Finite Group Databasebeta

← #513805

Subgroups — #513805

Order 512.

3898 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2325C212yesmaximal
2C2×C4×#121C211yesmaximal, characteristic
3#13833C212yesmaximal, complemented
4#13832C212yesmaximal, complemented
5#13819C212yesmaximal, complemented
6#13813C214yesmaximal, complemented
7#13813C212yesmaximal, complemented
8#13813C212yesmaximal, complemented
9D4×#95C212yesmaximal, complemented
10#13789C214yesmaximal, complemented
11#13788C212yesmaximal, complemented
12#13782C212yesmaximal, complemented
13#13778C212yesmaximal, complemented
14C4×#380C211yesmaximal, characteristic, complemented
15C2×#1434C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2214yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2325C4132yescomplemented
22C4×#121C4116yescomplemented
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388C2214yes—
26C2×#388C2212yes—
27C2×#387C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2214yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2214yes—
40C2×C4×C22⋊C4C2212yes—
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1437C2214yes—
47#1437C2214yes—
48#1437C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C4⋊#95C2214yes—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2212yes—
60#1434C2212yes—
61#1434C2212yes—
62#1434C2214yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2214yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2314yes—
85C24×C4C2311yescharacteristic
86C2×#121C2×C4116yes—
87C2×#121C2×C418yes—
88C2×C4×D4—44no—
89C2×C4×D4—48no—
90C2×C4×D4—48no—
91C2×C4×D4—48no—
92C2×C4×D4C2×C4132yescomplemented
93C2×C4×D4C2×C418yescomplemented
94C2×C4×D4C2×C4116yescomplemented
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C42—44no—
112C22×C42C2×C418yescomplemented
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2314yes—
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2314yes—
122#390C2×C418yescomplemented
123#389C2×C418yescomplemented
124#388C2×C4132yescomplemented
125#388C2×C4116yescomplemented
126#387C2×C4132yescomplemented
127#386C2×C4132yescomplemented
128#386C2×C4132yescomplemented
129#386C2×C4116yescomplemented
130#385C2×C4132yescomplemented
131#380C2×C4116yescomplemented
132#380C2×C418yescomplemented
133C4×C4⋊C4—44no—
134C4×C4⋊C4C2×C418yescomplemented
135C4×C22⋊C4—44no—
136C4×C22⋊C4—48no—
137C4×C22⋊C4—44no—
138C4×C22⋊C4—48no—
139C4×C22⋊C4C2×C4116yescomplemented
140C2×#95—42no—
141C2×#95—44no—
142C2×#95—42no—
143C2×#95—44no—
144C2×#95C2×C4116yescomplemented
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—42no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4C22×C4116yes—
179C22×D4C22×C418yes—
180C22×D4C22×C414yes—
181C22×D4C22×C418yes—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4C22×C414yes—
216C23×C4C22×C414yes—
217C23×C4C22×C418yes—
218C23×C4C22×C418yes—
219C23×C4C22×C418yes—
220C23×C4C22×C4116yes—
221C23×C4C22×C418yes—
222C23×C4C22×C414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236#121C42164yescomplemented
237C4×D4—416no—
238C4×D4—432no—
239C4×D4—464no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C4116yes—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C22×C418yes—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—44no—
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⋊C4C22×C4116yes—
271C2×C22⋊C4C22×C418yes—
272C2×C22⋊C4C22×C418yes—
273C2×C22⋊C4C22×C4116yes—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C4116yes—
277C2×C22⋊C4C22×C4116yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C4116yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—416no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C22×C414yes—
298C2×C42C22×C414yes—
299C2×C42C22×C418yes—
300C2×C42C22×C4116yes—
301C2×C42C22×C414yes—
302C2×C42C22×C414yes—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C22×C4116yes—
321#95C22×C4116yes—
322#95C22×C418yes—
323#95C22×C418yes—
324#95C22×C418yes—
325#95C22×C4116yes—
326#95C22×C4116yes—
327#95C22×C4116yes—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24C23×C414yes—
344C24C23×C418yes—
345C24C23×C412yes—
346C24C23×C414yes—
347C24H(5,𝔽2)12yes—
