Finite Group Databasebeta

← #1423513

Subgroups — #1423513

Order 512.

4662 conjugacy classes of subgroups.

The subgroup lattice has 794 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2433C211yesmaximal, characteristic
2C2×#2366C211yesmaximal, characteristic, complemented
3#15984C212yesmaximal, complemented
4#15496C214yesmaximal, complemented
5#15313C212yesmaximal, complemented
6#15283C214yesmaximal, complemented
7#15112C212yesmaximal, complemented
8#14730C214yesmaximal, complemented
9#14513C212yesmaximal, complemented
10#14419C211yesmaximal, characteristic, complemented
11#14240C212yesmaximal, complemented
12#13967C214yesmaximal, complemented
13#13867C212yesmaximal, complemented
14C24×D4C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#120C2214yes—
19C22×C4×D4C2214yescomplemented
20C23×C22⋊C4C2212yes—
21#2433—28no—
22C2×#399C2212yes—
23C2×#397C2214yescomplemented
24C2×#394C2214yescomplemented
25C2×#394C2214yescomplemented
26C2×#392C2214yes—
27#2366—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2214yescomplemented
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yescomplemented
37C2×#386C2214yescomplemented
38C2×#386C2214yescomplemented
39C2×#381C2212yescomplemented
40C2×#379C2214yes—
41C2×#379C2212yes—
42C2×C4×C4⋊C4C2212yescomplemented
43C2×C4×C22⋊C4C2211yescharacteristic
44#1445C2212yescomplemented
45#1442C2214yescomplemented
46#1442C2212yescomplemented
47#1440C2212yescomplemented
48#1440C2214yescomplemented
49#1439C2212yes—
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yes—
57#1438C2212yescomplemented
58#1438C2214yescomplemented
59#1438C2214yescomplemented
60#1437C2214yes—
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1434C2212yescomplemented
66C22⋊#95C2214yes—
67C22⋊#95C2214yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2312yes—
70C23×D4—24no—
71C23×D4—28no—
72C23×D4—42no—
73C23×D4—42no—
74C23×D4—44no—
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2314yes—
79C23×D4C2314yes—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×#127—28no—
87C2×#127—24no—
88C2×#127—44no—
89C2×#127D418yes—
90C2×#121—24no—
91C2×#121—42no—
92C2×#121—42no—
93C2×#120—216no—
94C2×#120—216no—
95C2×#120—48no—
96C2×C4×D4—216no—
97C2×C4×D4—48no—
98C2×C4×D4—48no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—48no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127#399—28no—
128#397—48no—
129#394—48no—
130#394—48no—
131#392—216no—
132#390—216no—
133#390—24no—
134#390—24no—
135#386—28no—
136#386—28no—
137#386—28no—
138#386—44no—
139#386—48no—
140#386—48no—
141#386—44no—
142#386—44no—
143#381—44no—
144#379—48no—
145#379—44no—
146C4×C4⋊C4—28no—
147C4×C22⋊C4—24no—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25—24no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—28no—
168C25—24no—
169C25—44no—
170C25—42no—
171C25—42no—
172C25—44no—
173C25—44no—
174C25C2412yes—
175C25C2412yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—22no—
191C22×D4—28no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—24no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—41no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—416no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—41no—
210C22×D4—41no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—432no—
217C22×D4—416no—
218C22×D4—432no—
219C22×D4—416no—
220C22×D4C2×D4116yes—
221C22×D4C2×D414yes—
222C22×D4C2×D414yes—
223C22×D4#4118yes—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—28no—
232C23×C4—28no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—44no—
243C23×C4—42no—
244C23×C4C2414yes—
245C23×C4C2414yes—
246C23×C4C2414yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253#127—232no—
254#127—416no—
255#121—416no—
256#120—464no—
257C4×D4—464no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
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—42no—
300C2×C22⋊C4—432no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4#4118yes—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—22no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—42no—
330C2×C42C2×D414yes—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346C24—21no—
347C24—24no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—21no—
359C24—22no—
360C24—216no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—22no—
372C24—24no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—48no—
380C24—48no—
381C24—48no—
382C24—48no—
383C24—416no—
384C24—416no—
385C24—416no—
386C24—416no—
387C24—48no—
388C24—48no—
389C24—48no—
390C24—48no—
391C24—88no—
392C24—88no—
393C24—88no—
394C24—88no—
395C24—88no—
396C24—84no—
397C24—84no—
398C24—88no—
399C24—88no—
400C24—88no—
401C24C22×D414yes—
402C24C22×D414yes—
403C24C2×#4114yes—
404C24C2×#4114yes—
405C24C2×#4114yes—
406C24H(5,𝔽2)14yes—
407C24H(5,𝔽2)12yes—
408C24C2511yescharacteristic
409C2×D4—28no—
410C2×D4—216no—
411C2×D4—232no—
412C2×D4—28no—
413C2×D4—232no—
414C2×D4—416no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—416no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—416no—
431C2×D4—416no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—4128no—
436C2×D4—48no—
437C2×D4—416no—
438C2×D4—48no—
439C2×D4—416no—
440C2×D4—48no—
441C2×D4—44no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—88no—
