Finite Group Databasebeta

← #6795180

Subgroups — #6795180

Order 512.

1823 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36154C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic, complemented
4#32261C214yesmaximal, complemented
5#31324C214yesmaximal, complemented
6#29895C212yesmaximal, complemented
7#28705C212yesmaximal, complemented
8#27603C212yesmaximal, complemented
9#26302C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25706C212yesmaximal, complemented
12#25340C212yesmaximal, complemented
13#24552C212yesmaximal, complemented
14#24302C211yesmaximal, characteristic, complemented
15D4×C22⋊C8C211yesmaximal, characteristic
16#21940C212yesmaximal, complemented
17C4×#484C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#500C2212yescomplemented
22C2×#484C2211yescharacteristic
23C2×#477C2212yescomplemented
24C2×C8×D4C2212yes—
25C22×C22⋊C8C2212yes—
26#2857C2211yescharacteristic
27#2848C2212yes—
28#2563—21no—
29#2563—22no—
30#2563—21no—
31#2563C2214yes—
32#2563C2211yescharacteristic
33#2563C2212yes—
34#2563C2212yes—
35#2538C2211yescharacteristic
36#2487C2212yes—
37#2454—21no—
38#2450—22no—
39#2442—21no—
40#2437—22no—
41#2417C2212yes—
42C2×#393—22no—
43#2349—21no—
44D4×C4⋊C4C2211yescharacteristic
45D4×C22⋊C4—21no—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#390—21no—
48C2×#382—21no—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2076C2214yescomplemented
54#2076C2214yescomplemented
55#2076C2214yescomplemented
56#2076C2214yescomplemented
57#2076C2214yescomplemented
58#2057C2212yescomplemented
59#2057C2214yescomplemented
60#2036C2214yescomplemented
61#2036C2214yescomplemented
62#2036C2214yescomplemented
63#2024C2212yescomplemented
64#1990C2212yescomplemented
65#1987C2212yescomplemented
66#1987C2212yescomplemented
67#1954C2212yes—
68#1947C2212yescomplemented
69#1928C2212yescomplemented
70#1921C2212yescomplemented
71#1921C2214yescomplemented
72#1892C2214yescomplemented
73#1872C2211yescharacteristic
74#1872C2212yes—
75#1869C2214yes—
76#1852C2211yescharacteristic
77#1852C2212yes—
78#1852C2212yes—
79#1852C2212yes—
80#1852C2212yes—
81#1852C2214yescomplemented
82#1852C2214yescomplemented
83#1852C2214yescomplemented
84#1824C2211yescharacteristic
85#1824C2212yes—
86#1824C2214yescomplemented
87#1808C2214yescomplemented
88#1808C2214yescomplemented
89#1808C2214yescomplemented
90#1797C2211yescharacteristic
91#1779C2212yes—
92C4×#107C2212yescomplemented
93C4×#103C2212yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4D411yescharacteristic
97C24×C4C2312yes—
98C23×C8C2314yes—
99#554—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—44no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D412yes—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4D412yes—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#500—44no—
136#484—24no—
137#477—44no—
138C8×D4—44no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×C22⋊C8—28no—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C4×C8C2312yes—
157#400—22no—
158#399—22no—
159#398—22no—
160#396—22no—
161#396—22no—
162#393—24no—
163#393—24no—
164#393—24no—
165#393—24no—
166#393—22no—
167#391—21no—
168#391C2311yescharacteristic
169#390—22no—
170#390—22no—
171#390—24no—
172#390—21no—
173#390—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389C2312yes—
178#389C2311yescharacteristic
179#387—21no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#385—21no—
186#385—22no—
187#385—22no—
188#384C2311yescharacteristic
189#382—22no—
190#382—22no—
191#382—24no—
192#382—21no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2314yes—
202#380C2314yes—
203#380C2314yes—
204#380C2314yes—
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210#380C2311yescharacteristic
211#379—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236C25C2×D412yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×Q8C2411yescharacteristic
244C22×D4—42no—
245C22×D4—42no—
246C22×D4Q1618yes—
247C22×D4C2×D411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D412yes—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C22×C8—28no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279#125—44no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—48no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—82no—
