Finite Group Databasebeta

← #6795179

Subgroups — #6795179

Order 512.

1785 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36161C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3#33146C211yesmaximal, characteristic, complemented
4#32266C214yesmaximal, complemented
5#31329C214yesmaximal, complemented
6#29894C212yesmaximal, complemented
7#28706C212yesmaximal, complemented
8#27602C212yesmaximal, complemented
9#26299C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic, complemented
11#25701C212yesmaximal, complemented
12#25340C212yesmaximal, complemented
13#24551C212yesmaximal, complemented
14#24302C211yesmaximal, characteristic, complemented
15D4×C22⋊C8C211yesmaximal, characteristic
16#21941C212yesmaximal, complemented
17C4×#483C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#498C2212yescomplemented
22C2×#483C2211yescharacteristic
23C2×#477C2212yescomplemented
24C2×C8×D4C2212yes—
25C22×C22⋊C8C2212yes—
26#2857C2211yescharacteristic
27#2848C2212yes—
28#2563—21no—
29#2563—21no—
30#2563—22no—
31#2563C2214yes—
32#2563C2212yes—
33#2563C2212yes—
34#2547C2211yescharacteristic
35#2538C2211yescharacteristic
36#2485C2212yes—
37#2480—22no—
38#2466—22no—
39#2447—21no—
40#2435—21no—
41#2415C2212yes—
42C2×#393—22no—
43#2367—21no—
44#2360—21no—
45#2354C2211yescharacteristic
46C4×C22⋊Q8C2211yescharacteristic
47C2×#390—21no—
48C2×#382—21no—
49#2280C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2077C2214yescomplemented
54#2077C2214yescomplemented
55#2077C2214yescomplemented
56#2076C2214yescomplemented
57#2075C2214yescomplemented
58#2058C2214yescomplemented
59#2057C2212yescomplemented
60#2036C2214yescomplemented
61#2035C2214yescomplemented
62#2035C2214yescomplemented
63#2024C2212yescomplemented
64#1990C2212yescomplemented
65#1988C2212yescomplemented
66#1987C2212yescomplemented
67#1954C2212yes—
68#1947C2212yescomplemented
69#1927C2212yescomplemented
70#1920C2212yescomplemented
71#1918C2214yescomplemented
72#1892C2214yescomplemented
73#1872C2211yescharacteristic
74#1872C2212yes—
75#1869C2214yes—
76#1854C2214yescomplemented
77#1854C2214yescomplemented
78#1854C2214yescomplemented
79#1853C2211yescharacteristic
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1825C2214yescomplemented
85#1823C2211yescharacteristic
86#1823C2212yes—
87#1808C2214yescomplemented
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#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—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—44no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4D412yes—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134#498—44no—
135#483—24no—
136#477—44no—
137C8×D4—44no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×C22⋊C8—28no—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156#397—22no—
157#397—22no—
158#397—22no—
159#396—22no—
160#394—22no—
161#394—22no—
162#393—24no—
163#393—24no—
164#393—24no—
165#393—24no—
166#391—21no—
167#391—21no—
168#391C2311yescharacteristic
169#390—22no—
170#390—24no—
171#390—22no—
172#390—21no—
173#390—21no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389C2311yescharacteristic
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—21no—
184#385—22no—
185#385—22no—
186#385—22no—
187#385—21no—
188#384C2311yescharacteristic
189#382—22no—
190#382—22no—
191#382—24no—
192#381—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380C2314yes—
199#380C2314yes—
200#380C2314yes—
201#380C2314yes—
202#380C2312yes—
203#380C2312yes—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207#380C2311yescharacteristic
208#379—22no—
209#379—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
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—24no—
245C22×D4—24no—
246C22×D4—42no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C22×C8—28no—
266C22×C8—44no—
