Finite Group Databasebeta

← #4308806

Subgroups — #4308806

Order 512.

2708 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic, complemented
3#16441C214yesmaximal, complemented
4#16430C214yesmaximal, complemented
5#16315C214yesmaximal, complemented
6#16275C214yesmaximal, complemented
7#14946C212yesmaximal, complemented
8#14912C212yesmaximal, complemented
9#14912C212yesmaximal, complemented
10#14776C212yesmaximal, complemented
11#14698C211yesmaximal, characteristic
12#14697C211yesmaximal, characteristic, complemented
13#14339C211yesmaximal, characteristic, complemented
14#14275C211yesmaximal, characteristic, complemented
15#14271C211yesmaximal, characteristic
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2618—28no—
22C2×#398C2212yes—
23C2×#397C2214yes—
24C2×#396C2212yes—
25C2×#396C2212yes—
26C2×#394C2212yes—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#387—28no—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1445C2212yes—
48#1445C2214yes—
49#1445C2214yes—
50#1445C2214yes—
51#1444C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1437C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#123—28no—
87C2×#123—44no—
88C2×#123—44no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42—24no—
122C22×C42—44no—
123C22×C42C2312yes—
124C22×C42C2312yes—
125#398—28no—
126#397—48no—
127#396—28no—
128#396—28no—
129#394—28no—
130#390—44no—
131#389—44no—
132#387—216no—
133#387—216no—
134#386—28no—
135#386—44no—
136#386—44no—
137#386—44no—
138#386—44no—
139#385—44no—
140#385—44no—
141#381—44no—
142#380—24no—
143#380—24no—
144#380—24no—
145C2×#95—24no—
146C2×#95—24no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—42no—
186C22×D4—42no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4#4114yes—
223C23×C4#4114yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237#123—432no—
238#121—416no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—416no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—416no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24C22×D412yes—
356C24C2×#4112yes—
357C24C2×#4112yes—
358C24H(5,𝔽2)12yes—
359C24H(5,𝔽2)12yes—
360C24H(5,𝔽2)12yes—
361C242−1+412yes—
362C24C2511yescharacteristic
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—84no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
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—22no—
395C22×C4—24no—
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—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4#123116yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C42−1+414yes—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—416no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—88no—
500C4⋊C4—88no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C42—48no—
514C42—48no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23C2×#12314yes—
555C23#53514yes—
556C23#53714yes—
557C23#54014yes—
558C23#54014yes—
559C23#54214yes—
560C23#54214yes—
561C23C23×D411yescharacteristic
562C23C22×#4111yescharacteristic
563C23C22×#4111yescharacteristic
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×H(5,𝔽2)12yes—
568C23C2×2−1+411yescharacteristic
569C23C2×2−1+412yes—
570C23#58512yes—
571C23#58512yes—
572D4—88no—
573D4—816no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
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—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
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—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—88no—
655C2×C4—88no—
656C2×C4C2×#12314yes—
657C2×C4C2×#12314yes—
658C22—28no—
659C22—28no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—28no—
666C22—28no—
667C22—24no—
668C22—82no—
669C22—84no—
670C22—84no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—84no—
677C22—84no—
678C22—82no—
679C22#262418yes—
680C22C22×#12311yescharacteristic
681C22C2×#53511yescharacteristic
682C22C2×#53711yescharacteristic
683C22C2×#54011yescharacteristic
684C22C2×#54011yescharacteristic
685C22C2×#54211yescharacteristic
686C22C2×#54211yescharacteristic
687C22#345912yes—
688C22#346112yes—
689C22#348411yescharacteristic
690C22#348511yescharacteristic
691C22#348611yescharacteristic
692C22#348711yescharacteristic
693C22#350512yes—
694C22#350712yes—
695C22#353412yes—
696C22#353512yes—
697C22#353612yes—
698C22#353612yes—
699C22#355912yes—
700C22#356012yes—
701C22#356012yes—
702C22#356312yes—
703C4—24no—
704C4—24no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—82no—
710C4—84no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—84no—
716C2—28no—
717C2—162no—
718C2C2×#262411yescharacteristic
719C2#3498311yescharacteristic
720C2#3518211yescharacteristic
721C2#3670711yescharacteristic
722C2#3672211yescharacteristic
723C2#3676112yes—
724C2#3676312yes—
725C2#3696111yescharacteristic
726C2#3696811yescharacteristic
727C2#3746412yes—
728C2#3746512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#387C2semidirect
3. #16441C2semidirect
4. #16430C2semidirect
5. #16315C2semidirect
6. #16275C2semidirect
7. #14946C2semidirect
8. #14912C2semidirect
9. #14912C2semidirect
10. #14776C2semidirect
12. #14697C2semidirect
13. #14339C2semidirect
14. #14275C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C22×#3872
#164412
#164302
#163152
#162752
#149462
#149122
#147762
#146982
#146972
#143392
#142752
#142712