Finite Group Databasebeta

← #657943

Subgroups — #657943

Order 512.

3118 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×#2484C212yesmaximal
2C2×#2344C211yesmaximal, characteristic
3C22×#384C211yesmaximal, characteristic
4#15434C214yesmaximal, complemented
5#15388C214yesmaximal, complemented
6#15146C212yesmaximal, complemented
7#15133C212yesmaximal, complemented
8#15073C212yesmaximal
9#15073C212yesmaximal
10#14697C211yesmaximal, characteristic
11#13981C212yesmaximal, complemented
12#13904C211yesmaximal, characteristic
13#13904C211yesmaximal, characteristic
14#13904C211yesmaximal, characteristic
15#13884C212yesmaximal, complemented
16#13870C212yesmaximal
17C4×#385C211yesmaximal, characteristic
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2484—216no—
25C2×#400C2214yes—
26C2×#398C2214yes—
27#2344—28no—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#387C2212yes—
32C2×#387C2214yes—
33C2×#387C2212yes—
34C2×#387C2214yes—
35C2×#385C2212yes—
36C2×#385C2211yescharacteristic
37C2×#385C2211yescharacteristic
38C2×#384—28no—
39C2×#384C2214yes—
40C2×#384C2212yes—
41C2×#384C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#1445C2212yes—
57#1445C2214yes—
58#1445C2214yes—
59#1445C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2212yes—
84C22⋊#95C2211yescharacteristic
85C4×#95C2212yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4—44no—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C42—24no—
130C22×C42—28no—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138#400—216no—
139#398—216no—
140#389—24no—
141#389—24no—
142#389—28no—
143#387—28no—
144#387—28no—
145#387—216no—
146#387—216no—
147#385—28no—
148#385—24no—
149#385D418yes—
150#384—28no—
151#384—216no—
152#384—28no—
153#381—28no—
154#381—28no—
155#381—28no—
156#381—28no—
157#381—44no—
158#381—44no—
159#381—44no—
160#380—24no—
161#380—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—28no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C25C2411yescharacteristic
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4C2×D418yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4#4114yes—
225C23×C4#4114yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243#117—432no—
244C2×C4⋊C4—232no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—432no—
291C2×C4⋊C4—432no—
292C2×C4⋊C4—416no—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D418yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D418yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C42—232no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42#4118yes—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95C2×D418yes—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C2×#4112yes—
372C24C2×#4112yes—
373C24C2×#4112yes—
374C24H(5,𝔽2)12yes—
375C24H(5,𝔽2)12yes—
376C24C2511yescharacteristic
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—216no—
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—28no—
396C22×C4—28no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4#123116yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4114yes—
499C22×C42−1+414yes—
500C22×C42−1+414yes—
501C22×C42−1+414yes—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—416no—
507C4⋊C4—416no—
508C4⋊C4—416no—
509C4⋊C4—416no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—416no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C22⋊C4—28no—
520C22⋊C4—28no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C42—216no—
526C42—216no—
527C42—48no—
528C42—48no—
529C42—44no—
530C42—44no—
531C42—48no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23C2×#12314yes—
549C23#53214yes—
550C23#54014yes—
551C23D4214yes—
552C23#54614yes—
553C23#55014yes—
554C23#55914yes—
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C22×#4111yescharacteristic
558C23C22×#4111yescharacteristic
559C23C22×#4112yes—
560C23C22×#4111yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×2−1+412yes—
564C23C2×2−1+412yes—
565C23C2×2−1+412yes—
566C2×C4—232no—
567C2×C4—232no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—232no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—432no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—816no—
659C2×C4C2×#12314yes—
660C2×C4C2×#12314yes—
661C2×C4#52514yes—
662C2×C4#53614yes—
663C2×C4#54718yes—
664C2×C4#55218yes—
665C2×C4#55218yes—
666C2×C4#55218yes—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—28no—
671C22—24no—
672C22—24no—
673C22—28no—
674C22—28no—
675C22—24no—
676C22—28no—
677C22#244518yes—
678C22C22×#12311yescharacteristic
679C22C2×#52511yescharacteristic
680C22C2×#53211yescharacteristic
681C22C2×#53611yescharacteristic
682C22C2×#54011yescharacteristic
683C22C2×D4211yescharacteristic
684C22C2×#54611yescharacteristic
685C22C2×#54712yes—
686C22C2×#55011yescharacteristic
687C22C2×#55212yes—
688C22C2×#55212yes—
689C22C2×#55212yes—
690C22C2×#55911yescharacteristic
691C22#348611yescharacteristic
692C22#348611yescharacteristic
693C22#349212yes—
694C22#349212yes—
695C22#349912yes—
696C22#350912yes—
697C22#350912yes—
698C22#350912yes—
699C22#354412yes—
700C22#354412yes—
701C4—28no—
702C4—28no—
703C4—24no—
704C4—42no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—42no—
709C4—82no—
710C4—81no—
711C4—81no—
712C4—84no—
713C4—82no—
714C4—84no—
715C4—82no—
716C4#248418yescomplemented
717C2—28no—
718C2C2×#244511yescharacteristic
719C2C2×#248411yescharacteristic
720C2#3567411yescharacteristic
721C2#3610311yescharacteristic
722C2#3616311yescharacteristic
723C2#3618112yes—
724C2#3788312yes—
725C2#3933711yescharacteristic
726C2#3946511yescharacteristic
727C2#5931212yes—
728C2#5932912yes—
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
4. #15434C2semidirect
5. #15388C2semidirect
6. #15146C2semidirect
7. #15133C2semidirect
11. #13981C2semidirect
15. #13884C2semidirect
716. C4#2484semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#23442
C22×#3842
#154342
#153882
#151462
#151332
#150732
#146972
#139812
#139042
#138842
#138702
C4×#3852