Finite Group Databasebeta

← #723276

Subgroups — #723276

Order 512.

3050 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic
3#16398C214yesmaximal, complemented
4#15958C214yesmaximal, complemented
5#15327C214yesmaximal, complemented
6#15232C212yesmaximal, complemented
7#15148C212yesmaximal, complemented
8#14921C214yesmaximal, complemented
9#14276C211yesmaximal, characteristic, complemented
10#14025C212yesmaximal
11#14018C212yesmaximal, complemented
12#13941C211yesmaximal, characteristic, complemented
13#13906C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic, complemented
15#13848C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C2×#399C2214yes—
21#2349—28no—
22#2328—28no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2214yes—
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×#376C2211yescharacteristic
46C22×#95C2212yes—
47#1445C2214yes—
48#1444C2212yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2212yes—
67#1438C2214yes—
68#1438C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2314yes—
80C24×C4C2312yes—
81C24×C4C2314yes—
82C24×C4C2312yes—
83C2×#127—24no—
84C2×#127—24no—
85C2×#127Q818yes—
86C2×C4×D4—28no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—44no—
93C2×C4×D4—44no—
94C2×C4×D4D418yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#399—48no—
124#390—24no—
125#390—28no—
126#390—24no—
127#390—44no—
128#390—44no—
129#389—28no—
130#389—44no—
131#386—216no—
132#386—216no—
133#386—48no—
134#386—48no—
135#385—216no—
136#385—48no—
137#384—24no—
138#382—24no—
139#382—24no—
140#380—28no—
141#380—48no—
142#380—48no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—44no—
146#376—24no—
147C2×#95—28no—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C25C2412yes—
174C25C2412yes—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4C2×D414yes—
192C22×D4C2×Q814yes—
193C22×D4C2×Q814yes—
194C22×D4C2×Q8116yes—
195C22×D4#4114yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C2×D418yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#127—232no—
227C4×D4—232no—
228C4×D4—432no—
229C4×D4—416no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
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—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
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⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
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⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4C2×D418yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42C2×D414yes—
309C2×C42C2×Q814yes—
310C2×C42#4114yes—
311C2×C42#4114yes—
312#95—22no—
313#95—24no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—24no—
319#95—28no—
320#95—22no—
321#95—22no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—42no—
327#95—42no—
328#95—44no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24C22×D414yes—
356C24C22×Q814yes—
357C24C22×Q814yes—
358C24C2×#4114yes—
359C24C2511yescharacteristic
360C2×D4—232no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4C22⋊Q8116yes—
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—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
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—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q812yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C4⋊C4—28no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C22⋊C4—216no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—28no—
505C42—44no—
506C42—48no—
507C42—48no—
508C42#123116yes—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23C2×C22⋊Q818yes—
555C23C23×D411yescharacteristic
556C23C23×Q811yescharacteristic
557C23C22×#4111yescharacteristic
558C23C22×#4111yescharacteristic
559C23C22×#4111yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)12yes—
563C23C2×H(5,𝔽2)12yes—
564C23#58512yes—
565C23#58512yes—
566D4—232no—
567D4—416no—
568D4—832no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—816no—
644C2×C4—816no—
645C2×C4—832no—
646C2×C4C2×C22⋊Q814yes—
647C2×C4C2×#12314yes—
648C2×C4C2×#12314yes—
649C2×C4C2×#12314yes—
650C2×C4#53514yes—
651C2×C4#55314yes—
652C2×C4#55314yes—
653C2×C4#55414yes—
654C2×C4#55414yes—
655C22—216no—
656C22—28no—
657C22—48no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—44no—
664C22—44no—
665C22—48no—
666C22—44no—
667C22—44no—
668C22—48no—
669C22—44no—
670C22—48no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22C22×C22⋊Q811yescharacteristic
678C22C22×#12311yescharacteristic
679C22C2×#53511yescharacteristic
680C22#346012yes—
681C22C2×#55311yescharacteristic
682C22C2×#55311yescharacteristic
683C22C2×#55411yescharacteristic
684C22C2×#55411yescharacteristic
685C22#348812yes—
686C22#349412yes—
687C22#349612yes—
688C22#350112yes—
689C22#350112yes—
690C22#350811yescharacteristic
691C22#350811yescharacteristic
692C22#351011yescharacteristic
693C22#351011yescharacteristic
694C22#351912yes—
695C22#352112yes—
696C22#354212yes—
697C22#354212yes—
698C22#354312yes—
699C22#356212yes—
700C4—24no—
701C4—24no—
702C4—44no—
703C4—44no—
704C4—44no—
705C4—44no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—84no—
711C4—84no—
712C4#248718yes—
713C2—48no—
714C2—84no—
715C2C2×#248711yescharacteristic
716C2#3525411yescharacteristic
717C2#3554111yescharacteristic
718C2#3577612yes—
719C2#3926212yes—
720C2#3967212yes—
721C2#4243812yes—
722C2#4494711yescharacteristic
723C2#4494711yescharacteristic
724C2#4504111yescharacteristic
725C2#4520011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2349C2semidirect
3. #16398C2semidirect
4. #15958C2semidirect
5. #15327C2semidirect
6. #15232C2semidirect
7. #15148C2semidirect
8. #14921C2semidirect
9. #14276C2semidirect
11. #14018C2semidirect
12. #13941C2semidirect
13. #13906C2semidirect
14. #13906C2semidirect
15. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#23282
#163982
#159582
#153272
#152322
#151482
#149212
#142762
#140252
#140182
#139412
#139062
#138482