Finite Group Databasebeta

← #4417617

Subgroups — #4417617

Order 512.

2148 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic, complemented
2C2×C4×#126C211yesmaximal, characteristic
3#17025C212yesmaximal, complemented
4#17023C212yesmaximal
5#16696C212yesmaximal, complemented
6#16665C212yesmaximal
7#16663C212yesmaximal, complemented
8#16656C212yesmaximal
9#15235C212yesmaximal, complemented
10#15234C212yesmaximal, complemented
11#15233C212yesmaximal
12#15223C212yesmaximal
13#15128C211yesmaximal, characteristic, complemented
14#15128C211yesmaximal, characteristic, complemented
15#15125C211yesmaximal, characteristic
16#15125C211yesmaximal, characteristic
17#14688C211yesmaximal, characteristic, complemented
18#14626C212yesmaximal, complemented
19#14625C212yesmaximal
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126C2211yescharacteristic
23#2862—28no—
24C2×#395C2212yes—
25C2×#395C2212yes—
26C2×#393C2212yes—
27C2×#393C2212yes—
28C4×#126—28no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2211yescharacteristic
36C2×#388C2211yescharacteristic
37C2×#388C2211yescharacteristic
38C2×#387C2211yescharacteristic
39C2×#387C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#383C2211yescharacteristic
42C2×#383C2212yes—
43C2×#382C2212yes—
44C2×#382C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C43C2211yescharacteristic
54#1446C2212yes—
55#1446C2212yes—
56#1446C2212yes—
57#1446C2212yes—
58#1445C2214yes—
59#1445C2214yes—
60#1444C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1443C2214yes—
64#1443C2214yes—
65#1443C2214yes—
66#1442C2214yes—
67#1442C2212yes—
68#1442C2212yes—
69#1442C2214yes—
70#1442C2214yes—
71#1442C2214yes—
72#1441C2212yes—
73#1441C2214yes—
74#1441C2212yes—
75#1441C2214yes—
76#1440C2214yes—
77#1440C2214yes—
78#1439C2212yes—
79#1439C2214yes—
80#1439C2212yes—
81#1439C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87C4⋊#95C2212yes—
88C4⋊#95C2212yes—
89C4⋊#95C2212yes—
90#1434C2212yes—
91#1434C2212yes—
92#1434C2212yes—
93C23×Q8C2312yes—
94C23×Q8C2312yes—
95C24×C4C2311yescharacteristic
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—44no—
99C2×C4⋊Q8—44no—
100C2×#126—24no—
101C2×#126—24no—
102C2×#126—24no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#395—28no—
136#395—44no—
137#393—28no—
138#393—44no—
139#391—28no—
140#391—28no—
141#391—28no—
142#391—28no—
143#389—44no—
144#389—44no—
145#388—24no—
146#388—24no—
147#388—24no—
148#387—24no—
149#387—24no—
150#387—24no—
151#383—24no—
152#383—44no—
153#382—28no—
154#382—28no—
155#380—44no—
156#380—44no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C43D418yes—
200C25C2411yescharacteristic
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—44no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—416no—
235C4⋊Q8—416no—
236#126—232no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
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—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4#4114yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4#4114yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C2×#4112yes—
368C24C2511yescharacteristic
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—88no—
376C2×Q8—88no—
377C2×Q8—88no—
378C2×Q8—88no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
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—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C42−1+412yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C22×C42−1+414yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C42—28no—
509C42—28no—
510C42—28no—
511C42—28no—
512C42—28no—
513C42—44no—
514C42—44no—
515C42—44no—
516C42—44no—
517C42—44no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23C23×D411yescharacteristic
542C23C22×#4111yescharacteristic
543C23C22×#4111yescharacteristic
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×2−1+411yescharacteristic
547C23C2×2−1+411yescharacteristic
548C23C2×2−1+412yes—
549C23C2×2−1+411yescharacteristic
550C23#58512yes—
551C23#58512yes—
552Q8—816no—
553Q8—816no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
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—44no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
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—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—88no—
650C2×C4#52514yes—
651C2×C4#53514yes—
652C2×C4#53614yes—
653C2×C4#53614yes—
654C2×C4#53714yes—
655C2×C4#56318yes—
656C2×C4#56318yes—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—82no—
667C22—82no—
668C22C2×#52511yescharacteristic
669C22C2×#53511yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#53611yescharacteristic
672C22C2×#53711yescharacteristic
673C22#344911yescharacteristic
674C22#344911yescharacteristic
675C22#350911yescharacteristic
676C22#350911yescharacteristic
677C22#351111yescharacteristic
678C22#351111yescharacteristic
679C22#351412yes—
680C22#351412yes—
681C22#351512yes—
682C22#351812yes—
683C22C2×#56312yes—
684C22C2×#56312yes—
685C22#354712yes—
686C22#355512yes—
687C22#355812yes—
688C22#356112yes—
689C22#356512yes—
690C22#356512yes—
691C4—24no—
692C4—24no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—81no—
701C4—84no—
702C4—81no—
703C4—81no—
704C4—84no—
705C4—81no—
706C4—84no—
707C4—84no—
708C4#282118yes—
709C2—82no—
710C2#3512911yescharacteristic
711C2#3520011yescharacteristic
712C2#3529711yescharacteristic
713C2C2×#282111yescharacteristic
714C2#3754611yescharacteristic
715C2#3756011yescharacteristic
716C2#3763311yescharacteristic
717C2#4604912yes—
718C2#4605112yes—
719C2#4606012yes—
720C2#4606212yes—
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
1. C2×#2862C2semidirect
3. #17025C2semidirect
5. #16696C2semidirect
7. #16663C2semidirect
9. #15235C2semidirect
10. #15234C2semidirect
13. #15128C2semidirect
14. #15128C2semidirect
17. #14688C2semidirect
18. #14626C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×C4×#1262
#170252
#170232
#166962
#166652
#166632
#166562
#152352
#152342
#152332
#152232
#151282
#151252
#146882
#146262
#146252