286C4×Q8—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—84no—
345C2×C22⋊C4—84no—
346C2×C22⋊C4—84no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—42no—
363#107—44no—
364#107—44no—
365#107—44no—
366#107—44no—
367C4⋊C8—24no—
368#103—24no—
369#103—42no—
370#103—44no—
371C22⋊C8—24no—
372C22⋊C8—22no—
373C22⋊C8—28no—
374C22⋊C8—44no—
375C22⋊C8—44no—
376C22⋊C8#4114yes—
377C4×C8—24no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395C24—24no—
396C24—24no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C2×Q1618yes—
402C24C22×D412yes—
403C24C22×D412yes—
404C24C22×D411yescharacteristic
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—81no—
417C2×Q8—81no—
418C2×Q8—816no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—416no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4C2×Q1614yes—
427C2×D4C2×Q1614yes—
428C2×D4C2×Q1618yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4C2×Q1614yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C42−1+412yes—
489C22×C4C2511yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—88no—
499C2×C8C2×#4114yes—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—81no—
530C4⋊C4—82no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—82no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—44no—
552C42—42no—
553C42—44no—
554C42—42no—
555C42H(5,𝔽2)14yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—216no—
573C23—48no—
574C23—48no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23#54712yes—
580C23C22×Q1614yes—
581C23C22×Q1614yes—
582C23C22×Q1612yes—
583C23#57612yes—
584C23C23×D411yescharacteristic
585Q8—42no—
586Q8—42no—
587Q8—88no—
588Q8—88no—
589Q8—162no—
590D4—28no—
591D4—28no—
592D4—48no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4—164no—
650C2×C4—164no—
651C2×C4#53412yes—
652C2×C4#53412yes—
653C2×C4#54712yes—
654C2×C4C22×Q1612yes—
655C2×C4C22×Q1612yes—
656C2×C4#57512yes—
657C2×C4C22×#4111yescharacteristic
658C2×C4C2×H(5,𝔽2)12yes—
659C2×C4C2×2−1+411yescharacteristic
660C2×C4#58512yes—
661C8—24no—
662C8—44no—
663C8—42no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—816no—
685C22#333814yes—
686C22C2×#53411yescharacteristic
687C22C2×#54711yescharacteristic
688C22#348511yescharacteristic
689C22C23×Q1611yescharacteristic
690C22C2×#57511yescharacteristic
691C22C2×#57611yescharacteristic
692C22#358511yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—42no—
697C4—42no—
698C4—41no—
699C4—81no—
700C4—82no—
701C4—82no—
702C4—81no—
703C4—81no—
704C4—82no—
705C4—84no—
706C4—81no—
707C4—82no—
708C4#318914yes—
709C4#351412yes—
710C4#356112yes—
711C2—24no—
712C2—44no—
713C2—44no—
714C2#3891811yescharacteristic
715C2C2×#318911yescharacteristic
716C2#6075611yescharacteristic
717C2C2×#333811yescharacteristic
718C2#6125611yescharacteristic
719C2#6169811yescharacteristic
720C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33149C2semidirect
4. #32261C2semidirect
5. #31324C2semidirect
6. #29895C2semidirect
7. #28705C2semidirect
8. #27603C2semidirect
9. #26302C2semidirect
10. #25813C2semidirect
11. #25706C2semidirect
12. #25340C2semidirect
13. #24552C2semidirect
14. #24302C2semidirect
16. #21940C2semidirect
17. C4×#484C2semidirect
21. C2×#500C22semidirect
23. C2×#477C22semidirect
53. #2076C22semidirect
54. #2076C22semidirect
55. #2076C22semidirect
56. #2076C22semidirect
57. #2076C22semidirect
58. #2057C22semidirect
59. #2057C22semidirect
60. #2036C22semidirect
61. #2036C22semidirect
62. #2036C22semidirect
63. #2024C22semidirect
64. #1990C22semidirect
65. #1987C22semidirect
66. #1987C22semidirect
68. #1947C22semidirect
69. #1928C22semidirect
70. #1921C22semidirect
71. #1921C22semidirect
72. #1892C22semidirect
81. #1852C22semidirect
82. #1852C22semidirect
83. #1852C22semidirect
86. #1824C22semidirect
87. #1808C22semidirect
88. #1808C22semidirect
89. #1808C22semidirect
92. C4×#107C22semidirect
93. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#361542
C2×#25632
#331492
#322612
#313242
#298952
#287052
#276032
#263022
#258132
#257062
#253402
#245522
#243022
D4×C22⋊C82
#219402
C4×#4842