Finite Group Databasebeta

← #6468403

Subgroups — #6468403

Order 512.

1698 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
1C4×#533C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic
3#29289C212yesmaximal, complemented
4#29275C212yesmaximal, complemented
5#29257C212yesmaximal, complemented
6#29244C212yesmaximal
7#29243C212yesmaximal
8#28287C212yesmaximal, complemented
9#27686C212yesmaximal
10#24307C212yesmaximal
11#23779C214yesmaximal, complemented
12C4×#479C214yesmaximal, complemented
13#21385C211yesmaximal, characteristic
14#21385C211yesmaximal, characteristic
15#21359C211yesmaximal, characteristic
16#21358C211yesmaximal, characteristic
17#20808C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C2×#479C2214yes—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×#125—21no—
28C4×#125—21no—
29C4×#125—21no—
30C4×#125—21no—
31C4×#125—21no—
32C4×#125—21no—
33C4×#125C2212yes—
34C4×#125C2212yes—
35C4×#125C2212yes—
36C4×#125C2212yes—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41#2296—21no—
42#2295—21no—
43#2295C2212yes—
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#387—21no—
47C2×#387—21no—
48#2285C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C42×Q8—21no—
51C42×Q8C2212yes—
52C4×#117—21no—
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C43C2211yescharacteristic
58#2101C2214yes—
59#2097C2214yes—
60#2088C2214yes—
61#2086C2214yes—
62#2068C2214yes—
63#2066C2214yes—
64#2059C2214yes—
65#2026C2214yes—
66#1989C2214yes—
67#1987C2212yes—
68#1987C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1949C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1920C2214yes—
77#1918C2214yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1865C2214yes—
83#1863C2214yes—
84#1853C2212yes—
85#1853C2212yes—
86#1852C2212yes—
87#1852C2212yes—
88#1842C2212yes—
89#1831C2212yes—
90#1805C2214yes—
91#1797C2212yes—
92#1797C2212yes—
93C4×#107C2214yes—
94C4×#106C2214yes—
95C4×C4⋊C8C2212yes—
96C4×C4⋊C8C2212yes—
97C4×#103C2214yes—
98C4×#103C2214yes—
99#533—24no—
100#533Q818yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—28no—
104C2×#125—21no—
105C2×#125—21no—
106C2×#125—21no—
107C2×#125C2312yes—
108C2×#125C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×#117—21no—
118C2×#117—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42—21no—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#479—216no—
137#431—24no—
138#431—24no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155#391—21no—
156#391—21no—
157#391C2312yes—
158#391C2312yes—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—21no—
192#382—21no—
193#382—21no—
194#382C2312yes—
195#382C2312yes—
196#380—21no—
197#380C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232#376—22no—
233C2×#95—21no—
234C2×#95—21no—
235C2×#95—21no—
236C2×#95—21no—
237C2×#95—21no—
238C2×#95—21no—
239C43—22no—
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8C2412yes—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271C22×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8C2×Q814yes—
276C4⋊Q8C2×Q814yes—
277#125—28no—
278#125—28no—
279#125—24no—
280#125—24no—
281#125—28no—
282#125—24no—
283#125#4118yes—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8C2×Q818yes—
287C4×Q8—28no—
288C4×Q8—24no—
289C4×Q8—28no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8C2×Q818yes—
294#117—28no—
295#117—44no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42—44no—
355C2×C42—44no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×Q814yes—
379C2×C42#4114yes—
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—28no—
387#107—44no—
388#106—28no—
389#106—44no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392C4⋊C8—28no—
393C4⋊C8—44no—
394#103—28no—
395#103—28no—
396#103—44no—
397#103—44no—
398C22⋊C8—24no—
399C22⋊C8—24no—
400C4×C8—24no—
401C4×C8—24no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424C24C22×D411yescharacteristic
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8C22×Q814yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4C22⋊Q814yes—
469C22×C4C22⋊Q814yes—
470C22×C4#13514yes—
471C22×C4#13714yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×Q812yes—
486C22×C4C2×#4112yes—
487C22×C4C2511yescharacteristic
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4C22×Q814yes—
529C4⋊C4C2×#4114yes—
530C4⋊C4C2×#4114yes—
531C22⋊C4—28no—
532C22⋊C4—44no—
533C42—28no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—44no—
558C42—44no—
559C42—44no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42C22⋊Q814yes—
567C42C22⋊Q814yes—
568C42C22×Q812yes—
569C42C22×Q812yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C42C2×#4114yes—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23C2×C22⋊Q812yes—
578C23C2×#13512yes—
579C23C2×#13712yes—
580C23C23×D411yescharacteristic
581Q8—28no—
582Q8—48no—
583Q8—44no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4#479116yescomplemented
646C2×C4C2×C22⋊Q812yes—
647C2×C4C2×C22⋊Q812yes—
648C2×C4C2×C22⋊Q812yes—
649C2×C4C2×C22⋊Q812yes—
650C2×C4C2×C22⋊Q812yes—
651C2×C4#52512yes—
652C2×C4#52512yes—
653C2×C4C2×#13512yes—
654C2×C4C2×#13512yes—
655C2×C4C2×#13712yes—
656C2×C4C2×#13712yes—
657C2×C4#57512yes—
658C2×C4#57512yes—
659C2×C4C23×Q811yescharacteristic
660C2×C4C22×#4112yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4#58512yes—
663C8—44no—
664C8—44no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22C2×#47914yes—
672C22C22×C22⋊Q811yescharacteristic
673C22C2×#52511yescharacteristic
674C22#344911yescharacteristic
675C22C22×#13511yescharacteristic
676C22C22×#13711yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57511yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—41no—
687C4—41no—
688C4—44no—
689C4—41no—
690C4—44no—
691C4—44no—
692C4—41no—
693C4—44no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4C2×#47914yes—
699C4C2×#47914yes—
700C4Q8×#4112yes—
701C4#348012yes—
702C2—24no—
703C2#3495211yescharacteristic
704C2C22×#47911yescharacteristic
705C2#6048411yescharacteristic
706C2#6053611yescharacteristic
707C2#6056211yescharacteristic
708C2#6094211yescharacteristic
709C2#6095011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29289C2semidirect
4. #29275C2semidirect
5. #29257C2semidirect
8. #28287C2semidirect
11. #23779C2semidirect
12. C4×#479C2semidirect
645. C2×C4#479semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C2×C4×#1252
#292892
#292752
#292572
#292442
#292432
#282872
#276862
#243072
#237792
C4×#4792
#213852
#213592
#213582
#208082