Finite Group Databasebeta

← #556094

Subgroups — #556094

Order 512.

3728 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#120C211yesmaximal, characteristic
2#14680C212yesmaximal, complemented
3#14410C212yesmaximal, complemented
4#14402C212yesmaximal, complemented
5#14397C214yesmaximal, complemented
6#14392C214yesmaximal, complemented
7#14366C212yesmaximal, complemented
8#14358C212yesmaximal
9#14345C211yesmaximal, characteristic, complemented
10#14344C211yesmaximal, characteristic, complemented
11#13920C212yesmaximal, complemented
12#13919C212yesmaximal, complemented
13#13915C212yesmaximal, complemented
14#13912C212yesmaximal, complemented
15#13788C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C4×#120C4116yescomplemented
21C2×#390C2214yes—
22C2×#389C2214yescomplemented
23C2×#388C2214yes—
24C2×#388C2214yes—
25C2×#388C2214yescomplemented
26C2×#386C2212yes—
27C2×#386C2211yescharacteristic
28C2×#385C2212yes—
29C2×#385C2211yescharacteristic
30C2×#383C2211yescharacteristic, complemented
31C2×#382C2211yescharacteristic, complemented
32C2×#381C2211yescharacteristic, complemented
33C2×#380C2214yes—
34C2×#380C2214yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic, complemented
38C2×#379C2214yes—
39C2×#379C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42#1445C2212yescomplemented
43#1444C2214yescomplemented
44#1442C2214yescomplemented
45#1441C2212yescomplemented
46#1441C2214yescomplemented
47#1440C2214yes—
48#1440C2212yescomplemented
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2212yescomplemented
54#1439C2214yescomplemented
55#1438C2212yes—
56#1438C2212yes—
57#1438C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64C4⋊#95C2212yescomplemented
65#1434C2212yescomplemented
66#1434C2214yescomplemented
67#1434C2212yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yescomplemented
74C4×#95C2212yescomplemented
75C4×#95C2212yescomplemented
76C26C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4—48no—
80C24×C4—42no—
81C24×C4—42no—
82C24×C4C2×C418yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C2×#120—28no—
89C2×#120C2×C418yes—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4—48no—
93C2×C4×D4C2×C418yes—
94C2×C4×D4C2×C4116yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123#390—48no—
124#389—48no—
125#388—216no—
126#388—48no—
127#388—48no—
128#386C2×C418yes—
129#386C2×C4116yes—
130#385C2×C4116yes—
131#385C2×C418yes—
132#383—42no—
133#382—42no—
134#381—42no—
135#380—44no—
136#380—44no—
137#380—48no—
138#380—42no—
139#380—48no—
140#379—48no—
141#379C2×C418yes—
142C4×C22⋊C4C2×C4116yes—
143C4×C22⋊C4C2×C418yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—44no—
171C25—41no—
172C25—41no—
173C25—42no—
174C25C22×C414yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C25C2412yes—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—44no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4C22×C418yes—
187C22×D4C22×C414yes—
188C23×C4—28no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—432no—
198C23×C4—416no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—432no—
208C23×C4—432no—
209C23×C4C22×C418yes—
210C23×C4C22×C414yes—
211C23×C4C22×C414yes—
212C23×C4C22×C414yes—
213C23×C4C22×C418yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226#120—232no—
227C4×D4—416no—
228C4×D4—432no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C414yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C22×C418yes—
283C2×C22⋊C4C22×C418yes—
284C2×C22⋊C4C22×C418yes—
285C2×C22⋊C4C22×C414yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C414yes—
288C2×C22⋊C4C22×C414yes—
289C2×C22⋊C4C22×C414yes—
290C2×C22⋊C4#4118yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—28no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C22×C414yes—
301C2×C42C22×C418yes—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C22×C418yes—
323#95C22×C418yes—
324#95C22×C418yes—
325C24—24no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—21no—
333C24—21no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—28no—
340C24—216no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—44no—
349C24—48no—
350C24—44no—
351C24—416no—
352C24—416no—
353C24—48no—
354C24—44no—
355C24—44no—
356C24—48no—
357C24—48no—
358C24—416no—
359C24—416no—
360C24—48no—
361C24—48no—
362C24C23×C412yes—
363C24C23×C412yes—
364C24C23×C414yes—
365C24C2×#4114yes—
366C24H(5,𝔽2)14yes—
367C242−1+414yes—
368C24C2511yescharacteristic
369C2×D4—28no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—48no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—48no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—4128no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4C23×C412yes—
456C22×C4C23×C414yes—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C23×C412yes—
460C22×C4C23×C414yes—
461C22×C4C23×C414yes—
462C22×C4C23×C412yes—
463C22×C4C2×#4118yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)18yes—
467C4⋊C4—216no—
468C4⋊C4—44no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C4⋊C4—88no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—216no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—88no—
490C22⋊C4#117116yes—
491C42—28no—
492C42—48no—
493C42—88no—
494C42—88no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—416no—
523C23—432no—
524C23—416no—
525C23—432no—
526C23—416no—
527C23—432no—
528C23—464no—
529C23—432no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—44no—
555C23—44no—
556C23C2×#11714yes—
557C23#52018yes—
558C23C24×C411yescharacteristic
559C23C22×#4112yes—
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)14yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×2−1+412yes—
565C23#58512yes—
566D4—432no—
567D4—416no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—432no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—432no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—88no—
635C2×C4C2×#11718yes—
636C2×C4#51814yes—
637C2×C4#51818yes—
638C22—24no—
639C22—28no—
640C22—28no—
641C22—28no—
642C22—28no—
643C22—464no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—48no—
649C22—48no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—48no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—82no—
664C22—84no—
665C22—82no—
666C22C22×#11711yescharacteristic
667C22C2×#51811yescharacteristic
668C22C2×#51812yes—
669C22C2×#52012yes—
670C22#342811yescharacteristic
671C22#346012yes—
672C22#346012yes—
673C22#351914yes—
674C22#352014yes—
675C22#353212yes—
676C22#353212yes—
677C22#353612yes—
678C22#353711yescharacteristic
679C22#353811yescharacteristic
680C22#354114yes—
681C22#354214yes—
682C4—24no—
683C4—28no—
684C4—28no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—44no—
690C4—84no—
691C4—84no—
692C4—84no—
693C4—84no—
694C2—44no—
695C2—48no—
696C2—82no—
697C2#3405811yescharacteristic
698C2#3419412yes—
699C2#3435611yescharacteristic
700C2#3436011yescharacteristic
701C2#3440412yes—
702C2#3690312yes—
703C2#3690414yes—
704C2#3693012yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#120C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14680C2semidirect
3. #14410C2semidirect
4. #14402C2semidirect
5. #14397C2semidirect
6. #14392C2semidirect
7. #14366C2semidirect
9. #14345C2semidirect
10. #14344C2semidirect
11. #13920C2semidirect
12. #13919C2semidirect
13. #13915C2semidirect
14. #13912C2semidirect
15. #13788C2semidirect
20. C4×#120C4semidirect
22. C2×#389C22semidirect
25. C2×#388C22semidirect
30. C2×#383C22semidirect
31. C2×#382C22semidirect
32. C2×#381C22semidirect
37. C2×#380C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1442C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
48. #1440C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. C4⋊#95C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
73. C22⋊#95C22semidirect
74. C4×#95C22semidirect
75. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1202
#146802
#144102
#144022
#143972
#143922
#143662
#143582
#143452
#143442
#139202
#139192
#139152
#139122
#137882