811yescharacteristic
548C23C23×Q811yescharacteristic
549C23C22×#4112yes—
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×H(5,𝔽2)12yes—
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)12yes—
554C23C2×2−1+412yes—
555C23#58511yescharacteristic
556C23#58511yescharacteristic
557D4—416no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—816no—
638C2×C4C2×C4⋊Q814yes—
639C2×C4#54018yes—
640C2×C4#54314yes—
641C2×C4#54718yes—
642C2×C4#55114yes—
643C2×C4#55418yes—
644C2×C4#55418yes—
645C2×C4#56014yes—
646C22—28no—
647C22—28no—
648C22—24no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22C22×C4⋊Q811yescharacteristic
669C22C2×#54012yes—
670C22C2×#54311yescharacteristic
671C22#345811yescharacteristic
672C22C2×#54712yes—
673C22C2×#55111yescharacteristic
674C22C2×#55412yes—
675C22C2×#55412yes—
676C22#349412yes—
677C22#349512yes—
678C22#349611yescharacteristic
679C22#349611yescharacteristic
680C22#349812yes—
681C22#350111yescharacteristic
682C22#350111yescharacteristic
683C22#350312yes—
684C22#350312yes—
685C22C2×#56011yescharacteristic
686C22#353711yescharacteristic
687C22#353811yescharacteristic
688C22#354612yes—
689C22#354812yes—
690C22#355412yes—
691C4—28no—
692C4—28no—
693C4—24no—
694C4—44no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—84no—
700C4—84no—
701C4—84no—
702C4—84no—
703C4#274818yescomplemented
704C2—44no—
705C2—44no—
706C2#3500211yescharacteristic
707C2#3618012yes—
708C2#3730711yescharacteristic
709C2#3787812yes—
710C2#3935212yes—
711C2#3960612yes—
712C2#4548711yescharacteristic
713C2#4550311yescharacteristic
714C2C2×#274811yescharacteristic
715C2#5919811yescharacteristic
716C2#5919911yescharacteristic
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
2. #17057C2semidirect
3. #16305C2semidirect
4. #15746C2semidirect
5. #15545C2semidirect
6. #15145C2semidirect
7. #15135C2semidirect
8. #15131C2semidirect
9. #14686C2semidirect
11. #14029C2semidirect
13. #13880C2semidirect
703. C4#2748semidirect

Maximal subgroups

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

subgroupindex
C2×#27482
#170572
#163052
#157462
#155452
#151452
#151352
#151312
#146862
#144012
#140292
#139062
#138802
#138722
C4×#3862