Finite Group Databasebeta

← #6795182

Subgroups — #6795182

Order 512.

1861 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36161C211yesmaximal, characteristic
2C2×#2547C211yesmaximal, characteristic
3#33147C211yesmaximal, characteristic, complemented
4#32267C214yesmaximal, complemented
5#31333C214yesmaximal, complemented
6#29894C212yesmaximal, complemented
7#28704C212yesmaximal, complemented
8#27604C212yesmaximal, complemented
9#26303C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25705C212yesmaximal, complemented
12#25341C212yesmaximal, complemented
13#24552C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic, complemented
15D4×C22⋊C8C211yesmaximal, characteristic
16#21941C212yesmaximal, complemented
17C4×#484C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#500C2212yescomplemented
23C2×#484C2211yescharacteristic
24C2×#475C2212yescomplemented
25C2×C8×D4C2212yes—
26C22×C22⋊C8C2212yes—
27#2857C2211yescharacteristic
28#2848C2212yes—
29#2547—21no—
30#2547—21no—
31#2547—22no—
32#2547C2214yes—
33#2547C2211yescharacteristic
34#2547C2212yes—
35#2547C2212yes—
36#2538C2211yescharacteristic
37#2485C2212yes—
38#2480—22no—
39#2466—22no—
40#2447—21no—
41#2435—21no—
42#2415C2212yes—
43C2×#397—22no—
44#2367—21no—
45#2360—21no—
46#2354C2211yescharacteristic
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391—21no—
49#2280C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2077C2214yescomplemented
54#2077C2214yescomplemented
55#2076C2214yescomplemented
56#2075C2214yescomplemented
57#2075C2214yescomplemented
58#2058C2212yescomplemented
59#2057C2214yescomplemented
60#2036C2214yescomplemented
61#2036C2214yescomplemented
62#2035C2214yescomplemented
63#2024C2212yescomplemented
64#1990C2212yescomplemented
65#1988C2212yescomplemented
66#1988C2212yescomplemented
67#1954C2212yes—
68#1948C2212yescomplemented
69#1928C2212yescomplemented
70#1921C2212yescomplemented
71#1919C2214yescomplemented
72#1892C2214yescomplemented
73#1872C2211yescharacteristic
74#1872C2212yes—
75#1869C2214yes—
76#1854C2214yescomplemented
77#1854C2214yescomplemented
78#1854C2214yescomplemented
79#1853C2212yes—
80#1853C2212yes—
81#1852C2211yescharacteristic
82#1852C2212yes—
83#1852C2212yes—
84#1825C2214yescomplemented
85#1824C2211yescharacteristic
86#1824C2212yes—
87#1808C2214yescomplemented
88#1808C2214yescomplemented
89#1807C2214yescomplemented
90#1797C2211yescharacteristic
91#1779C2212yes—
92C4×#107C2212yescomplemented
93C4×#103C2212yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4D411yescharacteristic
97C24×C4C2312yes—
98C23×C8C2314yes—
99#556—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#126—22no—
107C2×#124—21no—
108C2×#124—22no—
109C2×#124—24no—
110C2×#124—22no—
111C2×#124—21no—
112C2×#124—48no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—44no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D412yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4D412yes—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#500—44no—
139#484—24no—
140#475—44no—
141C8×D4—44no—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×C22⋊C8—28no—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160#397—24no—
161#397—24no—
162#397—24no—
163#397—24no—
164#397—22no—
165#397—22no—
166#397—22no—
167#396—22no—
168#394—22no—
169#394—22no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—24no—
174#391—21no—
175#391C2311yescharacteristic
176#390—21no—
177#390—21no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2311yescharacteristic
183#388—22no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—21no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—21no—
192#384C2311yescharacteristic
193#381—22no—
194#380—21no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203#380C2314yes—
204#380C2312yes—
205#380C2314yes—
206#380C2314yes—
207#380C2312yes—
208#380C2311yescharacteristic
209#379—22no—
210#379—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—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—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2411yescharacteristic
245C22×D4—24no—
246C22×D4—24no—
247C22×D4—42no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2412yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C22×C8—28no—
