Finite Group Databasebeta

← #793204

Subgroups — #793204

Order 512.

2728 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2495C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#16693C212yesmaximal, complemented
4#15781C214yesmaximal, complemented
5#15775C214yesmaximal, complemented
6#15195C211yesmaximal, characteristic, complemented
7#15181C211yesmaximal, characteristic, complemented
8#15145C211yesmaximal, characteristic, complemented
9#15100C212yesmaximal, complemented
10#14900C212yesmaximal
11#13994C212yesmaximal, complemented
12#13952C212yesmaximal, complemented
13#13906C212yesmaximal, complemented
14#13879C212yesmaximal, complemented
15#13871C212yesmaximal, complemented
16C4×#392C211yesmaximal, characteristic
17#13810C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×#117C2212yes—
21#2495—28no—
22C2×#400C2212yes—
23C2×#398C2214yes—
24C2×#397C2212yes—
25C2×#392C2211yescharacteristic
26#2354—28no—
27C2×#390C2212yes—
28C2×#390C2214yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#388C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2212yes—
36C2×#383C2212yes—
37C2×#382C2211yescharacteristic
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⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×#376C2212yes—
50#1444C2211yescharacteristic
51#1444C2212yes—
52#1444C2212yes—
53#1442C2212yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2212yes—
60#1439C2211yescharacteristic
61#1439C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1438C2211yescharacteristic
69#1438C2212yes—
70#1438C2214yes—
71#1438C2212yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2214yes—
81#1434C2212yes—
82#1434C2214yes—
83#1434C2214yes—
84C4×#95C2211yescharacteristic
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⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2314yes—
131#400—28no—
132#398—48no—
133#397—28no—
134#392—24no—
135#390—28no—
136#390—48no—
137#389—24no—
138#389—24no—
139#388—28no—
140#388—216no—
141#388—48no—
142#384—24no—
143#384—44no—
144#383—28no—
145#382—24no—
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—
157#376—28no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C25C2412yes—
186C25C2411yescharacteristic
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4#4118yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C4×D4—232no—
227C4×D4—432no—
228#117—432no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
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—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
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—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
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—44no—
304C2×C22⋊C4—44no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—22no—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—28no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319C2×C42C2×D418yes—
320C2×C42#4118yes—
321#95—22no—
322#95—22no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—22no—
339#95—44no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95C2×D418yes—
347#95#4114yes—
348C24—21no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24C2×#4114yes—
373C242−1+412yes—
374C24C2511yescharacteristic
375C2×D4—28no—
376C2×D4—216no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—48no—
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—22no—
396C22×C4—24no—
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—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—216no—
420C22×C4—216no—
421C22×C4—216no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—88no—
500C22⋊C4—216no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—28no—
515C42—216no—
516C42—216no—
517C42—416no—
518C42—44no—
519C42—416no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23C23×D411yescharacteristic
579C23C22×#4112yes—
580C23C22×#4112yes—
581C23C22×#4111yescharacteristic
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×H(5,𝔽2)12yes—
585C23C2×2−1+411yescharacteristic
586C23C2×2−1+411yescharacteristic
587C23#58512yes—
588C23#58511yescharacteristic
589D4—432no—
590D4—416no—
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—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—432no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
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—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—88no—
673C2×C4#52514yes—
674C2×C4#53214yes—
675C2×C4#53514yes—
676C2×C4#54714yes—
677C2×C4#55118yes—
678C2×C4#55318yes—
679C2×C4#55518yes—
680C22—42no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—44no—
690C22—44no—
691C22—42no—
692C22—82no—
693C22—81no—
694C22—82no—
695C22—81no—
696C22—81no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—81no—
704C22—82no—
705C22—82no—
706C22C2×#52511yescharacteristic
707C22C2×#53211yescharacteristic
708C22C2×#53511yescharacteristic
709C22C2×#54711yescharacteristic
710C22C2×#55112yes—
711C22C2×#55312yes—
712C22C2×#55512yes—
713C22#347512yes—
714C22#348711yescharacteristic
715C22#348911yescharacteristic
716C22#349212yes—
717C22#349312yes—
718C22#349512yes—
719C22#349512yes—
720C22#349811yescharacteristic
721C22#350612yes—
722C22#351112yes—
723C22#351711yescharacteristic
724C22#352011yescharacteristic
725C22#354212yes—
726C22#354612yes—
727C22#354711yescharacteristic
728C22#355211yescharacteristic
729C4—28no—
730C4—24no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—44no—
735C4—42no—
736C4—81no—
737C4—81no—
738C4—84no—
739C4—82no—
740C4—84no—
741C4—82no—
742C4—81no—
743C4#260318yes—
744C2—82no—
745C2—84no—
746C2C2×#260311yescharacteristic
747C2#3568911yescharacteristic
748C2#3606511yescharacteristic
749C2#3618711yescharacteristic
750C2#3642711yescharacteristic
751C2#3892011yescharacteristic
752C2#4287611yescharacteristic
753C2#4489812yes—
754C2#4497212yes—
755C2#5920312yes—
756C2#5922812yes—
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×#2495C2semidirect
2. C2×#2354C2semidirect
3. #16693C2semidirect
4. #15781C2semidirect
5. #15775C2semidirect
6. #15195C2semidirect
7. #15181C2semidirect
8. #15145C2semidirect
9. #15100C2semidirect
11. #13994C2semidirect
12. #13952C2semidirect
13. #13906C2semidirect
14. #13879C2semidirect
15. #13871C2semidirect
17. #13810C2semidirect
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×#24952
C2×#23542
#166932
#157812
#157752
#151952
#151812
#151452
#151002
#149002
#139942
#139522
#139062
#138792
#138712
C4×#3922
#138102