Finite Group Databasebeta

← #4320003

Subgroups — #4320003

Order 512.

4006 conjugacy classes of subgroups.

The subgroup lattice has 775 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2539C212yesmaximal, complemented
2#16824C211yesmaximal, characteristic, complemented
3#16338C212yesmaximal, complemented
4#16252C211yesmaximal, characteristic, complemented
5#16250C214yesmaximal, complemented
6#15334C214yesmaximal, complemented
7#15320C214yesmaximal, complemented
8#14964C214yesmaximal, complemented
9#14716C212yesmaximal, complemented
10#14716C212yesmaximal, complemented
11#14521C212yesmaximal, complemented
12#14418C212yesmaximal, complemented
13#13794C211yesmaximal, characteristic, complemented
14C22×#123C2214yes—
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×#120C2214yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2539—216no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#394C2214yescomplemented
26C2×#392C2212yes—
27C2×#392C2212yes—
28C2×#392C2212yes—
29C2×#392C2212yes—
30C2×#392C2212yes—
31C2×#392C2212yes—
32C2×#392C2214yescomplemented
33C2×#390C2214yes—
34C2×#390C2214yes—
35C2×#388C2214yescomplemented
36C2×#387C2214yescomplemented
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×C4×C22⋊C4C2212yes—
40#1442C2214yescomplemented
41#1441C2214yescomplemented
42#1441C2214yescomplemented
43#1440C2214yescomplemented
44#1439C2212yes—
45#1439C2212yes—
46#1439C2212yes—
47#1439C2212yes—
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yescomplemented
57#1438C2214yescomplemented
58#1437C2214yescomplemented
59#1436C2214yescomplemented
60C22⋊#95C2211yescharacteristic
61C22⋊#95C2212yes—
62C22⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2214yes—
68C26C2311yescharacteristic
69C23×D4C2314yes—
70C23×D4C2314yes—
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×#123—216no—
80C2×#123—48no—
81C2×#123—48no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—48no—
85C2×#121—48no—
86C2×#121—44no—
87C2×#121—44no—
88C2×#121—44no—
89C2×#120—216no—
90C2×#120—48no—
91C2×#120—48no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—42no—
100C22×C22⋊C4—42no—
101C22×C22⋊C4—42no—
102C22×C22⋊C4—42no—
103C22×C22⋊C4—42no—
104C22×C22⋊C4—42no—
105C22×C22⋊C4D418yescomplemented
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C42C2314yes—
127#399—28no—
128#399—28no—
129#394—48no—
130#392—28no—
131#392—28no—
132#392—28no—
133#392—48no—
134#392—44no—
135#392—44no—
136#392—44no—
137#390—216no—
138#390—216no—
139#388—48no—
140#387—48no—
141#386—48no—
142#386—48no—
143C4×C22⋊C4—28no—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25—22no—
161C25—22no—
162C25—22no—
163C25—22no—
164C25—22no—
165C25—22no—
166C25—44no—
167C25—44no—
168C25C2×D414yes—
169C25C2414yes—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—44no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—41no—
215C23×C4—41no—
216C23×C4—42no—
217C23×C4—41no—
218C23×C4—41no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D418yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236#123—464no—
237#121—432no—
238#121—432no—
239#121—432no—
240#120—464no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—416no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
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—416no—
304C2×C22⋊C4—416no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4#4114yes—
317C2×C42—28no—
318C2×C42—44no—
319C2×C42—44no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—44no—
336C24—28no—
337C24—28no—
338C24—28no—
339C24—28no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—21no—
354C24—24no—
355C24—24no—
356C24—21no—
357C24—21no—
358C24—28no—
359C24—28no—
360C24—28no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—416no—
377C24—84no—
378C24—84no—
379C24—88no—
380C24—84no—
381C24—88no—
382C24—88no—
383C24—84no—
384C24—84no—
385C24—84no—
386C24C22×D412yes—
387C24C22×D412yes—
388C24C22×D412yes—
389C24C2×#4112yes—
390C24C2×#4112yes—
391C24H(5,𝔽2)12yes—
