Finite Group Databasebeta

← #676815

Subgroups — #676815

Order 512.

2202 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic
3#15816C214yesmaximal
4#15806C214yesmaximal
5#15238C212yesmaximal
6#15232C212yesmaximal
7#15187C212yesmaximal
8#15186C212yesmaximal
9#14142C212yesmaximal
10#14054C212yesmaximal
11#14003C212yesmaximal
12#13933C212yesmaximal
13#13926C211yesmaximal, characteristic
14#13926C211yesmaximal, characteristic
15C4⋊C42C211yesmaximal, characteristic
16C4×#391C211yesmaximal, characteristic
17C4×#391C211yesmaximal, characteristic
18C22×#125C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2859—28no—
22#2582—28no—
23C2×#398C2214yes—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#382C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C43C2211yescharacteristic
46#1445C2214yes—
47#1445C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2214yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79#1434C2214yes—
80#1434C2214yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C2×#125—28no—
88C2×#125—28no—
89C2×#125—44no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#398—216no—
126#391—28no—
127#391—24no—
128#391—28no—
129#391—28no—
130#391—24no—
131#391—24no—
132#391—44no—
133#391Q818yescomplemented
134#389—28no—
135#389—28no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—28no—
140#384—24no—
141#384D418yes—
142#382—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—44no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C43—24no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8C2×Q814yes—
195C22×Q8#4114yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213#125—432no—
214C4×Q8—416no—
215C4×Q8—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×Q814yes—
266C2×C4⋊C4#4114yes—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—28no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42—41no—
295C2×C42—41no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42C2×D414yes—
299C2×C42C2×Q814yes—
300C2×C42#4114yes—
301C2×C42#4114yes—
302C2×C42#4114yes—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95C2×Q818yes—
336#95C2×Q818yes—
337C24C2511yescharacteristic
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4C22×D418yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+412yes—
469C22×C42−1+412yes—
470C22×C42−1+412yes—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—88no—
488C42—28no—
489C42—28no—
490C42—44no—
491C42—44no—
492C42—48no—
493C42—48no—
494C42—48no—
495C42—48no—
496C42—44no—
497C42—44no—
498C42—88no—
499C42#124116yes—
500C23C23×D411yescharacteristic
501C23C23×Q811yescharacteristic
502C23C22×#4111yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23C2×2−1+411yescharacteristic
511C23#58511yescharacteristic
512C23#58511yescharacteristic
513C23#58511yescharacteristic
514C23#58511yescharacteristic
515Q8—416no—
516Q8—416no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4C2×#12414yes—
622C2×C4C2×#12414yes—
623C2×C4C2×#12414yes—
624C2×C4#52914yes—
625C2×C4#53314yes—
626C2×C4#54114yes—
627C2×C4#54214yes—
628C2×C4#54614yes—
629C2×C4#54714yes—
630C2×C4#54814yes—
631C2×C4D4×Q814yes—
632C2×C4#55214yes—
633C2×C4#55214yes—
634C2×C4#55414yes—
635C2×C4#55614yes—
636C22C22×#12411yescharacteristic
637C22C2×#52911yescharacteristic
638C22C2×#53311yescharacteristic
639C22C2×#54111yescharacteristic
640C22C2×#54211yescharacteristic
641C22C2×#54611yescharacteristic
642C22C2×#54711yescharacteristic
643C22C2×#54811yescharacteristic
644C22C2×D4×Q811yescharacteristic
645C22C2×#55211yescharacteristic
646C22C2×#55211yescharacteristic
647C22C2×#55411yescharacteristic
648C22C2×#55611yescharacteristic
649C22Q8×#4111yescharacteristic
650C22Q8×#4111yescharacteristic
651C22#348011yescharacteristic
652C22#348011yescharacteristic
653C22#348511yescharacteristic
654C22#348711yescharacteristic
655C22#350211yescharacteristic
656C22#350211yescharacteristic
657C22#350711yescharacteristic
658C22#350711yescharacteristic
659C22#351211yescharacteristic
660C22#351211yescharacteristic
661C22#351211yescharacteristic
662C22#351211yescharacteristic
663C22#351411yescharacteristic
664C22#351511yescharacteristic
665C22#351711yescharacteristic
666C22#351811yescharacteristic
667C22#355411yescharacteristic
668C22#355411yescharacteristic
669C22#355511yescharacteristic
670C22#355511yescharacteristic
671C4—24no—
672C4—44no—
673C4—42no—
674C4—48no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—84no—
685C4—82no—
686C4#248918yes—
687C4#249218yes—
688C2C2×#249211yescharacteristic
689C2C2×#248911yescharacteristic
690C2#3564911yescharacteristic
691C2#3628511yescharacteristic
692C2#3628511yescharacteristic
693C2#3655511yescharacteristic
694C2#3659011yescharacteristic
695C2#3897711yescharacteristic
696C2#3897811yescharacteristic
697C2#3963511yescharacteristic
698C2#3976211yescharacteristic
699C2#5906411yescharacteristic
700C2#5906711yescharacteristic
701C2#5907811yescharacteristic
702C2#5908111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
133. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#25822
#158162
#158062
#152382
#152322
#151872
#151862
#141422
#140542
#140032
#139332
#139262
C4⋊C422
C4×#3912