445C2×D4—88no—
446C2×D4—88no—
447C2×D4—88no—
448C2×D4—84no—
449C2×D4—88no—
450C2×D4#123116yes—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—28no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—22no—
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—22no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—416no—
503C22×C4—416no—
504C22×C4—44no—
505C22×C4—48no—
506C22×C4—88no—
507C22×C4—88no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—84no—
511C22×C4—88no—
512C22×C4—84no—
513C22×C4—84no—
514C22×C4—84no—
515C22×C4—88no—
516C22×C4C22×D412yes—
517C22×C4C22×D414yes—
518C22×C4C2×#4114yes—
519C22×C4C2×#4114yes—
520C22×C4C2×#4114yes—
521C22×C4H(5,𝔽2)12yes—
522C22×C4H(5,𝔽2)18yes—
523C22×C4H(5,𝔽2)14yes—
524C4⋊C4—216no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—88no—
528C22⋊C4—216no—
529C22⋊C4—416no—
530C22⋊C4—416no—
531C22⋊C4—416no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—416no—
535C22⋊C4—416no—
536C22⋊C4—88no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C22⋊C4—88no—
544C22⋊C4—84no—
545C22⋊C4—88no—
546C22⋊C4—84no—
547C42—28no—
548C42—416no—
549C42—44no—
550C23—24no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—24no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—24no—
569C23—22no—
570C23—24no—
571C23—28no—
572C23—28no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—28no—
579C23—28no—
580C23—432no—
581C23—432no—
582C23—41no—
583C23—42no—
584C23—41no—
585C23—42no—
586C23—42no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—42no—
592C23—42no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—44no—
601C23—42no—
602C23—42no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—42no—
608C23—44no—
609C23—42no—
610C23—42no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—42no—
623C23—44no—
624C23—42no—
625C23—42no—
626C23—44no—
627C23—42no—
628C23—42no—
629C23—44no—
630C23—44no—
631C23—44no—
632C23—42no—
633C23—816no—
634C23—832no—
635C23—816no—
636C23—816no—
637C23—832no—
638C23—832no—
639C23—816no—
640C23—816no—
641C23—832no—
642C23C2×#12318yes—
643C23#55914yes—
644C23C23×D411yescharacteristic
645C23C22×#4112yes—
646C23C22×#4112yes—
647C23C2×H(5,𝔽2)12yes—
648C23C2×H(5,𝔽2)11yescharacteristic
649C23C2×H(5,𝔽2)14yes—
650C23C2×H(5,𝔽2)11yescharacteristic
651C23C2×H(5,𝔽2)12yes—
652D4—416no—
653D4—464no—
654D4—416no—
655D4—432no—
656D4—88no—
657D4—832no—
658D4—816no—
659C2×C4—24no—
660C2×C4—28no—
661C2×C4—24no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—28no—
667C2×C4—24no—
668C2×C4—28no—
669C2×C4—22no—
670C2×C4—24no—
671C2×C4—28no—
672C2×C4—28no—
673C2×C4—28no—
674C2×C4—416no—
675C2×C4—41no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—41no—
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—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—42no—
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—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—42no—
711C2×C4—816no—
712C2×C4—816no—
713C2×C4—816no—
714C2×C4C2×#12314yes—
715C2×C4#54618yes—
716C22—24no—
717C22—28no—
718C22—216no—
719C22—28no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—42no—
725C22—44no—
726C22—42no—
727C22—48no—
728C22—44no—
729C22—44no—
730C22—48no—
731C22—48no—
732C22—48no—
733C22—44no—
734C22—44no—
735C22—44no—
736C22—44no—
737C22—44no—
738C22—44no—
739C22—44no—
740C22—48no—
741C22—48no—
742C22—48no—
743C22—48no—
744C22—44no—
745C22—864no—
746C22—82no—
747C22—84no—
748C22—82no—
749C22—82no—
750C22—84no—
751C22—84no—
752C22—84no—
753C22—84no—
754C22—84no—
755C22—82no—
756C22C22×#12311yescharacteristic
757C22C2×#54612yes—
758C22C2×#55911yescharacteristic
759C22#348114yes—
760C22#348411yescharacteristic
761C22#348411yescharacteristic
762C22#348412yes—
763C22#349012yes—
764C22#349012yes—
765C22#350812yes—
766C22#351914yes—
767C22#351914yes—
768C22#353212yes—
769C22#353712yes—
770C22#354311yescharacteristic
771C22#354314yes—
772C4—24no—
773C4—28no—
774C4—42no—
775C4—48no—
776C4—44no—
777C4—82no—
778C4—84no—
779C4—84no—
780C4—84no—
781C2—48no—
782C2—44no—
783C2—84no—
784C2—84no—
785C2#3535011yescharacteristic
786C2#3623612yes—
787C2#4347614yes—
788C2#4493612yes—
789C2#4814312yes—
790C2#5200411yescharacteristic
791C2#5914311yescharacteristic
792C2#5962312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2433C2
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×#2366C2semidirect
3. #15984C2semidirect
4. #15496C2semidirect
5. #15313C2semidirect
6. #15283C2semidirect
7. #15112C2semidirect
8. #14730C2semidirect
9. #14513C2semidirect
10. #14419C2semidirect
11. #14240C2semidirect
12. #13967C2semidirect
13. #13867C2semidirect
19. C22×C4×D4C22semidirect
23. C2×#397C22semidirect
24. C2×#394C22semidirect
25. C2×#394C22semidirect
30. C2×#390C22semidirect
36. C2×#386C22semidirect
37. C2×#386C22semidirect
38. C2×#386C22semidirect
39. C2×#381C22semidirect
42. C2×C4×C4⋊C4C22semidirect
44. #1445C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
57. #1438C22semidirect
58. #1438C22semidirect
59. #1438C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1434C22semidirect
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×#24332
C2×#23662
#159842
#154962
#153132
#152832
#151122
#147302
#145132
#144192
#142402
#139672
#138672