Finite Group Databasebeta

← #4930468

Subgroups — #4930468

Order 512.

2886 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2562C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic, complemented
4C2×#2276C211yesmaximal, characteristic, complemented
5#15093C214yesmaximal, complemented
6#15074C214yesmaximal, complemented
7#14908C212yesmaximal, complemented
8#14898C212yesmaximal
9#14869C212yesmaximal, complemented
10#14813C212yesmaximal
11#14283C211yesmaximal, characteristic, complemented
12#13969C212yesmaximal, complemented
13#13969C212yesmaximal, complemented
14#13911C211yesmaximal, characteristic, complemented
15#13876C212yesmaximal, complemented
16#13875C212yesmaximal, complemented
17#13847C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2565—28no—
24#2562—28no—
25#2538—28no—
26C2×#400C2212yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#396C2214yes—
30C2×#394C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46#2276—28no—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2214yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×#376C2211yescharacteristic
55C2×#376C2212yes—
56#1445C2211yescharacteristic
57#1445C2211yescharacteristic
58#1443C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1440C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2211yescharacteristic
74#1438C2211yescharacteristic
75#1437C2214yes—
76#1437C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86#1434C2214yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×#125—24no—
91C2×#125—24no—
92C2×#125—24no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—24no—
99C2×C4×Q8—28no—
100C2×C4×Q8—24no—
101C2×C4×Q8D418yes—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#400—28no—
140#398—28no—
141#398—28no—
142#396—216no—
143#394—28no—
144#391—24no—
145#391—24no—
146#390—28no—
147#389—28no—
148#388—216no—
149#388—48no—
150#387—28no—
151#387—28no—
152#384—24no—
153#384—24no—
154#382—24no—
155#382—44no—
156#380—28no—
157#380—28no—
158#380—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—216no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4D418yes—
164C4×C22⋊C4—28no—
165C4×C22⋊C4—28no—
166#376—24no—
167#376—28no—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C25C2412yes—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8C2×D414yes—
201C22×Q8#4118yes—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—42no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230#125—416no—
231C22⋊Q8—432no—
232C4×Q8—264no—
233C4×Q8—232no—
234C4×D4—416no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42C2×D418yes—
329C2×C42C2×D418yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42#4118yes—
333C2×C42#4114yes—
334#95—22no—
335#95—28no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95C2×D418yes—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C242−1+414yes—
377C24C2511yescharacteristic
378C2×Q8—28no—
379C2×Q8—216no—
380C2×Q8—216no—
381C2×Q8—28no—
382C2×Q8#124116yes—
383C2×D4—28no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4C22×D414yes—
473C22×C4C22×D418yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4118yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C4⋊C4—232no—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—416no—
507C4⋊C4—84no—
508C22⋊C4—216no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—232no—
518C42—216no—
519C42—216no—
520C42—48no—
521C42—44no—
522C42—44no—
523C42—88no—
524C42#121116yes—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C22×#4111yescharacteristic
563C23C22×#4112yes—
564C23C22×#4112yes—
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×2−1+412yes—
567C23C2×2−1+411yescharacteristic
568C23#58512yes—
569C23#58511yescharacteristic
570C23#58511yescharacteristic
571Q8—232no—
572D4—416no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
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—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—816no—
662C2×C4C2×#12114yes—
663C2×C4C2×#12114yes—
664C2×C4C2×#12114yes—
665C2×C4#52518yes—
666C2×C4C2×#12418yes—
667C2×C4C2×#12414yes—
668C2×C4#52918yes—
669C2×C4#53214yes—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22C22×#12111yescharacteristic
686C22C2×#52512yes—
687C22C22×#12411yescharacteristic
688C22C2×#52912yes—
689C22C2×#53211yescharacteristic
690C22#348611yescharacteristic
691C22#348711yescharacteristic
692C22#348811yescharacteristic
693C22#348911yescharacteristic
694C22#348912yes—
695C22#349212yes—
696C22#349312yes—
697C22#349512yes—
698C22#349512yes—
699C22#349512yes—
700C22#349611yescharacteristic
701C22#349711yescharacteristic
702C22#351111yescharacteristic
703C22#351211yescharacteristic
704C22#351612yes—
705C22#351712yes—
706C22#352312yes—
707C22#352412yes—
708C4—28no—
709C4—24no—
710C4—24no—
711C4—42no—
712C4—48no—
713C4—44no—
714C4—42no—
715C4—44no—
716C4—44no—
717C4—82no—
718C4—82no—
719C4—81no—
720C4—84no—
721C4—81no—
722C4—82no—
723C4#285018yes—
724C2—84no—
725C2#3556411yescharacteristic
726C2#3560912yes—
727C2#3565712yes—
728C2#3566111yescharacteristic
729C2#3576312yes—
730C2#3583212yes—
731C2#3591011yescharacteristic
732C2#3592611yescharacteristic
733C2#3604911yescharacteristic
734C2#3606511yescharacteristic
735C2C2×#285011yescharacteristic
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
3. C2×#2565C2semidirect
4. C2×#2276C2semidirect
5. #15093C2semidirect
6. #15074C2semidirect
7. #14908C2semidirect
9. #14869C2semidirect
11. #14283C2semidirect
12. #13969C2semidirect
13. #13969C2semidirect
14. #13911C2semidirect
15. #13876C2semidirect
16. #13875C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25622
C2×#25652
C2×#22762
#150932
#150742
#149082
#148982
#148692
#148132
#142832
#139692
#139112
#138762
#138752
#138472