Finite Group Databasebeta

← #6686287

Subgroups — #6686287

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3374C211yesmaximal, characteristic, complemented
2C2×#3356C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic
4#36562C211yesmaximal, characteristic
5#33196C211yesmaximal, characteristic, complemented
6#30652C214yesmaximal, complemented
7#30620C214yesmaximal, complemented
8#30280C211yesmaximal, characteristic, complemented
9#29830C211yesmaximal, characteristic, complemented
10#28684C211yesmaximal, characteristic, complemented
11#28606C211yesmaximal, characteristic
12#26574C214yesmaximal, complemented
13#26360C214yesmaximal, complemented
14#25836C211yesmaximal, characteristic, complemented
15#25702C211yesmaximal, characteristic
16#25647C211yesmaximal, characteristic, complemented
17#25570C211yesmaximal, characteristic
18#22214C211yesmaximal, characteristic
19#21119C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3374—28no—
25#3356—28no—
26C2×#498C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#491C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#483C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#470C2212yes—
38C2×#462C2211yescharacteristic
39C2×#462C2212yes—
40C2×#446C2211yescharacteristic
41C2×#441C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#428C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2859—21no—
48#2721—24no—
49#2658—21no—
50#2567C2211yescharacteristic
51#2566—21no—
52#2565—21no—
53#2556—24no—
54#2547C2211yescharacteristic
55#2546—21no—
56#2546C2211yescharacteristic
57#2545—21no—
58#2542C2211yescharacteristic
59#2441C2214yes—
60#2429C2214yes—
61#2423C2214yes—
62#2360C2214yes—
63#2327—21no—
64C4×#124—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2096C2214yes—
69#2096C2214yes—
70#2088C2214yes—
71#2068C2214yes—
72#2059C2214yes—
73#2056C2214yes—
74#2043C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1948C2214yes—
85#1943C2214yes—
86#1928C2212yes—
87#1924C2214yes—
88#1921C2212yes—
89#1894C2214yes—
90#1878C2212yes—
91#1873C2214yes—
92#1865C2214yes—
93#1865C2214yes—
94#1863C2212yes—
95#1853C2214yes—
96#1851C2214yes—
97#1842C2211yescharacteristic
98#1803C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2312yes—
104C22×Q16C2311yescharacteristic
105C22×M16C2312yes—
106#557—24no—
107#557—24no—
108#557—42no—
109#557—42no—
110#556—24no—
111#556—42no—
112#554—24no—
113#554—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×#126—22no—
124C2×#125—21no—
125C2×#125—21no—
126C2×#125—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#124—21no—
130C2×#124—21no—
131C2×#124D411yescharacteristic
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×C22⋊Q8C2314yes—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×D4—21no—
146C2×C4×D4C2311yescharacteristic
147C2×#117C2312yes—
148C22×C4⋊C4C2312yes—
149#498—24no—
150#498—42no—
151#491—28no—
152#487—24no—
153#487—42no—
154#485—28no—
155#483—28no—
156#479—28no—
157#477—24no—
158#475—24no—
159#475—24no—
160#470—28no—
161#462—28no—
162#462—24no—
163#446—24no—
164#441—28no—
165C4×Q16D418yes—
166#436—24no—
167#428—28no—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2314yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2314yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2314yes—
189C2×#103C2312yes—
190C2×#103C2311yescharacteristic
191C2×#103C2311yescharacteristic
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2314yes—
194C2×#97C2312yes—
195C2×C4×C8C2311yescharacteristic
196C2×C4×C8C2311yescharacteristic
197#399—24no—
198#399—24no—
199#399—22no—
200#398—22no—
201#398—24no—
202#398—24no—
203#397—22no—
204#396—24no—
205#396—24no—
206#395—22no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391D411yescharacteristic
212#391C2312yes—
213#391C2314yes—
214#390—21no—
215#390D411yescharacteristic
216#389—24no—
217#389—24no—
218#389—24no—
219#389C2314yes—
220#389C2312yes—
221#389C2314yes—
222#388—24no—
223#388—24no—
224#388D414yes—
225#387—22no—
226#387—24no—
227#387—24no—
228#387—24no—
229#385—24no—
230#384—21no—
231#384—21no—
232#384—22no—
233#384C2314yes—
234#382—21no—