392C24H(5,𝔽2)12yes—
393C24H(5,𝔽2)12yes—
394C242−1+412yes—
395C24C2511yescharacteristic
396C2×D4—416no—
397C2×D4—416no—
398C2×D4—416no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—416no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—416no—
405C2×D4—416no—
406C2×D4—48no—
407C2×D4—416no—
408C2×D4—416no—
409C2×D4—416no—
410C2×D4—88no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—88no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—84no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—84no—
429C2×D4—88no—
430C2×D4—88no—
431C2×D4—84no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—216no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—88no—
485C22×C4—88no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4—88no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4C22×D414yes—
496C22×C4C22×D414yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4114yes—
499C22×C4H(5,𝔽2)18yes—
500C22×C4H(5,𝔽2)18yes—
501C4⋊C4—216no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C22⋊C4—232no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—44no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C42—88no—
527C23—232no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—28no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
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—44no—
609C23—44no—
610C23—44no—
611C23—816no—
612C23—816no—
613C23—816no—
614C23—816no—
615C23—816no—
616C23—88no—
617C23—816no—
618C23—88no—
619C23#53214yes—
620C23#53414yes—
621C23#53714yes—
622C23C23×D411yescharacteristic
623C23C22×#4111yescharacteristic
624C23C22×#4111yescharacteristic
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)11yescharacteristic
627C23C2×H(5,𝔽2)11yescharacteristic
628C23C2×H(5,𝔽2)14yes—
629C23C2×H(5,𝔽2)14yes—
630C23C2×2−1+411yescharacteristic
631D4—832no—
632D4—832no—
633D4—816no—
634D4—816no—
635D4—816no—
636D4—832no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—28no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—28no—
647C2×C4—24no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—816no—
689C2×C4—88no—
690C2×C4—816no—
691C2×C4—88no—
692C22—28no—
693C22—28no—
694C22—28no—
695C22—24no—
696C22—24no—
697C22—24no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—832no—
720C22—84no—
721C22—84no—
722C22—84no—
723C22—84no—
724C22—84no—
725C22—84no—
726C22—84no—
727C22—84no—
728C22—84no—
729C22—84no—
730C22—84no—
731C22—84no—
732C22—84no—
733C22—84no—
734C22—84no—
735C22C2×#53211yescharacteristic
736C22C2×#53411yescharacteristic
737C22C2×#53711yescharacteristic
738C22#348414yes—
739C22#349012yes—
740C22#349011yescharacteristic
741C22#349011yescharacteristic
742C22#349211yescharacteristic
743C22#349211yescharacteristic
744C22#350812yes—
745C22#351914yes—
746C22#353212yes—
747C22#353614yes—
748C22#354314yes—
749C22#354312yes—
750C22#354312yes—
751C22#354312yes—
752C4—44no—
753C4—44no—
754C4—48no—
755C4—44no—
756C4—44no—
757C4—48no—
758C4—82no—
759C4—84no—
760C4—82no—
761C2—44no—
762C2—44no—
763C2—44no—
764C2—164no—
765C2#3505111yescharacteristic
766C2#3569412yes—
767C2#3723212yes—
768C2#4058612yes—
769C2#4638711yescharacteristic
770C2#4650712yes—
771C2#4703111yescharacteristic
772C2#4788412yes—
773C2#4814312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2539C2semidirect
2. #16824C2semidirect
3. #16338C2semidirect
4. #16252C2semidirect
5. #16250C2semidirect
6. #15334C2semidirect
7. #15320C2semidirect
8. #14964C2semidirect
9. #14716C2semidirect
10. #14716C2semidirect
11. #14521C2semidirect
12. #14418C2semidirect
13. #13794C2semidirect
25. C2×#394C22semidirect
32. C2×#392C22semidirect
35. C2×#388C22semidirect
36. C2×#387C22semidirect
40. #1442C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
56. #1438C22semidirect
57. #1438C22semidirect
58. #1437C22semidirect
59. #1436C22semidirect
105. C22×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25392
#168242
#163382
#162522
#162502
#153342
#153202
#149642
#147162
#145212
#144182
#137942