Finite Group Databasebeta

← #513809

Subgroups — #513809

Order 512.

3438 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C212yesmaximal
2C2×C4×#124C211yesmaximal, characteristic
3#13831C214yesmaximal, complemented
4#13822C212yesmaximal, complemented
5#13820C212yesmaximal
6#13820C211yesmaximal, characteristic
7#13817C212yesmaximal, complemented
8#13817C211yesmaximal, characteristic, complemented
9#13814C212yesmaximal, complemented
10Q8×#95C211yesmaximal, characteristic, complemented
11D4×#95C211yesmaximal, characteristic
12#13789C214yesmaximal, complemented
13#13788C212yesmaximal, complemented
14#13782C212yesmaximal, complemented
15#13779C212yesmaximal, complemented
16C4×#384C211yesmaximal, characteristic, complemented
17#13757C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2326C4132yescomplemented
24C4×#124C4116yescomplemented
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2211yescharacteristic
29C2×#388C2212yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2212yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#384C2212yes—
36C2×#382C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2214yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2212yes—
47C22×#95C2212yes—
48C2×C43C2211yescharacteristic
49#1437C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2212yes—
68#1434C2214yes—
69#1434C2211yescharacteristic
70#1434C2211yescharacteristic
71#1434C2212yes—
72#1434C2211yescharacteristic
73#1434C2211yescharacteristic
74#1434C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2214yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×#124C2×C4116yes—
90C2×#124C2×C418yes—
91C2×C4×Q8—44no—
92C2×C4×Q8—44no—
93C2×C4×Q8—44no—
94C2×C4×Q8C2×C4116yescomplemented
95C2×C4×Q8C2×C418yescomplemented
96C2×C4×D4—44no—
97C2×C4×D4—44no—
98C2×C4×D4—44no—
99C2×C4×D4C2×C4116yes—
100C2×C4×D4C2×C418yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2314yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#391C2×C4116yescomplemented
132#391C2×C418yescomplemented
133#390C2×C4116yes—
134#390C2×C418yes—
135#388C2×C4132yescomplemented
136#388C2×C4132yescomplemented
137#388C2×C4116yescomplemented
138#388C2×C4116yescomplemented
139#385C2×C4132yescomplemented
140#385C2×C4132yescomplemented
141#384C2×C4116yescomplemented
142#382C2×C418yescomplemented
143C4×C4⋊C4—44no—
144C4×C4⋊C4—44no—
145C4×C4⋊C4—44no—
146C4×C4⋊C4—44no—
147C4×C4⋊C4—44no—
148C4×C22⋊C4—44no—
149C4×C22⋊C4—48no—
150C4×C22⋊C4C2×C4116yescomplemented
151C4×C22⋊C4C2×C4116yescomplemented
152#376C2×C4116yescomplemented
153C2×#95—44no—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C43C2×C418yescomplemented
186C25C2412yes—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8C22×C418yes—
191C22×Q8C22×C414yes—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4C22×C418yes—
196C22×D4C22×C414yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C22×C418yes—
208C23×C4C22×C4116yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225#124C42164yescomplemented
226C4×Q8—432no—
227C4×Q8—416no—
228C4×D4—432no—
229C4×D4—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C22×C4116yes—
248C2×C4⋊C4C22×C418yes—
249C2×C4⋊C4C22×C4116yes—
250C2×C4⋊C4C22×C4116yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4C22×C418yes—
262C2×C22⋊C4C22×C418yes—
263C2×C22⋊C4C22×C4116yes—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C4116yes—
266C2×C22⋊C4C22×C4116yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C4116yes—
270C2×C22⋊C4C22×C4116yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42C22×C414yes—
297C2×C42C22×C418yes—
298C2×C42C22×C414yes—
299C2×C42C22×C414yes—
300C2×C42C22×C414yes—
301C2×C42C22×C418yes—
302C2×C42C22×C418yes—
303C2×C42C22×C4116yes—
304C2×C42C22×C414yes—
305C2×C42C22×C414yes—
306C2×C42C22×C414yes—
307C2×C42C22×C418yes—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C22×C418yes—
330#95C22×C418yes—
331#95C22×C418yes—
332#95C22×C418yes—
333#95C22×C418yes—
334#95C22×C418yes—
335#95C22×C418yes—
336#95C22×C418yes—
337#95C22×C4116yes—
338#95C22×C4116yes—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24C23×C414yes—
