Finite Group Databasebeta

← #517994

Subgroups — #517994

Order 512.

2566 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3C2×#2355C211yesmaximal, characteristic
4#14354C211yesmaximal, characteristic
5#14327C212yesmaximal
6#14244C212yesmaximal
7#14241C212yesmaximal
8#14237C214yesmaximal
9#14214C214yesmaximal
10#13926C211yesmaximal, characteristic
11#13925C211yesmaximal, characteristic
12#13911C212yesmaximal
13#13909C212yesmaximal
14C4×#398C212yesmaximal
15#13804C212yesmaximal
16C4⋊C42C212yesmaximal
17#13750C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#2399C4116yescomplemented
21C2×#398C2211yescharacteristic
22C2×#398C2212yes—
23C2×#398C2211yescharacteristic
24C2×#395C2212yes—
25C2×#395C2211yescharacteristic
26C2×#395C2211yescharacteristic
27#2355—28no—
28#2355—28no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#387C2214yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#384C2211yescharacteristic
36C2×#384C2212yes—
37C2×#384C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#382C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2214yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×#376C2211yescharacteristic
55#1446C2212yes—
56#1445C2212yes—
57#1444C2212yes—
58#1444C2214yes—
59#1444C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81#1434C2214yes—
82C4×#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2214yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8C2×C418yes—
93C2×C4×Q8C2×C418yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#398C2×C418yes—
128#398C2×C4116yes—
129#398C2×C418yes—
130#395C2×C4116yes—
131#395C2×C418yes—
132#395C2×C418yes—
133#391—28no—
134#391—28no—
135#391—44no—
136#387—28no—
137#387—28no—
138#387—216no—
139#384—24no—
140#384—24no—
141#384—28no—
142#382—24no—
143#382—28no—
144#382—24no—
145#382—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—48no—
154C4×C4⋊C4C2×C4116yes—
155C4×C4⋊C4C2×C4116yes—
156C4×C4⋊C4D418yes—
157C4×C4⋊C4Q8116yescomplemented
158#376C2×C418yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—41no—
196C22×Q8—41no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8C22×C414yes—
200C22×Q8C22×C414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C4×Q8—232no—
219C4×Q8—232no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
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—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
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—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C22×C418yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×Q818yes—
276C2×C4⋊C4C2×Q818yes—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—28no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—42no—
300C2×C42C22×C418yes—
301C2×C42C22×C414yes—
302C2×C42C22×C414yes—
303C2×C42C22×C414yes—
304C2×C42C22×C418yes—
305C2×C42C2×D418yes—
306C2×C42C2×D414yes—
307C2×C42C2×Q8116yes—
308C2×C42C2×Q818yes—
309C2×C42#4114yes—
310C2×C42#4118yes—
311#95—24no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C22×C418yes—
333#95C22×C414yes—
334#95C22×C414yes—
335#95C22×C418yes—
336#95C22×C418yes—
337#95C22×C418yes—
338#95C22×C418yes—
339#95C2×D418yes—
340#95C2×Q8116yes—
341C24C2511yescharacteristic
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—44no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—48no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4C23×C414yes—
429C22×C4C23×C412yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C414yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C23×C414yes—
436C22×C4C23×C414yes—
437C22×C4C23×C412yes—
438C22×C4C23×C412yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×Q818yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q818yes—
448C22×C4C22×Q814yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4114yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—28no—
464C4⋊C4—216no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C4⋊C4—84no—
477C4⋊C4C4⋊Q8116yes—
478C42—28no—
479C42—28no—
480C42—216no—
481C42—48no—
482C42—416no—
483C42—48no—
484C42—84no—
485C42—88no—
486C42—84no—
487C42—88no—
488C42C4×D4116yes—
489C42C4×Q8132yes—
490C23C24×C411yescharacteristic
491C23C23×D411yescharacteristic
492C23C23×Q812yes—
493C23C22×#4111yescharacteristic
494C23C22×#4112yes—
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×2−1+411yescharacteristic
497C23#58512yes—
498C23#58512yes—
499C23#58511yescharacteristic
500C23#58511yescharacteristic
501Q8—416no—
502Q8—416no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4C2×C4×D414yes—
596C2×C4C2×C4×D414yes—
597C2×C4C2×C4×D414yes—
598C2×C4C2×C4×Q818yes—
599C2×C4C2×C4×Q818yes—
600C2×C4C2×C4×Q818yes—
601C2×C4#52014yes—
602C2×C4C2×C4⋊Q818yes—
603C2×C4C2×C4⋊Q814yes—
604C2×C4#53214yes—
605C2×C4#53318yes—
606C22C22×C4×D411yescharacteristic
607C22C22×C4×Q812yes—
608C22C2×#52011yescharacteristic
609C22#342812yes—
610C22#342811yescharacteristic
611C22C22×C4⋊Q811yescharacteristic
612C22C2×#53211yescharacteristic
613C22C2×#53312yes—
614C22#348511yescharacteristic
615C22#348711yescharacteristic
616C22#348811yescharacteristic
617C22#348911yescharacteristic
618C22#349312yes—
619C22#349411yescharacteristic
620C22#349511yescharacteristic
621C22#350112yes—
622C22#350212yes—
623C22#350412yes—
624C22#350412yes—
625C22#350612yes—
626C22#350712yes—
627C22#351012yes—
628C22#351012yes—
629C4—28no—
630C4—24no—
631C4—24no—
632C4—24no—
633C4—28no—
634C4—28no—
635C4—48no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—84no—
643C4—82no—
644C4—82no—
645C4C4×C4⋊Q818yes—
646C2C2×C4×C4⋊Q811yescharacteristic
647C2#3420111yescharacteristic
648C2#3420211yescharacteristic
649C2#3421312yes—
650C2#3421512yes—
651C2#3595011yescharacteristic
652C2#3595212yes—
653C2#3595611yescharacteristic
654C2#3595711yescharacteristic
655C2#3595812yes—
656C2#3595911yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #2399C4semidirect
157. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
C2×#23552
#143542
#143272
#142442
#142412
#142372
#142142
#139262
#139252
#139112
#139092
C4×#3982
#138042
C4⋊C422
#137502