Finite Group Databasebeta

← #6686272

Subgroups — #6686272

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3364C211yesmaximal, characteristic, complemented
2C2×#3335C211yesmaximal, characteristic
3#39173C211yesmaximal, characteristic, complemented
4#36406C211yesmaximal, characteristic
5#33201C211yesmaximal, characteristic, complemented
6#30639C214yesmaximal, complemented
7#30590C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29831C211yesmaximal, characteristic, complemented
10#28677C211yesmaximal, characteristic, complemented
11#28600C211yesmaximal, characteristic, complemented
12#26558C214yesmaximal, complemented
13#26346C214yesmaximal, complemented
14#25840C211yesmaximal, characteristic, complemented
15#25703C211yesmaximal, characteristic
16#25646C211yesmaximal, characteristic, complemented
17#25573C211yesmaximal, characteristic, complemented
18#22208C211yesmaximal, characteristic, complemented
19#21117C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#547C2211yescharacteristic
24#3364—28no—
25#3335—28no—
26C2×#500C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#488C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#481C2212yes—
32C2×#483C2212yes—
33C2×#479C2212yes—
34C2×#475C2211yescharacteristic
35C2×#476C2211yescharacteristic
36C2×#474C2211yescharacteristic
37C2×#465C2212yes—
38C2×#464C2212yes—
39C2×#462C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C22×#106C2212yes—
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2857—21no—
48#2690—24no—
49#2642—21no—
50#2567C2211yescharacteristic
51#2565—21no—
52#2553—24no—
53#2547—21no—
54#2547C2211yescharacteristic
55#2546C2211yescharacteristic
56#2542C2211yescharacteristic
57#2499—21no—
58#2442C2214yes—
59#2432—21no—
60#2427C2214yes—
61#2421C2214yes—
62#2360C2214yes—
63#2326—21no—
64C4×#124C2211yescharacteristic
65#2297—21no—
66#2295C2211yescharacteristic
67C42×D4C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2088C2214yes—
71#2066C2214yes—
72#2059C2214yes—
73#2056C2214yes—
74#2042C2214yes—
75#2035C2214yes—
76#2033C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1941C2214yes—
86#1927C2212yes—
87#1923C2214yes—
88#1920C2212yes—
89#1892C2214yes—
90#1877C2212yes—
91#1872C2214yes—
92#1865C2214yes—
93#1864C2212yes—
94#1863C2214yes—
95#1852C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1804C2214yes—
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41—21no—
104C22×Q16C2312yes—
105C22×SD16C2311yescharacteristic
106C23×C8C2312yes—
107#557—24no—
108#557—42no—
109#556—24no—
110#556—42no—
111#554—24no—
112#554—42no—
113#547—24no—
114#547—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×#126—22no—
122C2×#125—21no—
123C2×#125—22no—
124C2×#125C2312yes—
125C2×#124—21no—
126C2×#124D411yescharacteristic
127C2×#124C2312yes—
128C2×#123—22no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8C2312yes—
133C2×C22⋊Q8C2314yes—
134C2×#121C2312yes—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2311yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C2×C4×D4C2311yescharacteristic
144C2×#117—21no—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2312yes—
147C22×C42—21no—
148#500—42no—
149#498—24no—
150#488—28no—
151#487—24no—
152#487—42no—
153#483—28no—
154#481—28no—
155#479—28no—
156#476—24no—
157#475—24no—
158#474—24no—
159#465—28no—
160#464—28no—
161#462—24no—
162C8×Q8—24no—
163C4×Q16—28no—
164C4×SD16D418yes—
165C8×D4—24no—
166C2×#107C2312yes—
167C2×#107C2314yes—
168C2×#106—28no—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2314yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2311yescharacteristic
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2311yescharacteristic
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2314yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195C2×C4×C8C2311yescharacteristic
196#400—24no—
197#400—24no—
198#399—22no—
199#398—22no—
200#398—24no—
201#397—24no—
202#397—22no—
203#397—24no—
204#395—22no—
205#394—24no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391D411yescharacteristic
215#391C2312yes—
216#391C2314yes—
217#390—21no—
218#390D411yescharacteristic
219#389—24no—
220#389C2314yes—
221#389C2314yes—
222#389C2312yes—
223#388—24no—
224#388D414yes—
225#387—22no—
226#387—24no—
227#387—24no—
228#386—24no—
229#385—24no—
230#385—24no—
231#384—21no—
232#384—21no—
233#382—22no—
234#382D411yescharacteristic
235#382C2314yes—
236#380—24no—
237#380—24no—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C22⋊C4—24no—
244C4×C22⋊C4—21no—
245C4×C22⋊C4—22no—
246C4×C22⋊C4C2314yes—
247#376—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C2×#41—44no—
259C2×#41—44no—