348C24H(5,𝔽2)14yes—
349C242−1+412yes—
350C24C2511yescharacteristic
351C2×D4—216no—
352C2×D4—232no—
353C2×D4—216no—
354C2×D4—232no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4C2×C42132yes—
360C2×D4C2×C42116yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—216no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4C2×C42116yes—
456C22×C4C23×C418yes—
457C22×C4C23×C412yes—
458C22×C4C23×C414yes—
459C22×C4C23×C418yes—
460C22×C4C23×C418yes—
461C22×C4C23×C414yes—
462C22×C4C23×C418yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C414yes—
467C22×C4C23×C412yes—
468C22×C4C23×C414yes—
469C22×C4C23×C418yes—
470C22×C4C23×C414yes—
471C22×C4C23×C412yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4C2×C42116yes—
482C22⋊C4—216no—
483C22⋊C4—232no—
484C22⋊C4—216no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4C2×C42132yes—
491C42—28no—
492C42—28no—
493C42—44no—
494C42—44no—
495C42—48no—
496C42—416no—
497C42—88no—
498C42—88no—
499C42—88no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23C22×C4218yes—
541C23C22×C4214yes—
542C23#51814yes—
543C23#51818yes—
544C23#52014yes—
545C23#52018yes—
546C23C24×C411yescharacteristic
547C23C24×C412yes—
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×2−1+411yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23#58511yescharacteristic
555C23#58511yescharacteristic
556C23#58512yes—
557D4—264no—
558D4—416no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
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—816no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4—832no—
634C2×C4C22×C4214yes—
635C2×C4C22×C4218yes—
636C2×C4C22×C4214yes—
637C2×C4#51814yes—
638C2×C4#52014yes—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22—28no—
646C22—28no—
647C22—24no—
648C22—24no—
649C22—28no—
650C22—28no—
651C22—28no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—48no—
657C22—48no—
658C22—48no—
659C22—48no—
660C22#227518yes—
661C22C23×C4211yescharacteristic
662C22C2×#51811yescharacteristic
663C22C2×#51811yescharacteristic
664C22C2×#51812yes—
665C22C2×#52011yescharacteristic
666C22C2×#52011yescharacteristic
667C22C2×#52012yes—
668C22#342811yescharacteristic
669C22#342811yescharacteristic
670C22C4×H(5,𝔽2)12yes—
671C22C4×H(5,𝔽2)12yes—
672C22C4×H(5,𝔽2)12yes—
673C22C4×2−1+412yes—
674C22#353412yes—
675C22#353512yes—
676C22#353612yes—
677C22#353712yes—
678C22#355811yescharacteristic
679C22#355911yescharacteristic
680C22#356012yes—
681C22#356012yes—
682C22#356111yescharacteristic
683C22#356311yescharacteristic
684C4—24no—
685C4—24no—
686C4—28no—
687C4—24no—
688C4—24no—
689C4—42no—
690C4—44no—
691C4—42no—
692C4—44no—
693C4—48no—
694C4—48no—
695C4—48no—
696C4—48no—
697C2—28no—
698C2—48no—
699C2C2×#227511yescharacteristic
700C2C4×#51811yescharacteristic
701C2C4×#52011yescharacteristic
702C2#3437911yescharacteristic
703C2#3438211yescharacteristic
704C2#3438611yescharacteristic
705C2#3438811yescharacteristic
706C2#3442612yes—
707C2#3442712yes—
708C2#3443612yes—
709C2#3443712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC22C23×C42
prime radicalC22×#121C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13833C2semidirect
4. #13832C2semidirect
5. #13819C2semidirect
6. #13813C2semidirect
7. #13813C2semidirect
8. #13813C2semidirect
9. D4×#95C2semidirect
10. #13789C2semidirect
11. #13788C2semidirect
12. #13782C2semidirect
13. #13778C2semidirect
14. C4×#380C2semidirect
15. C2×#1434C2semidirect
21. #2325C4semidirect
22. C4×#121C4semidirect
92. C2×C4×D4C2×C4semidirect
93. C2×C4×D4C2×C4semidirect
94. C2×C4×D4C2×C4semidirect
112. C22×C42C2×C4semidirect
122. #390C2×C4semidirect
123. #389C2×C4semidirect
124. #388C2×C4semidirect
125. #388C2×C4semidirect
126. #387C2×C4semidirect
127. #386C2×C4semidirect
128. #386C2×C4semidirect
129. #386C2×C4semidirect
130. #385C2×C4semidirect
131. #380C2×C4semidirect
132. #380C2×C4semidirect
134. C4×C4⋊C4C2×C4semidirect
139. C4×C22⋊C4C2×C4semidirect
144. C2×#95C2×C4semidirect
236. #121C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23252
C2×C4×#1212
#138332
#138322
#138192
#138132
D4×#952
#137892
#137882
#137822
#137782
C4×#3802
C2×#14342