Finite Group Databasebeta

← #364014

Subgroups — #364014

Order 512.

1332 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34026C211yesmaximal, characteristic
2#20364C211yesmaximal, characteristic, complemented
3C4×#382C211yesmaximal, characteristic, complemented
4#11226C212yesmaximal, complemented
5#11075C212yesmaximal
6#10933C212yesmaximal
7#10886C212yesmaximal, complemented
8#10808C212yesmaximal, complemented
9#10790C212yesmaximal
10C2×C4×#41C2211yescharacteristic
11C22×#104C2214yes—
12C2×C4×M16C2212yes—
13#2298—21no—
14#2298—21no—
15#2297C2211yescharacteristic
16#2297C2211yescharacteristic
17#2296C2211yescharacteristic
18#2296C2211yescharacteristic
19#2295C411yescharacteristic, complemented
20#2295C411yescharacteristic, complemented
21#2295C411yescharacteristic, complemented
22#2295C411yescharacteristic, complemented
23#2293C411yescharacteristic, complemented
24#2293C411yescharacteristic, complemented
25#2293C411yescharacteristic, complemented
26#2293C411yescharacteristic, complemented
27#2290—21no—
28#2290—21no—
29#2290—21no—
30#2290—21no—
31C2×#382—21no—
32C4×#117—21no—
33C4×#117—21no—
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C43C2211yescharacteristic
37#1745—21no—
38#1745—21no—
39#1744—21no—
40#1744—21no—
41C2×#339C2214yes—
42C2×#339C2214yes—
43C2×#339C2214yes—
44#1728C2212yes—
45#1728C2212yes—
46C4⋊#95—21no—
47#1434—21no—
48#1434—21no—
49C4×#95C2212yes—
50C4×#95C2212yes—
51C22×#41C2311yescharacteristic
52C22×M16C2314yes—
53C4×#41—24no—
54C4×#41—24no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—21no—
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C4⋊C4—22no—
70C22×C42—21no—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2314yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#104—28no—
80C2×#104—28no—
81C2×#104—44no—
82C4×M16—28no—
83C2×#97—22no—
84C2×C4×C8—22no—
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#389C2×C411yescharacteristic
94#389C2×C411yescharacteristic
95#389C2×C411yescharacteristic
96#389C2×C411yescharacteristic
97#389C2×C411yescharacteristic
98#389C2×C411yescharacteristic
99#389C2×C411yescharacteristic
100#389C2×C411yescharacteristic
101#388—21no—
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#387—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#386—21no—
118#386—21no—
119#386—21no—
120#386—21no—
121#385—21no—
122#385—21no—
123#385—21no—
124#385—21no—
125#382—28no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—44no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149#376—21no—
150#376—21no—
151#376—21no—
152#376—21no—
153#376—21no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95—22no—
159C43—21no—
160C43C2×C411yescharacteristic
161C43C2×C411yescharacteristic
162C43C2×C411yescharacteristic
163C43C2×C411yescharacteristic
164C43D411yescharacteristic
165C43D411yescharacteristic
166C43D411yescharacteristic
167C43D411yescharacteristic
168#339—28no—
169#339—28no—
170#339—44no—
171#336—22no—
172#336—22no—
173#336—22no—
174#336—22no—
175C2×#41—22no—
176C2×#41—24no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41#4114yes—
180C22×Q8C22×C411yescharacteristic
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2411yescharacteristic
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16#4118yes—
200C22×C8—24no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×D4—28no—
206C4×D4—28no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211#117—28no—
212#117—22no—
213#117—24no—
214#117—22no—
215#117—22no—
216#117—44no—
217#117—44no—
218#117—44no—
219#117—44no—
220#117#4114yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C42—28no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306#104—416no—
307#97—28no—
308C4×C8—28no—
309#95—28no—
310#95—28no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—44no—
332#95—44no—
333#95—44no—
334C24—21no—
335C24C22×D411yescharacteristic
336#41—28no—
337#41—28no—
338#41—44no—
339#41—44no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8#11712yes—
344C2×D4—24no—
345C2×D4—21no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—42no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4#11712yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4114yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C42−1+412yes—
414M16—28no—
415M16—28no—
416M16—44no—
417M16—44no—
418C2×C8—28no—
419C2×C8—44no—
420C2×C8—44no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—416no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4#11712yes—
439C4⋊C4#11712yes—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C42—24no—
455C42—24no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—44no—
493C42—44no—
494C42—44no—
495C42—44no—
496C42#10414yescomplemented
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C22×C22⋊C411yescharacteristic
509C23#52512yes—
510C23#53212yes—
511C23#53511yescharacteristic
512C23#53611yescharacteristic
513Q8—24no—
514Q8—42no—
515D4—24no—
516D4—44no—
517D4—44no—
518D4—42no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4C2×#10412yes—
589C2×C4C2×#10412yes—
590C2×C4C2×#10412yes—
591C2×C4#42112yes—
592C2×C4#42112yes—
593C2×C4#42112yes—
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×#11711yescharacteristic
596C2×C4C4×#4112yes—
597C2×C4#52012yes—
598C2×C4#52512yes—
599C2×C4#52512yes—
600C2×C4#53712yes—
601C2×C4#55114yes—
602C2×C4#55214yes—
603C2×C4#55214yes—
604C2×C4#55514yes—
605C8—48no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22#228011yescharacteristic
622C22#229711yescharacteristic
623C22#231111yescharacteristic
624C22#261012yes—
625C22#261112yes—
626C22C22×#10411yescharacteristic
627C22C2×#42111yescharacteristic
628C22C2×#42111yescharacteristic
629C22C2×#42111yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—88no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4#234312yes—
651C4#234612yes—
652C4#260914yes—
653C2—22no—
654C2—22no—
655C2—82no—
656C2#1395711yescharacteristic
657C2#2074711yescharacteristic
658C2#2074811yescharacteristic
659C2#2138211yescharacteristic
660C2#2138311yescharacteristic
661C2#2189911yescharacteristic
662C2#2190011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20364C2semidirect
3. C4×#382C2semidirect
4. #11226C2semidirect
7. #10886C2semidirect
8. #10808C2semidirect
19. #2295C4semidirect
20. #2295C4semidirect
21. #2295C4semidirect
22. #2295C4semidirect
23. #2293C4semidirect
24. #2293C4semidirect
25. #2293C4semidirect
26. #2293C4semidirect
496. C42#104semidirect

Maximal subgroups

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

subgroupindex
#340262
#203642
C4×#3822
#112262
#110752
#109332
#108862
#108082
#107902