Finite Group Databasebeta

← #793166

Subgroups — #793166

Order 512.

2842 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic, complemented
2C2×#2345C211yesmaximal, characteristic, complemented
3#16886C212yesmaximal, complemented
4#15442C214yesmaximal, complemented
5#15402C214yesmaximal, complemented
6#15213C212yesmaximal, complemented
7#15181C211yesmaximal, characteristic, complemented
8#15153C211yesmaximal, characteristic, complemented
9#15135C211yesmaximal, characteristic, complemented
10#14977C212yesmaximal
11#14240C212yesmaximal, complemented
12#14129C212yesmaximal, complemented
13#13974C212yesmaximal, complemented
14#13936C211yesmaximal, characteristic, complemented
15#13872C212yesmaximal, complemented
16#13871C212yesmaximal, complemented
17C4×#392C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C2×#400C2214yes—
23C2×#399C2212yes—
24C2×#392C2211yescharacteristic
25#2354—28no—
26#2345—28no—
27C2×#390C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#386C2214yes—
36C2×#384C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1444C2212yes—
53#1444C2211yescharacteristic
54#1443C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2211yescharacteristic
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1438C2211yescharacteristic
69#1438C2214yes—
70#1438C2212yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—44no—
95C2×#117—28no—
96C2×#117—44no—
97C2×#117—44no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134#400—48no—
135#399—28no—
136#392—24no—
137#390—28no—
138#389—24no—
139#389—24no—
140#389—44no—
141#388—216no—
142#388—48no—
143#387—28no—
144#387—28no—
145#386—48no—
146#384—24no—
147#380—28no—
148#380—24no—
149#380—28no—
150#380—28no—
151#380—28no—
152#380—44no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4D418yescomplemented
156C4×C22⋊C4—24no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—44no—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
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—44no—
214C23×C4—44no—
215C23×C4#4114yes—
216C23×C4#4118yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C4×D4—232no—
233C4×D4—432no—
234#117—432no—
235C2×C4⋊C4—232no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
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—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4#4114yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4#4114yes—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D414yes—
324C2×C42C2×D418yes—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—22no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—22no—
341#95—44no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D418yes—
352C24—21no—
353C24—22no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24C2×#4114yes—
375C24C2×#4112yes—
376C24C2×#4112yes—
377C242−1+412yes—
378C24C2511yescharacteristic
379C2×D4—28no—
380C2×D4—216no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—44no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
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—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—216no—
430C22×C4—28no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—216no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C42−1+412yes—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—88no—
507C22⋊C4—216no—
508C22⋊C4—28no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C42—28no—
521C42—48no—
522C42—416no—
523C42—44no—
524C42—48no—
525C42—84no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23#55614yes—
584C23C23×D411yescharacteristic
585C23C22×#4111yescharacteristic
586C23C22×#4112yes—
587C23C22×#4111yescharacteristic
588C23C22×#4111yescharacteristic
589C23C2×H(5,𝔽2)12yes—
590C23C2×H(5,𝔽2)12yes—
591C23C2×2−1+411yescharacteristic
592C23C2×2−1+411yescharacteristic
593C23#58512yes—
594C23#58511yescharacteristic
595D4—416no—
596D4—432no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—416no—
646C2×C4—432no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—816no—
679C2×C4#52514yes—
680C2×C4#52514yes—
681C2×C4#53514yes—
682C2×C4#54814yes—
683C2×C4#55118yes—
684C2×C4#55318yes—
685C22—28no—
686C22—24no—
687C22—42no—
688C22—42no—
689C22—44no—
690C22—42no—
691C22—44no—
692C22—42no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22C2×#52511yescharacteristic
712C22C2×#52511yescharacteristic
713C22C2×#53511yescharacteristic
714C22C2×#54811yescharacteristic
715C22C2×#55112yes—
716C22C2×#55312yes—
717C22C2×#55611yescharacteristic
718C22#347512yes—
719C22#348011yescharacteristic
720C22#348511yescharacteristic
721C22#348911yescharacteristic
722C22#349212yes—
723C22#349412yes—
724C22#349512yes—
725C22#349811yescharacteristic
726C22#350612yes—
727C22#350711yescharacteristic
728C22#351412yes—
729C22#351912yes—
730C22#352311yescharacteristic
731C22#353912yes—
732C22#354112yes—
733C22#354812yes—
734C4—24no—
735C4—42no—
736C4—44no—
737C4—42no—
738C4—44no—
739C4—42no—
740C4—44no—
741C4—44no—
742C4—44no—
743C4—84no—
744C4—82no—
745C4—82no—
746C4—84no—
747C4—81no—
748C4#259818yes—
749C2—44no—
750C2—84no—
751C2C2×#259811yescharacteristic
752C2#3568911yescharacteristic
753C2#3602211yescharacteristic
754C2#3651611yescharacteristic
755C2#3657911yescharacteristic
756C2#3916911yescharacteristic
757C2#4267911yescharacteristic
758C2#4490012yes—
759C2#4497712yes—
760C2#5920412yes—
761C2#5975312yes—
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×#2354C2semidirect
2. C2×#2345C2semidirect
3. #16886C2semidirect
4. #15442C2semidirect
5. #15402C2semidirect
6. #15213C2semidirect
7. #15181C2semidirect
8. #15153C2semidirect
9. #15135C2semidirect
11. #14240C2semidirect
12. #14129C2semidirect
13. #13974C2semidirect
14. #13936C2semidirect
15. #13872C2semidirect
16. #13871C2semidirect
155. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#23452
#168862
#154422
#154022
#152132
#151812
#151532
#151352
#149772
#142402
#141292
#139742
#139362
#138722
#138712
C4×#3922