Finite Group Databasebeta

← #6684238

Subgroups — #6684238

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3380C211yesmaximal, characteristic, complemented
2C2×#3303C211yesmaximal, characteristic
3#39489C211yesmaximal, characteristic, complemented
4#36164C211yesmaximal, characteristic, complemented
5#33197C211yesmaximal, characteristic, complemented
6#30784C214yesmaximal, complemented
7#30667C214yesmaximal, complemented
8#30279C211yesmaximal, characteristic, complemented
9#29828C211yesmaximal, characteristic, complemented
10#28705C211yesmaximal, characteristic, complemented
11#28638C211yesmaximal, characteristic, complemented
12#25956C214yesmaximal, complemented
13#25934C211yesmaximal, characteristic, complemented
14#25886C211yesmaximal, characteristic, complemented
15#25839C211yesmaximal, characteristic, complemented
16#25704C211yesmaximal, characteristic, complemented
17#24899C214yesmaximal, complemented
18#22208C211yesmaximal, characteristic, complemented
19#21113C211yesmaximal, characteristic, complemented
20C2×#555C2211yescharacteristic
21C2×#554C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#547C2211yescharacteristic
24#3380—28no—
25#3303—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#499C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#478C2212yes—
33C2×#478C2212yes—
34C2×#476C2211yescharacteristic
35C2×#476C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#465C2212yes—
38C2×#461C2211yescharacteristic
39C2×#449C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C4×D8C2212yes—
42C2×C4×D8C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#416C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2850—21no—
48#2674—24no—
49#2620—21no—
50#2568C2211yescharacteristic
51#2567C2211yescharacteristic
52#2563C2211yescharacteristic
53#2563C2211yescharacteristic
54#2504—21no—
55#2503—21no—
56#2501—21no—
57#2499—21no—
58#2489C2214yes—
59#2488—24no—
60#2441C2214yes—
61#2422C2214yes—
62#2339C2214yes—
63#2317—21no—
64C4×#124C2211yescharacteristic
65#2293C2211yescharacteristic
66C2×#382—21no—
67C42×D4C2211yescharacteristic
68#2095C2214yes—
69#2095C2214yes—
70#2076C2214yes—
71#2068C2214yes—
72#2066C2214yes—
73#2059C2214yes—
74#2055C2214yes—
75#2035C2214yes—
76#2033C2214yes—
77#2026C2214yes—
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1948C2214yes—
85#1933C2214yes—
86#1925C2212yes—
87#1923C2214yes—
88#1918C2212yes—
89#1879C2212yes—
90#1877C2214yes—
91#1862C2214yes—
92#1862C2212yes—
93#1854C2214yes—
94#1850C2214yes—
95#1842C2211yescharacteristic
96#1808C2214yes—
97#1802C2214yes—
98#1788C2214yes—
99C4×#107C2211yescharacteristic
100C4×#102C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41C2312yes—
104C22×SD16C2312yes—
105C22×D8C2311yescharacteristic
106C23×C8C2312yes—
107#555—24no—
108#555—42no—
109#554—24no—
110#554—24no—
111#554—42no—
112#554—42no—
113#547—24no—
114#547—42no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×#125C2312yes—
121C2×#125C2312yes—
122C2×#124—22no—
123C2×#124D411yescharacteristic
124C2×#123—22no—
125C2×#123—24no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8C2312yes—
131C2×#121—21no—
132C2×#121—21no—
133C2×#121C2312yes—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2311yescharacteristic
139C2×C4×D4C2311yescharacteristic
140C2×C4×D4C2311yescharacteristic
141C2×#117—21no—
142C2×#117—21no—
143C2×#117C2312yes—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C42—21no—
148#500—24no—
149#500—42no—
150#499—28no—
151#487—24no—
152#487—42no—
153#485—28no—
154#478—28no—
155#478—28no—
156#476—24no—
157#476—24no—
158#476—24no—
159#465—28no—
160#461—24no—
161#449—28no—
162C8×Q8—24no—
163C4×D8—28no—
164C4×D8D418yes—
165C8×D4—24no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2314yes—
169C2×#107C2312yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2314yes—
178#416—28no—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×#102C2311yescharacteristic
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2314yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195C2×C4×C8C2311yescharacteristic
196#400—22no—
197#400—24no—
198#399—24no—
199#397—24no—
200#397—24no—
201#396—24no—
202#395—22no—
203#394—24no—
204#393—22no—
205#393—22no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391D411yescharacteristic
210#391C2314yes—
211#391C2314yes—
212#390—21no—
213#390—21no—
214#390D411yescharacteristic
215#390C2312yes—
216#389—24no—
217#389C2314yes—
218#389C2312yes—
219#388—24no—
220#388—22no—
221#388D414yes—
222#387—24no—
223#386—24no—
224#386—24no—
225#385—24no—
226#384—21no—
227#384—21no—
228#384—21no—
229#384C2314yes—
230#382—21no—
231#382—21no—
232#382—21no—
233#382—21no—
234#382—21no—
235#382—24no—
236#382D411yescharacteristic
237#381—24no—
238#380C2314yes—
239#380C2314yes—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—21no—
245C4×C22⋊C4—21no—
246C4×C22⋊C4C2314yes—
