Finite Group Databasebeta

← #793212

Subgroups — #793212

Order 512.

2656 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#16799C212yesmaximal
4#15835C214yesmaximal, complemented
5#15733C214yesmaximal, complemented
6#15242C211yesmaximal, characteristic, complemented
7#15212C212yesmaximal, complemented
8#15181C211yesmaximal, characteristic, complemented
9#15150C211yesmaximal, characteristic, complemented
10#14982C212yesmaximal, complemented
11#14249C212yesmaximal, complemented
12#13994C212yesmaximal, complemented
13#13905C212yesmaximal, complemented
14#13879C212yesmaximal, complemented
15#13871C212yesmaximal, complemented
16C4×#392C211yesmaximal, characteristic
17#13812C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×#117C2212yes—
21#2497—28no—
22C2×#400C2212yes—
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#396C2212yes—
26C2×#392C2211yescharacteristic
27#2354—28no—
28C2×#390C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#388C2214yes—
32C2×#388C2212yes—
33C2×#388C2214yes—
34C2×#387C2212yes—
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1445C2212yes—
53#1445C2212yes—
54#1444C2211yescharacteristic
55#1442C2212yes—
56#1442C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2211yescharacteristic
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1438C2211yescharacteristic
67#1438C2214yes—
68#1438C2212yes—
69#1438C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—28no—
93C2×C4×D4—44no—
94C2×#117—28no—
95C2×#117—44no—
96C2×#117—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131#400—28no—
132#397—28no—
133#397—44no—
134#396—28no—
135#392—24no—
136#390—28no—
137#389—24no—
138#389—24no—
139#388—28no—
140#388—216no—
141#388—48no—
142#387—28no—
143#384—24no—
144#384—24no—
145#382—44no—
146#380—28no—
147#380—28no—
148#380—24no—
149#380—24no—
150#380—44no—
151#380—44no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4D418yescomplemented
155C4×C22⋊C4—24no—
156C4×C22⋊C4—28no—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2411yescharacteristic
185C25C2412yes—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—42no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4#4118yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C4×D4—232no—
226C4×D4—432no—
227#117—432no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
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—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4#4114yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C2×D414yes—
317C2×C42C2×D418yes—
318C2×C42#4118yes—
319#95—22no—
320#95—22no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—22no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—22no—
335#95—44no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—42no—
345#95C2×D418yes—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—21no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24C2×#4112yes—
371C24C2×#4114yes—
372C24C2511yescharacteristic
373C2×D4—28no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—44no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+412yes—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C22⋊C4—216no—
497C22⋊C4—28no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—84no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—28no—
511C42—216no—
512C42—48no—
513C42—416no—
514C42—44no—
515C42—88no—
516C23—24no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—42no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23C23×D411yescharacteristic
575C23C22×#4112yes—
576C23C22×#4112yes—
577C23C22×#4111yescharacteristic
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)11yescharacteristic
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×2−1+411yescharacteristic
582C23#58512yes—
583C23#58512yes—
584C23#58511yescharacteristic
585D4—416no—
586D4—432no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—432no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—88no—
668C2×C4#52514yes—
669C2×C4#53214yes—
670C2×C4#53514yes—
671C2×C4#54814yes—
672C2×C4#55118yes—
673C2×C4#55318yes—
674C22—44no—
675C22—42no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—42no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—42no—
686C22—81no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—81no—
695C22—82no—
696C22—81no—
697C22—82no—
698C22—81no—
699C22—82no—
700C22C2×#52511yescharacteristic
701C22C2×#53211yescharacteristic
702C22C2×#53511yescharacteristic
703C22C2×#54811yescharacteristic
704C22C2×#55112yes—
705C22C2×#55312yes—
706C22#347512yes—
707C22#347512yes—
708C22#348511yescharacteristic
709C22#348811yescharacteristic
710C22#349312yes—
711C22#349312yes—
712C22#349412yes—
713C22#349512yes—
714C22#350011yescharacteristic
715C22#350611yescharacteristic
716C22#350612yes—
717C22#350712yes—
718C22#350811yescharacteristic
719C22#353912yes—
720C22#354811yescharacteristic
721C22#355411yescharacteristic
722C22#356212yes—
723C4—24no—
724C4—44no—
725C4—44no—
726C4—44no—
727C4—44no—
728C4—42no—
729C4—42no—
730C4—44no—
731C4—42no—
732C4—81no—
733C4—84no—
734C4—82no—
735C4—84no—
736C4—82no—
737C4#260318yes—
738C2—84no—
739C2—82no—
740C2C2×#260311yescharacteristic
741C2#3575611yescharacteristic
742C2#3606211yescharacteristic
743C2#3651611yescharacteristic
744C2#3653411yescharacteristic
745C2#3972511yescharacteristic
746C2#4265211yescharacteristic
747C2#4490012yes—
748C2#4513012yes—
749C2#5920012yes—
750C2#5930712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2497C2semidirect
2. C2×#2354C2semidirect
4. #15835C2semidirect
5. #15733C2semidirect
6. #15242C2semidirect
7. #15212C2semidirect
8. #15181C2semidirect
9. #15150C2semidirect
10. #14982C2semidirect
11. #14249C2semidirect
12. #13994C2semidirect
13. #13905C2semidirect
14. #13879C2semidirect
15. #13871C2semidirect
17. #13812C2semidirect
154. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#23542
#167992
#158352
#157332
#152422
#152122
#151812
#151502
#149822
#142492
#139942
#139052
#138792
#138712
C4×#3922
#138122