Finite Group Databasebeta

← #674796

Subgroups — #674796

Order 512.

3646 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2C2×#2420C212yesmaximal, complemented
3C2×#2422C212yesmaximal, complemented
4C2×#2349C211yesmaximal, characteristic, complemented
5#15539C214yesmaximal, complemented
6#15370C214yesmaximal, complemented
7#15223C212yesmaximal, complemented
8#15135C212yesmaximal, complemented
9#14839C214yesmaximal, complemented
10#14687C211yesmaximal, characteristic, complemented
11#13996C214yesmaximal, complemented
12#13954C211yesmaximal, characteristic, complemented
13#13905C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic, complemented
16C22×#123C2214yes—
17C22×#121C2212yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2535—28no—
22#2422—216no—
23#2420—216no—
24C2×#399C2214yes—
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#397C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#392C2212yes—
31#2349—28no—
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2212yes—
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2212yes—
46C2×#381C2214yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2214yes—
56C22×#95C2212yes—
57C2×C43C2211yescharacteristic
58#1445C2214yes—
59#1445C2212yes—
60#1445C2214yes—
61#1445C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2212yes—
71#1437C2212yes—
72C4⋊#95C2214yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C23×D4C2311yescharacteristic
80C23×D4C2314yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2314yes—
84C24×C4C2312yes—
85C2×#123—216no—
86C2×#123—216no—
87C2×#123—48no—
88C2×#121—28no—
89C2×#121—28no—
90C2×#121—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D418yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#399—48no—
133#397—216no—
134#397—216no—
135#397—28no—
136#394—216no—
137#394—216no—
138#392—28no—
139#390—28no—
140#390D418yescomplemented
141#390D418yescomplemented
142#389—28no—
143#389—28no—
144#389—28no—
145#386—216no—
146#386—216no—
147#385—216no—
148#385—216no—
149#384—28no—
150#382—24no—
151#382—28no—
152#382—24no—
153#381—216no—
154#380—28no—
155#380—28no—
156#380—28no—
157#380—28no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—48no—
163C2×#95—28no—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C43—24no—
186C25C2412yes—
187C25C2412yes—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4C2×D414yes—
205C22×D4C2×D414yes—
206C22×D4C2×D414yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D418yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239#123—464no—
240#121—432no—
241C4×D4—232no—
242C4×D4—432no—
243C4×D4—416no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
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—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D4116yes—
317C2×C22⋊C4C2×D4116yes—
318C2×C22⋊C4#4118yes—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—44no—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42#4114yes—
338C2×C42#4114yes—
339#95—22no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—24no—
356#95—28no—
357#95—24no—
358#95—28no—
359#95—44no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24C22×D414yes—
383C24C22×D414yes—
384C24C22×D414yes—
385C24C2×#4114yes—
386C24C2511yescharacteristic
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—44no—
397C2×D4—416no—
398C2×D4—416no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—44no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4#127116yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—216no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—88no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4—88no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4114yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4H(5,𝔽2)14yes—
504C22×C42−1+412yes—
505C22×C42−1+412yes—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—416no—
515C4⋊C4—88no—
516C4⋊C4—88no—
517C4⋊C4—88no—
518C22⋊C4—216no—
519C22⋊C4—232no—
520C22⋊C4—216no—
521C22⋊C4—416no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C42—28no—
532C42—28no—
533C42—48no—
534C42—48no—
535C42—48no—
536C42—84no—
537C42#123116yescomplemented
538C23—24no—
539C23—24no—
540C23—28no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
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—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23C2×#12718yes—
596C23C23×D411yescharacteristic
597C23C23×D411yescharacteristic
598C23C23×D411yescharacteristic
599C23C22×#4111yescharacteristic
600C23C22×#4111yescharacteristic
601C23C22×#4112yes—
602C23C2×H(5,𝔽2)12yes—
603C23C2×2−1+411yescharacteristic
604C23C2×2−1+411yescharacteristic
605C23#58512yes—
606C23#58512yes—
607D4—232no—
608D4—416no—
609D4—832no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—28no—
635C2×C4—28no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—816no—
682C2×C4—832no—
683C2×C4—816no—
684C2×C4C2×#12314yes—
685C2×C4C2×#12314yes—
686C2×C4C2×#12314yes—
687C2×C4C2×#12714yes—
688C2×C4#53614yes—
689C2×C4#54618yes—
690C2×C4#54714yes—
691C2×C4#54714yes—
692C2×C4#54714yes—
693C2×C4#54714yes—
694C22—216no—
695C22—28no—
696C22—48no—
697C22—44no—
698C22—44no—
699C22—48no—
700C22—44no—
701C22—48no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C22×#12311yescharacteristic
721C22C22×#12711yescharacteristic
722C22#344411yescharacteristic
723C22C2×#53611yescharacteristic
724C22#345511yescharacteristic
725C22C2×#54612yes—
726C22C2×#54711yescharacteristic
727C22C2×#54711yescharacteristic
728C22C2×#54711yescharacteristic
729C22C2×#54711yescharacteristic
730C22D4×#4112yes—
731C22D4×#4112yes—
732C22#347512yes—
733C22#349412yes—
734C22#349612yes—
735C22#349612yes—
736C22#350012yes—
737C22#350811yescharacteristic
738C22#351011yescharacteristic
739C22#351112yes—
740C22#351512yes—
741C22#355912yes—
742C22#356512yes—
743C4—24no—
744C4—28no—
745C4—24no—
746C4—44no—
747C4—44no—
748C4—44no—
749C4—44no—
750C4—44no—
751C4—82no—
752C4—84no—
753C4—84no—
754C4—82no—
755C4#250118yes—
756C2—48no—
757C2—84no—
758C2#3474211yescharacteristic
759C2C2×#250111yescharacteristic
760C2#3554111yescharacteristic
761C2#3613511yescharacteristic
762C2#3614911yescharacteristic
763C2#3626012yes—
764C2#3937211yescharacteristic
765C2#3937411yescharacteristic
766C2#3995212yes—
767C2#5931512yes—
768C2#5931812yes—
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
2. C2×#2420C2semidirect
3. C2×#2422C2semidirect
4. C2×#2349C2semidirect
5. #15539C2semidirect
6. #15370C2semidirect
7. #15223C2semidirect
8. #15135C2semidirect
9. #14839C2semidirect
10. #14687C2semidirect
11. #13996C2semidirect
12. #13954C2semidirect
13. #13905C2semidirect
14. C4×#390C2semidirect
15. C4×#390C2semidirect
140. #390D4semidirect
141. #390D4semidirect
537. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
C2×#24202
C2×#24222
C2×#23492
#155392
#153702
#152232
#151352
#148392
#146872
#139962
#139542
#139052
C4×#3902