Finite Group Databasebeta

← #4387627

Subgroups — #4387627

Order 512.

3890 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic, complemented
2C2×#2308C211yesmaximal, characteristic
3#16667C214yesmaximal, complemented
4#15439C214yesmaximal, complemented
5#15429C214yesmaximal, complemented
6#15328C214yesmaximal, complemented
7#15327C214yesmaximal, complemented
8#14726C214yesmaximal, complemented
9#14338C211yesmaximal, characteristic, complemented
10#14338C211yesmaximal, characteristic, complemented
11#14127C211yesmaximal, characteristic, complemented
12#13867C211yesmaximal, characteristic
13#13867C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18#2842—28no—
19C2×#399C2214yescomplemented
20C2×#396C2214yes—
21C2×#396C2214yescomplemented
22C2×#396C2214yescomplemented
23#2308—28no—
24C2×#386C2214yes—
25C2×#386C2214yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yescomplemented
31C2×#385C2212yescomplemented
32C2×#385C2214yescomplemented
33C2×#385C2214yescomplemented
34C2×#381C2212yes—
35C2×#380C2212yescomplemented
36C2×#380C2212yescomplemented
37C2×#380C2214yescomplemented
38C2×#379C2211yescharacteristic
39C2×#379C2211yescharacteristic
40C2×#379C2211yescharacteristic
41C2×#379C2212yes—
42C2×#379C2212yes—
43C2×#379C2211yescharacteristic
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1443C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1438C2214yes—
61#1438C2214yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yescomplemented
65C22⋊#95C2212yescomplemented
66C22⋊#95C2212yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×D4—24no—
72C23×D4—24no—
73C23×D4—44no—
74C23×D4—44no—
75C23×D4Q818yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118#399—216no—
119#396—216no—
120#396—216no—
121#396—48no—
122#386—216no—
123#386—216no—
124#386—44no—
125#386—44no—
126#386—44no—
127#386—44no—
128#385—48no—
129#385—44no—
130#385—48no—
131#385—44no—
132#381—28no—
133#380—44no—
134#380—44no—
135#380—48no—
136#379—24no—
137#379—24no—
138#379—24no—
139#379—44no—
140#379—44no—
141#379D418yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25—22no—
159C25—22no—
160C25—24no—
161C25—24no—
162C25—42no—
163C25—42no—
164C25—42no—
165C25—42no—
166C25—42no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25—42no—
171C25—42no—
172C25C2×D414yes—
173C25C2×Q814yes—
174C25C2×Q814yes—
175C25#4114yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C25C2412yes—
180C22×D4—232no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—432no—
194C22×D4—432no—
195C22×D4—416no—
196C22×D4—416no—
197C22×D4C2×Q818yes—
198C22×D4C2×Q818yes—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4C2×Q814yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—416no—
295C2×C22⋊C4—416no—
296C2×C22⋊C4—432no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4C2×D418yes—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4C2×D418yes—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336C24—216no—
337C24—28no—
338C24—28no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—28no—
366C24—216no—
367C24—44no—
368C24—44no—
369C24—48no—
370C24—48no—
371C24—44no—
372C24—44no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—44no—
378C24—44no—
379C24—48no—
380C24—48no—
381C24—82no—
382C24—82no—
383C24—82no—
384C24—82no—
385C24—84no—
386C24—84no—
387C24—88no—
388C24—88no—
389C24—88no—
390C24—88no—
391C24—88no—
392C24—88no—
393C24—88no—
394C24—88no—
395C24C22⋊Q8116yes—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C22×D412yes—
399C24C22×Q812yes—
400C24C22×Q812yes—
401C24C22×Q812yes—
402C24C22×Q814yes—
403C24C2×#4112yes—
404C24C2×#4112yes—
405C24C2×#4112yes—
406C24H(5,𝔽2)14yes—
407C24H(5,𝔽2)14yes—
408C24C2511yescharacteristic
409C2×D4—216no—
