Finite Group Databasebeta

← #6684260

Subgroups — #6684260

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3386C211yesmaximal, characteristic, complemented
2C2×#3331C211yesmaximal, characteristic
3#39173C211yesmaximal, characteristic, complemented
4#36303C211yesmaximal, characteristic
5#33201C211yesmaximal, characteristic, complemented
6#30793C214yesmaximal, complemented
7#30688C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29831C211yesmaximal, characteristic, complemented
10#28712C211yesmaximal, characteristic, complemented
11#28647C211yesmaximal, characteristic, complemented
12#25972C214yesmaximal, complemented
13#25931C211yesmaximal, characteristic, complemented
14#25882C211yesmaximal, characteristic
15#25834C211yesmaximal, characteristic, complemented
16#25702C211yesmaximal, characteristic, complemented
17#24923C214yesmaximal, complemented
18#22214C211yesmaximal, characteristic, complemented
19#21117C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#554C2211yescharacteristic
22C2×D4×Q8C2211yescharacteristic
23C2×#547C2211yescharacteristic
24#3386—28no—
25#3331—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#483C2212yes—
32C2×#477C2212yes—
33C2×#475C2211yescharacteristic
34C2×#478C2212yes—
35C2×#476C2211yescharacteristic
36C2×#474C2211yescharacteristic
37C2×#470C2212yes—
38C2×#462C2211yescharacteristic
39C2×#452C2212yes—
40C2×#446C2211yescharacteristic
41C2×#440C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#417C2212yes—
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2850—21no—
48#2690—24no—
49#2642—21no—
50#2568C2211yescharacteristic
51#2567C2211yescharacteristic
52#2565—21no—
53#2547—21no—
54#2547C2211yescharacteristic
55#2547C2211yescharacteristic
56#2503—21no—
57#2492C2214yes—
58#2488—24no—
59#2442C2214yes—
60#2436—21no—
61#2421C2214yes—
62#2339C2214yes—
63#2326—21no—
64C4×#124C2211yescharacteristic
65#2295C2211yescharacteristic
66C2×#391—21no—
67C42×D4C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2076C2214yes—
71#2066C2214yes—
72#2066C2214yes—
73#2058C2214yes—
74#2056C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2026C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1935C2214yes—
86#1927C2212yes—
87#1924C2214yes—
88#1920C2212yes—
89#1878C2214yes—
90#1877C2212yes—
91#1864C2214yes—
92#1864C2212yes—
93#1853C2214yes—
94#1851C2214yes—
95#1842C2211yescharacteristic
96#1809C2214yes—
97#1804C2214yes—
98#1788C2214yes—
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41C2312yes—
104C23×Q8—21no—
105C22×Q16C2312yes—
106C22×SD16C2311yescharacteristic
107C22×M16C2312yes—
108#557—24no—
109#557—42no—
110#554—24no—
111#554—42no—
112D4×Q8—24no—
113D4×Q8—42no—
114#547—24no—
115#547—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×#125—21no—
123C2×#125—22no—
124C2×#125C2312yes—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124D411yescharacteristic
128C2×#123—22no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8C2312yes—
133C2×C22⋊Q8C2312yes—
134C2×#121—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×D4—21no—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C2×C4×D4C2311yescharacteristic
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2312yes—
146C22×C42—21no—
147#501—28no—
148#500—24no—
149#498—42no—
150#487—24no—
151#487—42no—
152#483—28no—
153#478—28no—
154#477—28no—
155#476—24no—
156#475—24no—
157#474—24no—
158#470—28no—
159#462—24no—
160#452—28no—
161#446—24no—
162#440—28no—
163C4×SD16D418yes—
164#436—24no—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2314yes—
171C2×#106C2312yes—
172C2×#106C2311yescharacteristic
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2314yes—
177#417—28no—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2312yes—
185C2×#103C2311yescharacteristic
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2314yes—
192C2×#97C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195#400—22no—
196#400—24no—
197#400—24no—
198#400—24no—
199#398—24no—
200#397—24no—
201#397—22no—
202#397—22no—
203#395—22no—
204#393—24no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—22no—
213#391—21no—
214#391—24no—
215#391D411yescharacteristic
216#391C2312yes—
217#391C2314yes—
218#391C2314yes—
219#390D411yescharacteristic
220#389C2314yes—
221#389C2312yes—
222#388—24no—
223#388—24no—
224#388D414yes—
225#387—22no—
226#387—24no—
227#386—24no—
228#386—24no—
229#385—24no—
230#384—21no—
231#382—21no—
232#382—22no—
233#382D411yescharacteristic
234#382C2314yes—
235#381—24no—
236#380—24no—
237#380C2314yes—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C22⋊C4—21no—
244C4×C22⋊C4—22no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C2×#95—24no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C2×#41—24no—
259C2×#41—28no—