267C22×C8—44no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277#126—48no—
278#124—44no—
279#124—44no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293#117—44no—
294#117—44no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—84no—
343C2×C22⋊C4—84no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—44no—
361#107—44no—
362#106—24no—
363#106—42no—
364#106—44no—
365#106—44no—
366C4⋊C8—24no—
367#103—24no—
368#103—42no—
369#103—44no—
370C22⋊C8—24no—
371C22⋊C8—22no—
372C22⋊C8—28no—
373C22⋊C8—44no—
374C22⋊C8—44no—
375C22⋊C8#4114yes—
376C4×C8—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C22×D412yes—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—816no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—416no—
426C2×D4—42no—
427C2×D4#13714yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4#13514yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4112yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—88no—
492C2×C8C2×#4114yes—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—82no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—44no—
545C42—42no—
546C42—44no—
547C42—44no—
548C42—42no—
549C42—82no—
550C42H(5,𝔽2)14yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—48no—
569C23—48no—
570C23—48no—
571C23—48no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23#54712yes—
576C23C2×#13512yes—
577C23C2×#13714yes—
578C23#57712yes—
579C23C23×D411yescharacteristic
580Q8—42no—
581Q8—42no—
582Q8—88no—
583Q8—88no—
584Q8—84no—
585D4—28no—
586D4—48no—
587D4—44no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—88no—
643C2×C4—84no—
644C2×C4#53412yes—
645C2×C4#53412yes—
646C2×C4#54712yes—
647C2×C4C2×#13512yes—
648C2×C4C2×#13512yes—
649C2×C4C2×#13712yes—
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)12yes—
652C2×C4C2×2−1+411yescharacteristic
653C2×C4#58512yes—
654C8—24no—
655C8—44no—
656C8—42no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—816no—
678C22#334014yes—
679C22C2×#53411yescharacteristic
680C22C2×#54711yescharacteristic
681C22#348511yescharacteristic
682C22C22×#13511yescharacteristic
683C22C22×#13711yescharacteristic
684C22C2×#57711yescharacteristic
685C22#358511yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—42no—
690C4—42no—
691C4—41no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—81no—
696C4—82no—
697C4—82no—
698C4—81no—
699C4—81no—
700C4—84no—
701C4#319014yes—
702C4#351412yes—
703C4#356112yes—
704C2—24no—
705C2—44no—
706C2—82no—
707C2#3891811yescharacteristic
708C2C2×#319011yescharacteristic
709C2#6075711yescharacteristic
710C2C2×#334011yescharacteristic
711C2#6125411yescharacteristic
712C2#6170011yescharacteristic
713C2#6173511yescharacteristic
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. #33146C2semidirect
4. #32266C2semidirect
5. #31329C2semidirect
6. #29894C2semidirect
7. #28706C2semidirect
8. #27602C2semidirect
9. #26299C2semidirect
10. #25811C2semidirect
11. #25701C2semidirect
12. #25340C2semidirect
13. #24551C2semidirect
14. #24302C2semidirect
16. #21941C2semidirect
17. C4×#483C2semidirect
21. C2×#498C22semidirect
23. C2×#477C22semidirect
53. #2077C22semidirect
54. #2077C22semidirect
55. #2077C22semidirect
56. #2076C22semidirect
57. #2075C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
60. #2036C22semidirect
61. #2035C22semidirect
62. #2035C22semidirect
63. #2024C22semidirect
64. #1990C22semidirect
65. #1988C22semidirect
66. #1987C22semidirect
68. #1947C22semidirect
69. #1927C22semidirect
70. #1920C22semidirect
71. #1918C22semidirect
72. #1892C22semidirect
76. #1854C22semidirect
77. #1854C22semidirect
78. #1854C22semidirect
84. #1825C22semidirect
87. #1808C22semidirect
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
#361612
C2×#25632
#331462
#322662
#313292
#298942
#287062
#276022
#262992
#258112
#257012
#253402
#245512
#243022
D4×C22⋊C82
#219412
C4×#4832