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—4128no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—24no—
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—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
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—48no—
479C22×C4—416no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—416no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4—88no—
489C22×C4—88no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—88no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4—88no—
500C22×C4—88no—
501C22×C4—88no—
502C22×C4—88no—
503C22×C4C22×D418yes—
504C22×C4C22×Q814yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4H(5,𝔽2)14yes—
507C4⋊C4—88no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—88no—
533C22⋊C4—88no—
534C23—232no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—28no—
578C23—28no—
579C23—28no—
580C23—28no—
581C23—28no—
582C23—28no—
583C23—416no—
584C23—416no—
585C23—416no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—88no—
609C23—88no—
610C23—832no—
611C23—88no—
612C23—88no—
613C23—88no—
614C23—88no—
615C23—832no—
616C23—88no—
617C23—88no—
618C23—88no—
619C23—88no—
620C23—832no—
621C23—832no—
622C23C2×C22⋊Q814yes—
623C23C2×C22⋊Q814yes—
624C23C2×C22⋊Q814yes—
625C23#54318yes—
626C23C23×D411yescharacteristic
627C23C23×Q811yescharacteristic
628C23C22×#4111yescharacteristic
629C23C2×H(5,𝔽2)12yes—
630C23C2×H(5,𝔽2)12yes—
631C23C2×H(5,𝔽2)12yes—
632C23C2×H(5,𝔽2)12yes—
633C23#58512yes—
634C23#58512yes—
635D4—432no—
636D4—432no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—416no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
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—88no—
694C2×C4—88no—
695C2×C4—832no—
696C2×C4—88no—
697C2×C4—88no—
698C22—24no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—24no—
704C22—216no—
705C22—24no—
706C22—28no—
707C22—24no—
708C22—24no—
709C22—44no—
710C22—44no—
711C22—42no—
712C22—42no—
713C22—44no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—44no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—44no—
722C22—42no—
723C22—42no—
724C22—42no—
725C22—44no—
726C22—42no—
727C22—48no—
728C22—48no—
729C22—48no—
730C22—48no—
731C22—48no—
732C22—48no—
733C22—48no—
734C22—48no—
735C22—48no—
736C22—832no—
737C22—832no—
738C22C22×C22⋊Q811yescharacteristic
739C22C2×#54312yes—
740C22#348412yes—
741C22#349011yescharacteristic
742C22#349011yescharacteristic
743C22#349011yescharacteristic
744C22#349011yescharacteristic
745C22#349111yescharacteristic
746C22#349111yescharacteristic
747C22#349612yes—
748C22#349612yes—
749C22#350112yes—
750C22#350112yes—
751C22#353414yes—
752C22#353714yes—
753C22#354214yes—
754C22#356014yes—
755C4—44no—
756C4—44no—
757C4—44no—
758C4—48no—
759C4—82no—
760C4—82no—
761C4—84no—
762C4—84no—
763C4—84no—
764C2—44no—
765C2—44no—
766C2—48no—
767C2—44no—
768C2#3498211yescharacteristic
769C2#3535911yescharacteristic
770C2#3535911yescharacteristic
771C2#4060914yes—
772C2#4638411yescharacteristic
773C2#4638611yescharacteristic
774C2#4638811yescharacteristic
775C2#4699711yescharacteristic
776C2#4784714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2842C2semidirect
3. #16667C2semidirect
4. #15439C2semidirect
5. #15429C2semidirect
6. #15328C2semidirect
7. #15327C2semidirect
8. #14726C2semidirect
9. #14338C2semidirect
10. #14338C2semidirect
11. #14127C2semidirect
19. C2×#399C22semidirect
21. C2×#396C22semidirect
22. C2×#396C22semidirect
30. C2×#385C22semidirect
31. C2×#385C22semidirect
32. C2×#385C22semidirect
33. C2×#385C22semidirect
35. C2×#380C22semidirect
36. C2×#380C22semidirect
37. C2×#380C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
C2×#23082
#166672
#154392
#154292
#153282
#153272
#147262
#143382
#141272
#138672