Finite Group Databasebeta

← #6795177

Subgroups — #6795177

Order 512.

1899 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36154C211yesmaximal, characteristic
2C2×#2547C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic, complemented
4#32260C214yesmaximal, complemented
5#31320C214yesmaximal, complemented
6#29893C212yesmaximal, complemented
7#28706C212yesmaximal, complemented
8#27601C212yesmaximal, complemented
9#26298C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic, complemented
11#25702C212yesmaximal, complemented
12#25341C212yesmaximal, complemented
13#24551C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic, complemented
15D4×C22⋊C8C211yesmaximal, characteristic
16#21938C212yesmaximal, complemented
17C4×#483C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#498C2212yescomplemented
23C2×#483C2211yescharacteristic
24C2×#475C2212yescomplemented
25C2×C8×D4C2212yes—
26C22×C22⋊C8C2212yes—
27#2857C2211yescharacteristic
28#2848C2212yes—
29#2563C2211yescharacteristic
30#2547—21no—
31#2547—22no—
32#2547—21no—
33#2547C2214yes—
34#2547C2212yes—
35#2547C2212yes—
36#2538C2211yescharacteristic
37#2487C2212yes—
38#2454—21no—
39#2450—22no—
40#2442—21no—
41#2437—22no—
42#2417C2212yes—
43C2×#397—22no—
44#2349—21no—
45D4×C4⋊C4C2211yescharacteristic
46D4×C22⋊C4—21no—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391—21no—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2077C2214yescomplemented
54#2077C2214yescomplemented
55#2077C2214yescomplemented
56#2077C2214yescomplemented
57#2077C2214yescomplemented
58#2058C2212yescomplemented
59#2058C2214yescomplemented
60#2035C2214yescomplemented
61#2035C2214yescomplemented
62#2035C2214yescomplemented
63#2024C2212yescomplemented
64#1990C2212yescomplemented
65#1988C2212yescomplemented
66#1987C2212yescomplemented
67#1954C2212yes—
68#1948C2212yescomplemented
69#1927C2212yescomplemented
70#1920C2212yescomplemented
71#1920C2214yescomplemented
72#1892C2214yescomplemented
73#1872C2211yescharacteristic
74#1872C2212yes—
75#1869C2214yes—
76#1853C2211yescharacteristic
77#1853C2212yes—
78#1853C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2214yescomplemented
82#1853C2214yescomplemented
83#1853C2214yescomplemented
84#1824C2214yescomplemented
85#1823C2211yescharacteristic
86#1823C2212yes—
87#1807C2214yescomplemented
88#1807C2214yescomplemented
89#1807C2214yescomplemented
90#1797C2211yescharacteristic
91#1779C2212yes—
92C4×#106C2212yescomplemented
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×#124—22no—
108C2×#124—22no—
109C2×#124—24no—
110C2×#124—48no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—44no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4D412yes—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#498—44no—
140#483—24no—
141#475—44no—
142C8×D4—44no—
143C2×#106C2312yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2312yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×C22⋊C8—28no—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161#400—22no—
162#399—22no—
163#398—22no—
164#397—24no—
165#397—24no—
166#397—24no—
167#397—24no—
168#396—22no—
169#396—22no—
170#393—22no—
171#391—22no—
172#391—22no—
173#391—24no—
174#391—21no—
175#391C2311yescharacteristic
176#390—21no—
177#390—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2311yescharacteristic
183#387—21no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—21no—
190#385—22no—
191#385—22no—
192#384C2311yescharacteristic
193#382—21no—
194#380—21no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2312yes—
204#380C2312yes—
205#380C2314yes—
206#380C2314yes—
207#380C2312yes—
208#380C2314yes—
209#380C2314yes—
210#380C2312yes—
211#380C2311yescharacteristic
212#379—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C2×#95—22no—
221C2×#95—21no—
222C2×#95—22no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25C2×D412yes—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2411yescharacteristic
245C22×D4—42no—
246C22×D4—42no—
247C22×D4SD1618yes—
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D411yescharacteristic
250C22×D4C2×D412yes—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C22×C8—28no—
269C22×C8—44no—
270C22×C8—44no—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280#125—44no—
