Finite Group Databasebeta

← #613112

Subgroups — #613112

Order 512.

3036 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3#15026C212yesmaximal, complemented
4#14945C212yesmaximal
5#14858C214yesmaximal, complemented
6#14785C214yesmaximal, complemented
7#14697C211yesmaximal, characteristic, complemented
8#14696C212yesmaximal
9#14695C212yesmaximal, complemented
10#14133C212yesmaximal, complemented
11#14030C212yesmaximal, complemented
12#13974C212yesmaximal, complemented
13#13924C211yesmaximal, characteristic, complemented
14#13881C211yesmaximal, characteristic, complemented
15#13875C212yesmaximal, complemented
16#13872C211yesmaximal, characteristic
17C4×#386C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C23×C42C2212yes—
23#2487—28no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#386C2212yes—
32C2×#386C2211yescharacteristic
33C2×#386C2212yes—
34C2×#386C2211yescharacteristic
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4—28no—
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C22×#95C2212yes—
50#1445C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2212yes—
62#1437C2214yes—
63#1437C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75#1434C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4D418yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42—24no—
124C22×C42—28no—
125C22×C42—28no—
126C22×C42—48no—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2314yes—
132C22×C42C2312yes—
133#400—28no—
134#399—28no—
135#389—24no—
136#389—24no—
137#389—28no—
138#388—48no—
139#387—48no—
140#386—28no—
141#386—24no—
142#386—28no—
143#386—24no—
144#385—28no—
145#385—28no—
146#382—28no—
147#380—24no—
148#380—24no—
149#380—24no—
150#380—24no—
151#380—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—216no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—28no—
157C2×#95—28no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2411yescharacteristic
188C25C2411yescharacteristic
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4#4114yes—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C2×D414yes—
222C23×C4C2×D418yes—
223C23×C4#4114yes—
224C23×C4#4114yes—
225C23×C4#4114yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C4×D4—416no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—232no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4#4118yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C42—232no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—28no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—432no—
325C2×C42#4118yes—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—44no—
347#95—44no—
348#95—42no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24C22×D412yes—
370C24C2×#4112yes—
371C24C2×#4112yes—
372C24C2×#4112yes—
373C24C2×#4112yes—
374C24C2×#4112yes—
375C242−1+412yes—
376C242−1+412yes—
377C24C2511yescharacteristic
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—28no—
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—28no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—416no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—48no—
477C22×C4—416no—
478C22×C4—44no—
479C22×C4—416no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4#123116yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4118yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4114yes—
502C22×C4C2×#4114yes—
503C22×C4H(5,𝔽2)14yes—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—416no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—216no—
528C42—216no—
529C42—416no—
530C42—416no—
531C42—44no—
532C42—48no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23C2×#12314yes—
573C23#52914yes—
574C23#54114yes—
575C23#54814yes—
576C23#54814yes—
577C23#55214yes—
578C23#55614yes—
579C23C23×D411yescharacteristic
580C23C22×#4111yescharacteristic
581C23C22×#4111yescharacteristic
582C23C22×#4112yes—
583C23C22×#4112yes—
584C23C22×#4111yescharacteristic
585C23C22×#4111yescharacteristic
586C23C2×H(5,𝔽2)12yes—
587C23C2×2−1+411yescharacteristic
588C23C2×2−1+411yescharacteristic
589C23#58512yes—
590D4—416no—
591C2×C4—232no—
592C2×C4—22no—
593C2×C4—24no—
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—232no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—432no—
619C2×C4—416no—
620C2×C4—432no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—88no—
679C2×C4C2×#12314yes—
680C2×C4C2×#12314yes—
681C2×C4#52514yes—
682C2×C4#52514yes—
683C2×C4#52918yes—
684C2×C4#52918yes—
685C2×C4#55318yes—
686C22—24no—
687C22—24no—
688C22—28no—
689C22—28no—
690C22—24no—
691C22—24no—
692C22—28no—
693C22—24no—
694C22—24no—
695C22—28no—
696C22—42no—
697C22—44no—
698C22—42no—
699C22—42no—
700C22—44no—
701C22—81no—
702C22—81no—
703C22—82no—
704C22—81no—
705C22—81no—
706C22—82no—
707C22#242118yes—
708C22C22×#12311yescharacteristic
709C22C2×#52511yescharacteristic
710C22C2×#52511yescharacteristic
711C22C2×#52912yes—
712C22C2×#52912yes—
713C22C2×#52911yescharacteristic
714C22C2×#54111yescharacteristic
715C22#346012yes—
716C22C2×#54811yescharacteristic
717C22C2×#54811yescharacteristic
718C22C2×#55211yescharacteristic
719C22C2×#55312yes—
720C22C2×#55611yescharacteristic
721C22#347512yes—
722C22#348012yes—
723C22#348112yes—
724C22#348411yescharacteristic
725C22#348511yescharacteristic
726C22#349212yes—
727C22#349512yes—
728C22#350612yes—
729C22#351612yes—
730C22#352012yes—
731C4—24no—
732C4—28no—
733C4—44no—
734C4—44no—
735C4—42no—
736C4—44no—
737C4—48no—
738C4—81no—
739C4—81no—
740C4—82no—
741C4—84no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#285318yes—
746C2—28no—
747C2—82no—
748C2C2×#242111yescharacteristic
749C2#3478812yes—
750C2#3515211yescharacteristic
751C2#3523112yes—
752C2#3625412yes—
753C2#3651611yescharacteristic
754C2#3652811yescharacteristic
755C2#3658111yescharacteristic
756C2#3658211yescharacteristic
757C2#3658612yes—
758C2C2×#285311yescharacteristic
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
2. C22×C4×C4⋊C4C2semidirect
3. #15026C2semidirect
5. #14858C2semidirect
6. #14785C2semidirect
7. #14697C2semidirect
9. #14695C2semidirect
10. #14133C2semidirect
11. #14030C2semidirect
12. #13974C2semidirect
13. #13924C2semidirect
14. #13881C2semidirect
15. #13875C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C22×C4×C4⋊C42
#150262
#149452
#148582
#147852
#146972
#146962
#146952
#141332
#140302
#139742
#139242
#138812
#138752
#138722
C4×#3862