260C2×#41—42no—
261C2×#41—44no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—22no—
265C22×Q8—21no—
266C22×Q8—44no—
267C22×Q8—44no—
268C22×Q8—82no—
269C22×Q8C2×D411yescharacteristic
270C22×Q8C2412yes—
271C22×Q8C2411yescharacteristic
272C22×D4—22no—
273C22×D4C2×D411yescharacteristic
274C22×D4C2411yescharacteristic
275C23×C4—21no—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4C2412yes—
281C2×Q16—24no—
282C2×Q16—44no—
283C2×SD16—24no—
284C2×SD16C2×D414yes—
285C2×M16—24no—
286C2×M16—44no—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2411yescharacteristic
291C22×C8C2411yescharacteristic
292C4⋊Q8—24no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—42no—
302C4⋊Q8—44no—
303#125—24no—
304#125—42no—
305#125—44no—
306#125—48no—
307#124—42no—
308#124—44no—
309#124—82no—
310#123—48no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—48no—
314C22⋊Q8—48no—
315C22⋊Q8—42no—
316C22⋊Q8—42no—
317C22⋊Q8—44no—
318#121—44no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—24no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8C2×D414yes—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—42no—
332C4×D4—42no—
333C4×D4—44no—
334C4×D4C2×D414yes—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D414yes—
372C2×C22⋊C4C2×D414yes—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—82no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D412yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385#107—24no—
386#107—24no—
387#107—24no—
388#106—24no—
389#106—24no—
390#106—42no—
391#106—44no—
392#106C2×D414yes—
393C4⋊C8—24no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396C4⋊C8—44no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—44no—
402#103—42no—
403#103—44no—
404#103C2×D414yes—
405#102—24no—
406#102—24no—
407#102—24no—
408#102C2×D414yes—
409C22⋊C8—24no—
410C22⋊C8—44no—
411#97—24no—
412C4×C8—22no—
413C4×C8C2×D414yes—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—21no—
421#95—22no—
422#95—21no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95C2×D414yes—
430#95C2×D414yes—
431C24—21no—
432C24C22×D412yes—
433#41—48no—
434#41—84no—
435#41—88no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×Q8—816no—
458C2×Q8C22×D412yes—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—44no—
465C2×D4—48no—
466C2×D4C22×D412yes—
467C2×D42−1+414yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—84no—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2511yescharacteristic
502Q16—84no—
503SD16—28no—
504M16—84no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8C22×D412yes—
515C2×C8C22×D412yes—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—82no—
545C4⋊C4—84no—
546C4⋊C4C22×D412yes—
547C4⋊C4C22×D412yes—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—48no—
557C22⋊C4—84no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—42no—
566C42—44no—
567C42—42no—
568C42—42no—
569C42—82no—
570C42C22×D412yes—
571C42H(5,𝔽2)12yes—
572C23—22no—
573C23—24no—
574C23—24no—
575C23—22no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—42no—
580C23—41no—
581C23—41no—
582C23C23×D411yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—44no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—44no—
590Q8—88no—
591Q8—84no—
592Q8—84no—
593Q8—162no—
594D4—22no—
595D4—24no—
596D4—44no—
597D4—88no—
598D4#53614yes—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—88no—
641C2×C4—88no—
642C2×C4—88no—
643C2×C4—164no—
644C2×C4D4212yes—
645C2×C4D4212yes—
646C2×C4#57512yes—
647C2×C4#57712yes—
648C2×C4C23×D411yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+412yes—
651C2×C4#58512yes—
652C8—22no—
653C8—22no—
654C8—42no—
655C8—42no—
656C8—42no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—44no—
667C22C2×D4211yescharacteristic
668C22#348912yes—
669C22C2×#57511yescharacteristic
670C22C2×#57711yescharacteristic
671C22#358612yes—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#327514yes—
688C4C2×#53611yescharacteristic
689C4#350011yescharacteristic
690C4#356112yes—
691C2—44no—
692C2—82no—
693C2#3767611yescharacteristic
694C2C2×#327511yescharacteristic
695C2#6108311yescharacteristic
696C2#6212112yes—
697C2#6218912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3386C2semidirect
3. #39173C2semidirect
5. #33201C2semidirect
6. #30793C2semidirect
7. #30688C2semidirect
8. #30281C2semidirect
9. #29831C2semidirect
10. #28712C2semidirect
11. #28647C2semidirect
12. #25972C2semidirect
13. #25931C2semidirect
15. #25834C2semidirect
16. #25702C2semidirect
17. #24923C2semidirect
18. #22214C2semidirect
19. #21117C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33862
C2×#33312
#391732
#363032
#332012
#307932
#306882
#302812
#298312
#287122
#286472
#259722
#259312
#258822
#258342
#257022
#249232
#222142
#211172