Finite Group Databasebeta

← #4327721

Subgroups — #4327721

Order 512.

3450 conjugacy classes of subgroups.

The subgroup lattice has 780 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2431C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic, complemented
3#16392C212yesmaximal, complemented
4#16363C212yesmaximal, complemented
5#16266C214yesmaximal, complemented
6#16110C212yesmaximal, complemented
7#16094C212yesmaximal, complemented
8#15307C214yesmaximal, complemented
9#15113C212yesmaximal, complemented
10#15063C212yesmaximal, complemented
11#14836C212yesmaximal, complemented
12#14777C212yesmaximal, complemented
13#14419C212yesmaximal, complemented
14#14401C212yesmaximal, complemented
15#13956C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2431—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#392C2212yes—
28C2×#392C2212yes—
29#2311—28no—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#388C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#386C2212yes—
43C2×#385C2214yes—
44C2×#381C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2214yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1446C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1438C2212yes—
72#1436C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22×#41—28no—
78C22×#41—42no—
79C22×#41—42no—
80C22×#41—44no—
81C22×#41D418yes—
82C23×Q8C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94C24×C4C2312yes—
95C2×#127—24no—
96C2×#127—28no—
97C2×#124—24no—
98C2×#124—44no—
99C2×#121—24no—
100C2×#121—42no—
101C2×#121—42no—
102C2×#117—24no—
103C2×#117—44no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—42no—
106C22×C4⋊C4—42no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#399—44no—
134#399—44no—
135#394—48no—
136#394—48no—
137#392—28no—
138#392—28no—
139#391—28no—
140#391—24no—
141#390—28no—
142#390—24no—
143#388—48no—
144#386—28no—
145#386—28no—
146#386—28no—
147#386—28no—
148#386—28no—
149#386—44no—
150#386—48no—
151#386—48no—
152#385—48no—
153#381—42no—
154#380—28no—
155#380—28no—
156#380—48no—
157C4×C22⋊C4—24no—
158C2×#95—42no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C25C2411yescharacteristic
182C25C2411yescharacteristic
183C2×#41—232no—
184C2×#41—416no—
185C2×#41—416no—
186C2×#41—432no—
187C2×#41—432no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8C2×D414yes—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—42no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4—41no—
209C22×D4—41no—
210C22×D4—41no—
211C22×D4—41no—
212C22×D4—42no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4—44no—
221C22×D4—44no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4C2×D414yes—
225C22×D4C2×D414yes—
226C22×D4C2×D414yes—
227C22×D4#4118yes—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—28no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—42no—
247C23×C4—44no—
248C23×C4—42no—
249C23×C4—44no—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2×D414yes—
253C23×C4C2414yes—
254C23×C4C2412yes—
255C23×C4C2414yes—
256C23×C4C2414yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2414yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264#127—416no—
265#124—416no—
266#121—416no—
267#117—416no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—416no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—28no—
338C2×C42—42no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—42no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—24no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24C22×D412yes—
390C24C22×D412yes—
391C24C22×D412yes—
392C24C2×#4114yes—
393C24C2×#4114yes—
394C24C2511yescharacteristic
395#41—4128no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—44no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—416no—
416C2×D4—416no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—44no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4—88no—
430C2×D4—88no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—88no—
434C2×D4—88no—
435C2×D4#123116yes—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—416no—
506C22×C4—48no—
507C22×C4—88no—
508C22×C4—84no—
509C22×C4—88no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—84no—
513C22×C4—88no—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C2×#4114yes—
519C22×C4H(5,𝔽2)18yes—
520C22×C4H(5,𝔽2)14yes—
521C22×C4H(5,𝔽2)12yes—
522C22×C4H(5,𝔽2)12yes—
523C22×C42−1+414yes—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—88no—
529C4⋊C4—88no—
530C4⋊C4—88no—
531C22⋊C4—48no—
532C22⋊C4—416no—
533C22⋊C4—416no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—84no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C42—44no—
548C42—44no—
549C42—48no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—28no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—24no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23—42no—
587C23—44no—
588C23—44no—
589C23—42no—
590C23—44no—
591C23—42no—
592C23—44no—
593C23—42no—
594C23—41no—
595C23—41no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—44no—
602C23—44no—
603C23—42no—
604C23—42no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—42no—
613C23—42no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—42no—
618C23—42no—
619C23C2×#12314yes—
620C23C2×#12314yes—
621C23C23×D411yescharacteristic
622C23C22×#4112yes—
623C23C2×H(5,𝔽2)11yescharacteristic
624C23C2×H(5,𝔽2)12yes—
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)14yes—
627C23C2×2−1+412yes—
628C23#58512yes—
629Q8—416no—
630D4—416no—
631D4—416no—
632D4—416no—
633D4—88no—
634D4—816no—
635D4—816no—
636D4—816no—
637C2×C4—232no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—28no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—28no—
658C2×C4—416no—
659C2×C4—416no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—816no—
701C2×C4—816no—
702C2×C4C2×#12314yes—
703C2×C4#53414yes—
704C2×C4#53414yes—
705C22—28no—
706C22—28no—
707C22—24no—
708C22—24no—
709C22—42no—
710C22—42no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—44no—
717C22—42no—
718C22—42no—
719C22—48no—
720C22—48no—
721C22—42no—
722C22—44no—
723C22—44no—
724C22—42no—
725C22—42no—
726C22—82no—
727C22—82no—
728C22—84no—
729C22—82no—
730C22—84no—
731C22—84no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22—84no—
739C22—84no—
740C22C22×#12311yescharacteristic
741C22C2×#53411yescharacteristic
742C22C2×#53411yescharacteristic
743C22#348912yes—
744C22#349012yes—
745C22#349012yes—
746C22#349812yes—
747C22#351314yes—
748C22#352014yes—
749C22#353712yes—
750C22#354214yes—
751C22#354712yes—
752C22#354814yes—
753C22#354912yes—
754C22#355011yescharacteristic
755C22#355111yescharacteristic
756C4—24no—
757C4—24no—
758C4—24no—
759C4—48no—
760C4—44no—
761C4—42no—
762C4—82no—
763C4—84no—
764C4—84no—
765C4—82no—
766C4—84no—
767C2—44no—
768C2—44no—
769C2—82no—
770C2—162no—
771C2#3499211yescharacteristic
772C2#3598712yes—
773C2#3729711yescharacteristic
774C2#3737611yescharacteristic
775C2#4290912yes—
776C2#4756914yes—
777C2#4968112yes—
778C2#4970312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
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×#2311C2semidirect
3. #16392C2semidirect
4. #16363C2semidirect
5. #16266C2semidirect
6. #16110C2semidirect
7. #16094C2semidirect
8. #15307C2semidirect
9. #15113C2semidirect
10. #15063C2semidirect
11. #14836C2semidirect
12. #14777C2semidirect
13. #14419C2semidirect
14. #14401C2semidirect
15. #13956C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24312
C2×#23112
#163922
#163632
#162662
#161102
#160942
#153072
#151132
#150632
#148362
#147772
#144192
#144012
#139562