260C2×#41—82no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2412yes—
266C22×Q8C2411yescharacteristic
267C22×D4—22no—
268C22×D4C2×D411yescharacteristic
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4C2412yes—
275C2×Q16—24no—
276C2×Q16—44no—
277C2×SD16—42no—
278C2×SD16C2×D414yes—
279C22×C8—24no—
280C22×C8—44no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2411yescharacteristic
285C22×C8C2411yescharacteristic
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—44no—
295C4⋊Q8—82no—
296#126—48no—
297#125—24no—
298#125—42no—
299#125—44no—
300#125—48no—
301#124—24no—
302#124—24no—
303#124—42no—
304#124—44no—
305#123—48no—
306C22⋊Q8—24no—
307C22⋊Q8—28no—
308C22⋊Q8—48no—
309C22⋊Q8—44no—
310C22⋊Q8—42no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313#121—24no—
314#121—42no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8—44no—
323C4×Q8C2×D414yes—
324C4×D4—24no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—42no—
329C4×D4—44no—
330C4×D4C2×D414yes—
331#117—44no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—42no—
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⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D414yes—
373C2×C22⋊C4C2×D414yes—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—82no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387#107—24no—
388#107—44no—
389#106—24no—
390#106—24no—
391#106—24no—
392#106—24no—
393#106—24no—
394#106C2×D414yes—
395C4⋊C8—24no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398C4⋊C8—44no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—44no—
405#103—44no—
406#103C2×D414yes—
407#102—24no—
408#102—24no—
409#102—42no—
410#102C2×D414yes—
411C22⋊C8—24no—
412C22⋊C8—44no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8C2×D414yes—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—21no—
420#95—24no—
421#95—24no—
422#95—22no—
423#95—21no—
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—816no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—82no—
452C2×Q8C22×D412yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—82no—
461C2×D4#13514yes—
462C2×D4C22×D412yes—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
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—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—84no—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C2511yescharacteristic
495Q16—48no—
496SD16—44no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—48no—
507C2×C8C22×D412yes—
508C2×C8C22×D412yes—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—82no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4C22×D412yes—
541C4⋊C4C22×D412yes—
542C4⋊C42−1+414yes—
543C22⋊C4—28no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—42no—
562C42—44no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42—82no—
567C42C22×D412yes—
568C42H(5,𝔽2)12yes—
569C23—24no—
570C23—22no—
571C23—24no—
572C23—22no—
573C23—22no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23C2×#13514yes—
579C23C23×D411yescharacteristic
580Q8—22no—
581Q8—22no—
582Q8—44no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—44no—
587Q8—84no—
588Q8—84no—
589Q8—162no—
590D4—22no—
591D4—22no—
592D4—28no—
593D4—42no—
594D4—88no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4—164no—
639C2×C4D4212yes—
640C2×C4D4212yes—
641C2×C4C2×#13512yes—
642C2×C4#57712yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+412yes—
646C2×C4#58512yes—
647C8—22no—
648C8—22no—
649C8—42no—
650C8—42no—
651C8—42no—
652C22—24no—
653C22—28no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—82no—
661C22—82no—
662C22C2×D4211yescharacteristic
663C22#348912yes—
664C22C22×#13511yescharacteristic
665C22C2×#57711yescharacteristic
666C22#358612yes—
667C4—22no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#328214yes—
683C4#344911yescharacteristic
684C4#349911yescharacteristic
685C4#355412yes—
686C2—44no—
687C2—82no—
688C2#3768011yescharacteristic
689C2C2×#328211yescharacteristic
690C2#6108311yescharacteristic
691C2#6210412yes—
692C2#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×#3364C2semidirect
3. #39173C2semidirect
5. #33201C2semidirect
6. #30639C2semidirect
7. #30590C2semidirect
8. #30281C2semidirect
9. #29831C2semidirect
10. #28677C2semidirect
11. #28600C2semidirect
12. #26558C2semidirect
13. #26346C2semidirect
14. #25840C2semidirect
16. #25646C2semidirect
17. #25573C2semidirect
18. #22208C2semidirect
19. #21117C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33642
C2×#33352
#391732
#364062
#332012
#306392
#305902
#302812
#298312
#286772
#286002
#265582
#263462
#258402
#257032
#256462
#255732
#222082
#211172