247C2×#95—24no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
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—21no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2412yes—
267C22×D4—22no—
268C22×D4—22no—
269C22×D4C2×D411yescharacteristic
270C22×D4C2411yescharacteristic
271C22×D4C2411yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4C2412yes—
279C2×SD16—24no—
280C2×SD16—44no—
281C2×D8—24no—
282C2×D8C2×D414yes—
283C22×C8—24no—
284C22×C8—44no—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2411yescharacteristic
289C22×C8C2411yescharacteristic
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—84no—
299#125—24no—
300#125—24no—
301#125—42no—
302#125—42no—
303#124—24no—
304#124—48no—
305#123—48no—
306#123—88no—
307C22⋊Q8—24no—
308C22⋊Q8—48no—
309C22⋊Q8—48no—
310C22⋊Q8—42no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313#121—24no—
314#121—42no—
315#121—44no—
316#121—44no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—42no—
329C4×D4C2×D414yes—
330C4×D4C2×D414yes—
331#117—24no—
332#117—42no—
333#117—44no—
334#117—44no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D414yes—
376C2×C22⋊C4C2×D414yes—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—44no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2×D411yescharacteristic
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388#107—24no—
389#107—24no—
390#107—42no—
391#107—44no—
392#107C2×D414yes—
393#106—24no—
394#106—24no—
395#106—24no—
396C4⋊C8—24no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—44no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—44no—
404#103—44no—
405#102—24no—
406#102—24no—
407#102—24no—
408#102—24no—
409#102—42no—
410#102C2×D414yes—
411#102C2×D414yes—
412C22⋊C8—24no—
413C22⋊C8—44no—
414C4×C8—24no—
415C4×C8—22no—
416C4×C8C2×D414yes—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—21no—
421#95—22no—
422#95—22no—
423#95—21no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95C2×D414yes—
429#95C2×D414yes—
430C24—21no—
431C24—21no—
432C24C22×D412yes—
433#41—48no—
434#41—84no—
435#41—88no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—44no—
458C2×D4—48no—
459C2×D4—44no—
460C2×D4—84no—
461C2×D4C22×D412yes—
462C2×D4C22×D412yes—
463C2×D42−1+414yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—82no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C2511yescharacteristic
499SD16—84no—
500D8—28no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—84no—
511C2×C8C22×D412yes—
512C2×C8C22×D412yes—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4C22×D412yes—
543C4⋊C4C22×D412yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—28no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—42no—
565C42—44no—
566C42—44no—
567C42—42no—
568C42—42no—
569C42#13514yes—
570C42C22×D412yes—
571C42H(5,𝔽2)12yes—
572C23—22no—
573C23—24no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—22no—
579C23—22no—
580C23—42no—
581C23—41no—
582C23—41no—
583C23—42no—
584C23—42no—
585C23—41no—
586C23—41no—
587C23C23×D411yescharacteristic
588Q8—44no—
589Q8—42no—
590Q8—42no—
591Q8—44no—
592Q8—162no—
593Q8—162no—
594Q8—162no—
595D4—22no—
596D4—22no—
597D4—22no—
598D4—24no—
599D4—44no—
600D4—42no—
601D4—88no—
602D4—164no—
603D4#53614yes—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4—164no—
645C2×C4—164no—
646C2×C4D4212yes—
647C2×C4D4212yes—
648C2×C4C2×#13514yes—
649C2×C4C2×#13512yes—
650C2×C4#57812yes—
651C2×C4C23×D411yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+412yes—
654C2×C4#58512yes—
655C8—22no—
656C8—22no—
657C8—42no—
658C8—42no—
659C8—42no—
660C22—24no—
661C22—42no—
662C22—44no—
663C22—42no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—82no—
674C22—82no—
675C22C2×D4211yescharacteristic
676C22#348912yes—
677C22C22×#13511yescharacteristic
678C22C2×#57811yescharacteristic
679C22#358512yes—
680C4—24no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#328314yes—
695C4C2×#53611yescharacteristic
696C4#350011yescharacteristic
697C4#356112yes—
698C2—44no—
699C2—82no—
700C2—82no—
701C2#3767611yescharacteristic
702C2C2×#328311yescharacteristic
703C2#6108211yescharacteristic
704C2#6209812yes—
705C2#6218112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3303C2
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×#3380C2semidirect
3. #39489C2semidirect
4. #36164C2semidirect
5. #33197C2semidirect
6. #30784C2semidirect
7. #30667C2semidirect
8. #30279C2semidirect
9. #29828C2semidirect
10. #28705C2semidirect
11. #28638C2semidirect
12. #25956C2semidirect
13. #25934C2semidirect
14. #25886C2semidirect
15. #25839C2semidirect
16. #25704C2semidirect
17. #24899C2semidirect
18. #22208C2semidirect
19. #21113C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33802
C2×#33032
#394892
#361642
#331972
#307842
#306672
#302792
#298282
#287052
#286382
#259562
#259342
#258862
#258392
#257042
#248992
#222082
#211132