267C22×C8—44no—
268C22×C8—44no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#126—48no—
279#124—416no—
280#124—48no—
281#124—44no—
282#124—44no—
283#124—48no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—44no—
297#117—44no—
298#117—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—44no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—42no—
366#107—44no—
367#107—44no—
368#106—44no—
369#106—44no—
370C4⋊C8—24no—
371#103—24no—
372#103—42no—
373#103—44no—
374C22⋊C8—24no—
375C22⋊C8—22no—
376C22⋊C8—28no—
377C22⋊C8—44no—
378C22⋊C8—44no—
379C22⋊C8#4114yes—
380C4×C8—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398C24—24no—
399C24—24no—
400C24—24no—
401C24—24no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24C22×D412yes—
406C24C22×D412yes—
407C24C22×D411yescharacteristic
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—816no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—416no—
430C2×D4—42no—
431C2×D4AGL(1,ℤ/8)14yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—41no—
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—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4#13514yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—88no—
496C2×C8C2×#4114yes—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—82no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—44no—
549C42—44no—
550C42—42no—
551C42—44no—
552C42—42no—
553C42—82no—
554C42H(5,𝔽2)14yes—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—22no—
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—48no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23#54712yes—
580C23C2×#13512yes—
581C23C2×AGL(1,ℤ/8)14yes—
582C23#57812yes—
583C23C23×D411yescharacteristic
584Q8—42no—
585Q8—42no—
586Q8—88no—
587Q8—84no—
588Q8—88no—
589D4—28no—
590D4—48no—
591D4—44no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4#53412yes—
649C2×C4#53412yes—
650C2×C4#54712yes—
651C2×C4C2×#13512yes—
652C2×C4C2×#13512yes—
653C2×C4C2×AGL(1,ℤ/8)12yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)12yes—
656C2×C4C2×2−1+411yescharacteristic
657C2×C4#58512yes—
658C8—24no—
659C8—44no—
660C8—42no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—816no—
682C22#334214yes—
683C22C2×#53411yescharacteristic
684C22C2×#54711yescharacteristic
685C22#348511yescharacteristic
686C22C22×#13511yescharacteristic
687C22C22×AGL(1,ℤ/8)11yescharacteristic
688C22C2×#57811yescharacteristic
689C22#358611yescharacteristic
690C4—22no—
691C4—22no—
692C4—22no—
693C4—42no—
694C4—42no—
695C4—41no—
696C4—82no—
697C4—81no—
698C4—81no—
699C4—82no—
700C4—84no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4—81no—
705C4#318814yes—
706C4#351412yes—
707C4#356112yes—
708C2—24no—
709C2—44no—
710C2—82no—
711C2#3891811yescharacteristic
712C2C2×#318811yescharacteristic
713C2#6075911yescharacteristic
714C2C2×#334211yescharacteristic
715C2#6125211yescharacteristic
716C2#6170111yescharacteristic
717C2#6173711yescharacteristic
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. #33147C2semidirect
4. #32267C2semidirect
5. #31333C2semidirect
6. #29894C2semidirect
7. #28704C2semidirect
8. #27604C2semidirect
9. #26303C2semidirect
10. #25813C2semidirect
11. #25705C2semidirect
12. #25341C2semidirect
13. #24552C2semidirect
14. #24301C2semidirect
16. #21941C2semidirect
17. C4×#484C2semidirect
22. C2×#500C22semidirect
24. C2×#475C22semidirect
53. #2077C22semidirect
54. #2077C22semidirect
55. #2076C22semidirect
56. #2075C22semidirect
57. #2075C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
60. #2036C22semidirect
61. #2036C22semidirect
62. #2035C22semidirect
63. #2024C22semidirect
64. #1990C22semidirect
65. #1988C22semidirect
66. #1988C22semidirect
68. #1948C22semidirect
69. #1928C22semidirect
70. #1921C22semidirect
71. #1919C22semidirect
72. #1892C22semidirect
76. #1854C22semidirect
77. #1854C22semidirect
78. #1854C22semidirect
84. #1825C22semidirect
87. #1808C22semidirect
88. #1808C22semidirect
89. #1807C22semidirect
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
#361612
C2×#25472
#331472
#322672
#313332
#298942
#287042
#276042
#263032
#258132
#257052
#253412
#245522
#243012
D4×C22⋊C82
#219412
C4×#4842