Finite Group Databasebeta

← #613848

Subgroups — #613848

Order 512.

3610 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic, complemented
3C2×#2535C211yesmaximal, characteristic
4C2×C42×D4C211yesmaximal, characteristic, complemented
5#15245C214yesmaximal, complemented
6#14833C214yesmaximal, complemented
7#14754C214yesmaximal, complemented
8#14709C211yesmaximal, characteristic, complemented
9#14696C214yesmaximal, complemented
10#13970C212yesmaximal, complemented
11#13970C212yesmaximal, complemented
12#13955C211yesmaximal, characteristic, complemented
13#13876C212yesmaximal, complemented
14#13875C212yesmaximal, complemented
15C4×#390C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×#121C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C42C2212yes—
22#2566—28no—
23#2562—28no—
24#2535—28no—
25C2×#400C2212yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#394C2212yes—
29C2×#392C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C42×D4—28no—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2214yes—
53C2×C43C2211yescharacteristic
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1438C2214yes—
62#1438C2214yes—
63#1437C2214yes—
64#1437C2212yes—
65#1437C2212yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2212yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76#1434C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C23×D4C2311yescharacteristic
80C23×D4C2314yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2314yes—
84C24×C4C2312yes—
85C2×#125—24no—
86C2×#125—24no—
87C2×#125—24no—
88C2×#121—28no—
89C2×#121—28no—
90C2×#121—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—28no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42—28no—
124C22×C42—48no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#400—28no—
135#397—28no—
136#396—216no—
137#394—28no—
138#392—28no—
139#390—24no—
140#390—24no—
141#390—28no—
142#388—216no—
143#388—48no—
144#386—216no—
145#386—216no—
146#386—48no—
147#385—216no—
148#384—24no—
149#382—24no—
150#382—24no—
151#380—28no—
152#380—28no—
153#380—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4D418yescomplemented
157C4×C22⋊C4—28no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—216no—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C43—24no—
183C25C2412yes—
184C25C2412yes—
185C22×D4—24no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4C2×D414yes—
201C22×D4#4118yes—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—28no—
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—28no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D418yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234#125—416no—
235#121—432no—
236C4×D4—264no—
237C4×D4—232no—
238C4×D4—416no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D414yes—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
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—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
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×C22⋊C4—44no—
306C2×C22⋊C4C2×D418yes—
307C2×C22⋊C4#4118yes—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—28no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—432no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42C2×D418yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42#4118yes—
330C2×C42#4114yes—
331#95—28no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D418yes—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
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—
373C24C22×D414yes—
374C24C2×#4114yes—
375C24C2×#4118yes—
376C24C2511yescharacteristic
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—216no—
380C2×D4—216no—
381C2×D4—28no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—44no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4#124116yes—
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—232no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—216no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4C22×D414yes—
475C22×C4C22×D418yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4H(5,𝔽2)14yes—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—88no—
502C22⋊C4—232no—
503C22⋊C4—216no—
504C22⋊C4—216no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C42—28no—
516C42—216no—
517C42—48no—
518C42—48no—
519C42—416no—
520C42—44no—
521C42—44no—
522C42—44no—
523C42#121116yes—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
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—
579C23C2×#12418yes—
580C23C23×D411yescharacteristic
581C23C23×D411yescharacteristic
582C23C22×#4111yescharacteristic
583C23C22×#4112yes—
584C23C22×#4112yes—
585C23C22×#4112yes—
586C23C2×H(5,𝔽2)12yes—
587C23#58512yes—
588C23#58512yes—
589D4—232no—
590D4—416no—
591D4—832no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—464no—
618C2×C4—432no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
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—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—816no—
666C2×C4C2×#12114yes—
667C2×C4C2×#12114yes—
668C2×C4C2×#12114yes—
669C2×C4#52518yes—
670C2×C4C2×#12414yes—
671C2×C4#52918yes—
672C2×C4#53214yes—
673C2×C4#55318yes—
674C22—28no—
675C22—216no—
676C22—48no—
677C22—44no—
678C22—48no—
679C22—48no—
680C22—48no—
681C22—48no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22C22×#12111yescharacteristic
699C22C2×#52512yes—
700C22C22×#12411yescharacteristic
701C22C2×#52912yes—
702C22C2×#53211yescharacteristic
703C22#344411yescharacteristic
704C22#345511yescharacteristic
705C22#346012yes—
706C22D4×#4112yes—
707C22D4×#4112yes—
708C22C2×#55312yes—
709C22#347512yes—
710C22#347512yes—
711C22#348112yes—
712C22#348411yescharacteristic
713C22#348711yescharacteristic
714C22#349612yes—
715C22#350612yes—
716C22#351212yes—
717C22#351312yes—
718C22#352112yes—
719C4—28no—
720C4—24no—
721C4—24no—
722C4—48no—
723C4—44no—
724C4—44no—
725C4—44no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—84no—
730C4—82no—
731C4#285018yes—
732C2—48no—
733C2—84no—
734C2#3474311yescharacteristic
735C2#3491712yes—
736C2#3516811yescharacteristic
737C2#3521812yes—
738C2#3622812yes—
739C2#3623812yes—
740C2#3644012yes—
741C2#3646812yes—
742C2C2×#285011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
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×#2562C2semidirect
2. C2×#2566C2semidirect
4. C2×C42×D4C2semidirect
5. #15245C2semidirect
6. #14833C2semidirect
7. #14754C2semidirect
8. #14709C2semidirect
9. #14696C2semidirect
10. #13970C2semidirect
11. #13970C2semidirect
12. #13955C2semidirect
13. #13876C2semidirect
14. #13875C2semidirect
15. C4×#390C2semidirect
156. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#25662
C2×#25352
C2×C42×D42
#152452
#148332
#147542
#147092
#146962
#139702
#139552
#138762
#138752
C4×#3902