Finite Group Databasebeta

← #556957

Subgroups — #556957

Order 512.

2634 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2820C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic, complemented
3C2×C4×#124C211yesmaximal, characteristic
4#14677C211yesmaximal, characteristic, complemented
5#14664C211yesmaximal, characteristic, complemented
6#14662C211yesmaximal, characteristic
7#14310C214yesmaximal, complemented
8#14306C212yesmaximal, complemented
9#14305C212yesmaximal, complemented
10#14302C212yesmaximal, complemented
11#14300C212yesmaximal
12#14241C212yesmaximal, complemented
13#14230C212yesmaximal, complemented
14#14229C212yesmaximal, complemented
15#14227C212yesmaximal
16#14222C214yesmaximal, complemented
17C4×#382C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2820C4116yescomplemented
23C2×#399C2212yes—
24C2×#397C2212yes—
25C2×#396C2214yes—
26C2×#395C2212yes—
27C2×#394C2212yes—
28C4×#125—28no—
29C4×#124—28no—
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#388C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#385C2214yes—
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×C43C2211yescharacteristic
53#1446C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1442C2212yes—
59#1442C2212yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1438C2212yes—
72#1437C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#125—24no—
90C2×#125—24no—
91C2×#125C2×C418yescomplemented
92C2×#124—24no—
93C2×#124—24no—
94C2×#124C2×C418yes—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C2×C4×D4—24no—
98C2×C4×D4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#399C2×C4116yescomplemented
135#397C2×C4116yescomplemented
136#396C2×C4132yescomplemented
137#395C2×C4116yescomplemented
138#394C2×C4116yes—
139#391—24no—
140#390—24no—
141#389—24no—
142#389—24no—
143#388—28no—
144#388—28no—
145#388—48no—
146#387—28no—
147#387—28no—
148#387—48no—
149#385—48no—
150#382—24no—
151#382—24no—
152#382—24no—
153#382C2×C418yescomplemented
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—44no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—28no—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C43D418yes—
189C25C2412yes—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8C22×C414yes—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4C22×C414yes—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#125—232no—
222#124—232no—
223C4×Q8—416no—
224C4×D4—416no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C4116yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4C22×C418yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C4116yes—
288C2×C22⋊C4C22×C4116yes—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—41no—
301C2×C42—41no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42C22×C414yes—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315C2×C42#4114yes—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C22×C418yes—
341#95C22×C418yes—
342#95C22×C4116yes—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24C23×C414yes—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×D4—28no—
361C2×D4—48no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—48no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4C23×C418yes—
438C22×C4C23×C414yes—
439C22×C4C23×C414yes—
440C22×C4C23×C414yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C23×C412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+414yes—
457C22×C42−1+412yes—
458C4⋊C4—216no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—88no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—28no—
483C42—28no—
484C42—28no—
485C42—44no—
486C42—44no—
487C42—44no—
488C42—48no—
489C42—48no—
490C42—48no—
491C42—88no—
492C42C4×D4116yes—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23C24×C411yescharacteristic
517C23C23×D411yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×2−1+412yes—
522C23C2×2−1+411yescharacteristic
523C23#58512yes—
524C23#58511yescharacteristic
525C23#58511yescharacteristic
526C23#58512yes—
527Q8—416no—
528D4—416no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—816no—
612C2×C4C2×C4×D414yes—
613C2×C4C2×C4×D414yes—
614C2×C4C2×C4×D414yes—
615C2×C4#53514yes—
616C2×C4#53614yes—
617C2×C4#53714yes—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—48no—
622C22—48no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—84no—
628C22—84no—
629C22C22×C4×D411yescharacteristic
630C22C4×H(5,𝔽2)12yes—
631C22C4×H(5,𝔽2)11yescharacteristic
632C22C4×2−1+412yes—
633C22C4×2−1+411yescharacteristic
634C22C2×#53511yescharacteristic
635C22C2×#53611yescharacteristic
636C22C2×#53711yescharacteristic
637C22#344911yescharacteristic
638C22#344911yescharacteristic
639C22#344911yescharacteristic
640C22#346012yes—
641C22#346011yescharacteristic
642C22#346112yes—
643C22#346111yescharacteristic
644C22#354512yes—
645C22#354512yes—
646C22#354512yes—
647C22#354812yes—
648C22#355012yes—
649C22#355312yes—
650C22#355312yes—
651C22#355312yes—
652C4—24no—
653C4—24no—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—24no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—48no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—84no—
668C2—84no—
669C2C4×#53511yescharacteristic
670C2C4×#53611yescharacteristic
671C2C4×#53711yescharacteristic
672C2#3452212yes—
673C2#3452312yes—
674C2#3760611yescharacteristic
675C2#3760711yescharacteristic
676C2#3760811yescharacteristic
677C2#3761012yes—
678C2#3761212yes—
679C2#3764011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#125C2semidirect
4. #14677C2semidirect
5. #14664C2semidirect
7. #14310C2semidirect
8. #14306C2semidirect
9. #14305C2semidirect
10. #14302C2semidirect
12. #14241C2semidirect
13. #14230C2semidirect
14. #14229C2semidirect
16. #14222C2semidirect
17. C4×#382C2semidirect
22. #2820C4semidirect
91. C2×#125C2×C4semidirect
134. #399C2×C4semidirect
135. #397C2×C4semidirect
136. #396C2×C4semidirect
137. #395C2×C4semidirect
153. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×C4×#1252
C2×C4×#1242
#146772
#146642
#146622
#143102
#143062
#143052
#143022
#143002
#142412
#142302
#142292
#142272
#142222
C4×#3822