349C24C23×C418yes—
350C24C2511yescharacteristic
351C2×Q8—28no—
352C2×Q8—216no—
353C2×Q8—28no—
354C2×Q8—216no—
355C2×Q8C2×C42116yes—
356C2×D4—28no—
357C2×D4—216no—
358C2×D4—28no—
359C2×D4—216no—
360C2×D4C2×C42116yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—216no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4C23×C414yes—
441C22×C4C23×C414yes—
442C22×C4C23×C414yes—
443C22×C4C23×C414yes—
444C22×C4C23×C412yes—
445C22×C4C23×C414yes—
446C22×C4C23×C414yes—
447C22×C4C23×C418yes—
448C22×C4C23×C412yes—
449C22×C4C23×C418yes—
450C22×C4C23×C412yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C23×C412yes—
454C22×C4C23×C414yes—
455C22×C4C23×C414yes—
456C22×C4C23×C414yes—
457C22×C4C23×C418yes—
458C22×C4C23×C412yes—
459C22×C4C23×C414yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—416no—
472C4⋊C4—416no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C22⋊C4—216no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4C2×C42132yes—
482C22⋊C4C2×C42132yes—
483C42—28no—
484C42—216no—
485C42—28no—
486C42—28no—
487C42—28no—
488C42—216no—
489C42—44no—
490C42—44no—
491C42—48no—
492C42—416no—
493C42—88no—
494C42—88no—
495C42—88no—
496C42C2×C42116yes—
497C23—24no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23C22×C4218yes—
520C23C24×C411yescharacteristic
521C23C24×C412yes—
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)12yes—
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×2−1+411yescharacteristic
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23#58512yes—
530C23#58512yes—
531Q8—232no—
532D4—232no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—816no—
618C2×C4—832no—
619C2×C4C22×C4214yes—
620C2×C4C22×C4214yes—
621C2×C4C22×C4218yes—
622C2×C4C22×C4214yes—
623C2×C4#51814yes—
624C2×C4#51818yes—
625C2×C4#52014yes—
626C2×C4#52014yes—
627C2×C4#52014yes—
628C2×C4#52018yes—
629C22—28no—
630C22—28no—
631C22—28no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22C23×C4211yescharacteristic
641C22C2×#51811yescharacteristic
642C22C2×#51812yes—
643C22C2×#52011yescharacteristic
644C22C2×#52011yescharacteristic
645C22C2×#52011yescharacteristic
646C22C2×#52012yes—
647C22#342811yescharacteristic
648C22#342811yescharacteristic
649C22C4×H(5,𝔽2)12yes—
650C22C4×H(5,𝔽2)12yes—
651C22C4×2−1+412yes—
652C22C4×2−1+412yes—
653C22#354311yescharacteristic
654C22#354411yescharacteristic
655C22#354612yes—
656C22#355211yescharacteristic
657C22#355211yescharacteristic
658C22#355412yes—
659C22#355812yes—
660C22#356112yes—
661C22#356212yes—
662C22#356512yes—
663C4—24no—
664C4—24no—
665C4—28no—
666C4—24no—
667C4—24no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—48no—
675C4—48no—
676C4—44no—
677C4—44no—
678C4#227518yes—
679C2—48no—
680C2C2×#227511yescharacteristic
681C2C4×#52011yescharacteristic
682C2C4×#52011yescharacteristic
683C2#3437711yescharacteristic
684C2#3438411yescharacteristic
685C2#3439211yescharacteristic
686C2#3439311yescharacteristic
687C2#3442812yes—
688C2#3442912yes—
689C2#3443812yes—
690C2#3443912yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC22C23×C42
prime radicalC23×D4C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13831C2semidirect
4. #13822C2semidirect
7. #13817C2semidirect
8. #13817C2semidirect
9. #13814C2semidirect
10. Q8×#95C2semidirect
12. #13789C2semidirect
13. #13788C2semidirect
14. #13782C2semidirect
15. #13779C2semidirect
16. C4×#384C2semidirect
17. #13757C2semidirect
23. #2326C4semidirect
24. C4×#124C4semidirect
94. C2×C4×Q8C2×C4semidirect
95. C2×C4×Q8C2×C4semidirect
131. #391C2×C4semidirect
132. #391C2×C4semidirect
135. #388C2×C4semidirect
136. #388C2×C4semidirect
137. #388C2×C4semidirect
138. #388C2×C4semidirect
139. #385C2×C4semidirect
140. #385C2×C4semidirect
141. #384C2×C4semidirect
142. #382C2×C4semidirect
150. C4×C22⋊C4C2×C4semidirect
151. C4×C22⋊C4C2×C4semidirect
152. #376C2×C4semidirect
185. C43C2×C4semidirect
225. #124C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×C4×#1242
#138312
#138222
#138202
#138172
#138142
Q8×#952
D4×#952
#137892
#137882
#137822
#137792
C4×#3842
#137572