Finite Group Databasebeta

← #1494651

Subgroups — #1494651

Order 512.

4302 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2679C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2472C214yesmaximal, complemented
4#15761C212yesmaximal, complemented
5#15643C212yesmaximal, complemented
6#15613C214yesmaximal, complemented
7#15283C214yesmaximal, complemented
8#15136C212yesmaximal, complemented
9#14743C214yesmaximal, complemented
10#14357C212yesmaximal, complemented
11#14227C212yesmaximal, complemented
12#13967C212yesmaximal, complemented
13#13824C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic, complemented
15C22×#124C2212yes—
16C22×C22⋊Q8C2214yescomplemented
17C22×#120C2212yes—
18C22×#120C2214yes—
19C22×C4×D4C2211yescharacteristic
20C23×C22⋊C4C2212yescomplemented
21#2679—28no—
22#2542—28no—
23#2472—232no—
24C2×#399C2214yescomplemented
25C2×#397C2214yescomplemented
26C2×#397C2214yescomplemented
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#394C2214yes—
30C2×#394C2212yes—
31C2×#394C2212yes—
32C2×#393C2214yescomplemented
33C2×#390C2211yescharacteristic
34C2×#390C2214yes—
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#388C2214yescomplemented
38C2×#388C2214yescomplemented
39C2×#386C2212yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#384C2211yescharacteristic, complemented
43C2×#384C2211yescharacteristic, complemented
44C2×#379C2214yescomplemented
45C2×#379C2212yescomplemented
46C2×C4×C4⋊C4C2211yescharacteristic, complemented
47C2×C4×C22⋊C4C2214yescomplemented
48#1446C2212yescomplemented
49#1442C2214yescomplemented
50#1442C2212yescomplemented
51#1441C2212yescomplemented
52#1440C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1438C2214yes—
58#1438C2214yes—
59#1438C2212yes—
60#1437C2214yescomplemented
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64#1436C2212yescomplemented
65#1436C2212yescomplemented
66C4⋊#95C2214yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C4×#95C2212yescomplemented
70C26C2312yes—
71C23×Q8C2312yes—
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×#124—216no—
82C2×#124—44no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—48no—
86C2×#120—28no—
87C2×#120—216no—
88C2×#120—28no—
89C2×#120—28no—
90C2×#120—216no—
91C2×#120—48no—
92C2×C4×D4—24no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—216no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#399—216no—
128#397—216no—
129#397—48no—
130#394—28no—
131#394—216no—
132#394—216no—
133#394—44no—
134#394D4116yescomplemented
135#393—216no—
136#390—216no—
137#390—24no—
138#390—28no—
139#390D418yes—
140#388—216no—
141#388—216no—
142#386—28no—
143#386—216no—
144#386—216no—
145#384—24no—
146#384—24no—
147#379—216no—
148#379—44no—
149C4×C4⋊C4—24no—
150C4×C22⋊C4—216no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25—28no—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—42no—
171C25—44no—
172C25—42no—
173C25—44no—
174C25—44no—
175C25—42no—
176C25—42no—
177C25C2414yes—
178C25C2412yes—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—24no—
182C22×Q8—44no—
183C22×Q8—42no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4C2×D418yes—
204C22×D4C2×D414yes—
205C22×D4#4118yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C4⋊Q8—416no—
228#124—432no—
229C22⋊Q8—464no—
230#120—464no—
231#120—432no—
232C4×D4—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×D414yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
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—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—432no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
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—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D4116yes—
306C2×C22⋊C4C2×D4116yes—
307C2×C22⋊C4C2×D4116yes—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—28no—
312C2×C42—22no—
313C2×C42—28no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42C2×D414yes—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95C2×D418yes—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—48no—
358C24—416no—
359C24—416no—
360C24—416no—
361C24—48no—