281#124—48no—
282#124—48no—
283#124—416no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—48no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—82no—
290C4×Q8—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—48no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4—84no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#106—24no—
366#106—42no—
367#106—44no—
368#106—44no—
369#106—44no—
370#106—44no—
371C4⋊C8—24no—
372#103—24no—
373#103—42no—
374#103—44no—
375C22⋊C8—24no—
376C22⋊C8—22no—
377C22⋊C8—28no—
378C22⋊C8—44no—
379C22⋊C8—44no—
380C22⋊C8#4114yes—
381C4×C8—24no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399C24—24no—
400C24—24no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24C2×SD1618yes—
406C24C22×D412yes—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—81no—
421C2×Q8—81no—
422C2×Q8—816no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—416no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4C2×SD1614yes—
431C2×D4C2×SD1614yes—
432C2×D4C2×SD1618yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4C2×SD1614yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C42−1+412yes—
493C22×C4C2511yescharacteristic
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—88no—
503C2×C8C2×#4114yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—81no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—81no—
534C4⋊C4—82no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—44no—
557C42—44no—
558C42—42no—
559C42H(5,𝔽2)14yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—216no—
577C23—48no—
578C23—48no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23#54712yes—
584C23C22×SD1614yes—
585C23C22×SD1614yes—
586C23C22×SD1612yes—
587C23#57712yes—
588C23C23×D411yescharacteristic
589Q8—42no—
590Q8—42no—
591Q8—88no—
592Q8—88no—
593Q8—162no—
594D4—28no—
595D4—28no—
596D4—48no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—88no—
652C2×C4—84no—
653C2×C4—164no—
654C2×C4—164no—
655C2×C4#53412yes—
656C2×C4#53412yes—
657C2×C4#54712yes—
658C2×C4C22×SD1612yes—
659C2×C4C22×SD1612yes—
660C2×C4#57512yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×2−1+411yescharacteristic
664C2×C4#58512yes—
665C8—24no—
666C8—44no—
667C8—42no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—816no—
689C22#333514yes—
690C22C2×#53411yescharacteristic
691C22C2×#54711yescharacteristic
692C22#348511yescharacteristic
693C22C23×SD1611yescharacteristic
694C22C2×#57511yescharacteristic
695C22C2×#57711yescharacteristic
696C22#358611yescharacteristic
697C4—22no—
698C4—22no—
699C4—22no—
700C4—42no—
701C4—42no—
702C4—41no—
703C4—81no—
704C4—82no—
705C4—82no—
706C4—81no—
707C4—84no—
708C4—81no—
709C4—81no—
710C4—82no—
711C4—82no—
712C4#318714yes—
713C4#351412yes—
714C4#356112yes—
715C2—24no—
716C2—44no—
717C2—44no—
718C2#3891811yescharacteristic
719C2C2×#318711yescharacteristic
720C2#6075811yescharacteristic
721C2C2×#333511yescharacteristic
722C2#6125711yescharacteristic
723C2#6169511yescharacteristic
724C2#6173911yescharacteristic
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. #33148C2semidirect
4. #32260C2semidirect
5. #31320C2semidirect
6. #29893C2semidirect
7. #28706C2semidirect
8. #27601C2semidirect
9. #26298C2semidirect
10. #25811C2semidirect
11. #25702C2semidirect
12. #25341C2semidirect
13. #24551C2semidirect
14. #24301C2semidirect
16. #21938C2semidirect
17. C4×#483C2semidirect
22. C2×#498C22semidirect
24. C2×#475C22semidirect
53. #2077C22semidirect
54. #2077C22semidirect
55. #2077C22semidirect
56. #2077C22semidirect
57. #2077C22semidirect
58. #2058C22semidirect
59. #2058C22semidirect
60. #2035C22semidirect
61. #2035C22semidirect
62. #2035C22semidirect
63. #2024C22semidirect
64. #1990C22semidirect
65. #1988C22semidirect
66. #1987C22semidirect
68. #1948C22semidirect
69. #1927C22semidirect
70. #1920C22semidirect
71. #1920C22semidirect
72. #1892C22semidirect
81. #1853C22semidirect
82. #1853C22semidirect
83. #1853C22semidirect
84. #1824C22semidirect
87. #1807C22semidirect
88. #1807C22semidirect
89. #1807C22semidirect
92. C4×#106C22semidirect
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×#25472
#331482
#322602
#313202
#298932
#287062
#276012
#262982
#258112
#257022
#253412
#245512
#243012
D4×C22⋊C82
#219382
C4×#4832