362C24—416no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—84no—
367C24—84no—
368C24—88no—
369C24—84no—
370C24—84no—
371C24—88no—
372C24—88no—
373C24—88no—
374C24—88no—
375C24—84no—
376C24—84no—
377C24—88no—
378C24—88no—
379C24—88no—
380C24C22×D418yes—
381C24C22×D414yes—
382C24C2×#4114yes—
383C24C2×#4114yes—
384C24C2511yescharacteristic
385C2×Q8—416no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—88no—
389C2×Q8—84no—
390C2×D4—216no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—416no—
396C2×D4—416no—
397C2×D4—416no—
398C2×D4—48no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—416no—
402C2×D4—84no—
403C2×D4—84no—
404C2×D4—88no—
405C2×D4—84no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—84no—
411C2×D4#124116yescomplemented
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
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—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—88no—
467C22×C4—84no—
468C22×C4—88no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4C22×D418yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C2×#4114yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)12yes—
482C4⋊C4—216no—
483C4⋊C4—44no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—416no—
489C4⋊C4—84no—
490C22⋊C4—232no—
491C22⋊C4—232no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—84no—
507C42—28no—
508C42—44no—
509C42—48no—
510C42—44no—
511C42—88no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—432no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—832no—
571C23—816no—
572C23—816no—
573C23—816no—
574C23—832no—
575C23—832no—
576C23—832no—
577C23—816no—
578C23—816no—
579C23—816no—
580C23C2×#12418yes—
581C23C23×D411yescharacteristic
582C23C23×D412yes—
583C23C22×#4112yes—
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×H(5,𝔽2)12yes—
586C23C2×H(5,𝔽2)11yescharacteristic
587C23#58514yes—
588C23#58512yes—
589Q8—816no—
590D4—416no—
591D4—816no—
592D4—816no—
593D4—832no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—816no—
652C2×C4—816no—
653C2×C4C2×#12414yes—
654C2×C4D4214yes—
655C2×C4D4218yes—
656C22—216no—
657C22—28no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—44no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—48no—
671C22—48no—
672C22—48no—
673C22—48no—
674C22—864no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22C22×#12411yescharacteristic
684C22C2×D4212yes—
685C22C2×D4211yescharacteristic
686C22D4×#4114yes—
687C22#348411yescharacteristic
688C22#348511yescharacteristic
689C22#348514yes—
690C22#349812yes—
691C22#350012yes—
692C22#350512yes—
693C22#350512yes—
694C22#351414yes—
695C22#354012yes—
696C22#354311yescharacteristic
697C22#354512yes—
698C22#356014yes—
699C4—24no—
700C4—28no—
701C4—24no—
702C4—42no—
703C4—48no—
704C4—44no—
705C4—44no—
706C4—84no—
707C4—84no—
708C4—82no—
709C4—82no—
710C2—48no—
711C2—88no—
712C2#3537711yescharacteristic
713C2D4×#12412yes—
714C2#4349212yes—
715C2#4350214yes—
716C2#5031112yes—
717C2#5174911yescharacteristic
718C2#5900012yes—
719C2#5912611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2679C2
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×#2542C2semidirect
3. C2×#2472C2semidirect
4. #15761C2semidirect
5. #15643C2semidirect
6. #15613C2semidirect
7. #15283C2semidirect
8. #15136C2semidirect
9. #14743C2semidirect
10. #14357C2semidirect
11. #14227C2semidirect
12. #13967C2semidirect
13. #13824C2semidirect
14. C22×C4⋊Q8C22semidirect
16. C22×C22⋊Q8C22semidirect
20. C23×C22⋊C4C22semidirect
24. C2×#399C22semidirect
25. C2×#397C22semidirect
26. C2×#397C22semidirect
32. C2×#393C22semidirect
37. C2×#388C22semidirect
38. C2×#388C22semidirect
42. C2×#384C22semidirect
43. C2×#384C22semidirect
44. C2×#379C22semidirect
45. C2×#379C22semidirect
46. C2×C4×C4⋊C4C22semidirect
47. C2×C4×C22⋊C4C22semidirect
48. #1446C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1440C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect
69. C4×#95C22semidirect
134. #394D4semidirect
411. C2×D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#26792
C2×#25422
C2×#24722
#157612
#156432
#156132
#152832
#151362
#147432
#143572
#142272
#139672
#138242