235#382—21no—
236#382D411yescharacteristic
237#380C2314yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2311yescharacteristic
244C4×C22⋊C4—24no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2412yes—
263C22×Q8C2411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×D4C2×D411yescharacteristic
266C23×C4—22no—
267C23×C4C2412yes—
268C2×Q16—24no—
269C2×Q16—42no—
270C2×Q16—44no—
271C2×Q16C2×D414yes—
272C2×M16—24no—
273C2×M16—44no—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—42no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293#126—48no—
294#125—24no—
295#125—42no—
296#125—44no—
297#125—44no—
298#125—48no—
299#125—48no—
300#124—24no—
301#124—44no—
302#124—82no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—28no—
306C22⋊Q8—48no—
307C22⋊Q8—44no—
308C22⋊Q8—42no—
309C22⋊Q8—42no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8C2×D414yes—
324C4×Q8C2×D414yes—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—44no—
328#117—24no—
329#117—42no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
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—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D414yes—
372C2×C22⋊C4C2×D414yes—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2×D411yescharacteristic
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388#107—24no—
389#107—24no—
390#107C2×D414yes—
391#106—24no—
392#106—24no—
393#106—24no—
394#106—24no—
395#106—44no—
396C4⋊C8—24no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—44no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—24no—
407#103—42no—
408#103—44no—
409#103—44no—
410#103C2×D414yes—
411#103C2×D414yes—
412C22⋊C8—24no—
413C22⋊C8—44no—
414#97—24no—
415C4×C8—22no—
416C4×C8C2×D414yes—
417#95—22no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—22no—
422#95—21no—
423#95—22no—
424#95—21no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95C2×D414yes—
429#95C2×D414yes—
430C24C22×D412yes—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
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×Q8C22×D412yes—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—84no—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C2511yescharacteristic
481Q16—44no—
482Q16—48no—
483M16—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8C22×D412yes—
494C2×C8C22×D412yes—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—48no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4C22×D412yes—
533C4⋊C4C22×D412yes—
534C4⋊C42−1+414yes—
535C22⋊C4—24no—
536C22⋊C4—28no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—84no—
543C42—24no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—42no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—44no—
555C42—42no—
556C42—42no—
557C42—82no—
558C42#13714yes—
559C42C22×D412yes—
560C42H(5,𝔽2)12yes—
561C23—24no—
562C23—22no—
563C23—24no—
564C23—22no—
565C23—22no—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—44no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—44no—
577Q8—84no—
578Q8—84no—
579Q8—162no—
580D4—44no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—164no—
622C2×C4D4212yes—
623C2×C4D4212yes—
624C2×C4C2×#13714yes—
625C2×C4C2×#13712yes—
626C2×C4#57712yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+412yes—
630C2×C4#58512yes—
631C8—22no—
632C8—22no—
633C8—42no—
634C8—42no—
635C8—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22C2×D4211yescharacteristic
642C22#348912yes—
643C22C22×#13711yescharacteristic
644C22C2×#57711yescharacteristic
645C22#358612yes—
646C4—22no—
647C4—24no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4#327814yes—
663C4#344911yescharacteristic
664C4#349911yescharacteristic
665C4#355412yes—
666C2—82no—
667C2#3768011yescharacteristic
668C2C2×#327811yescharacteristic
669C2#6108311yescharacteristic
670C2#6211312yes—
671C2#6218212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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×#3374C2semidirect
5. #33196C2semidirect
6. #30652C2semidirect
7. #30620C2semidirect
8. #30280C2semidirect
9. #29830C2semidirect
10. #28684C2semidirect
12. #26574C2semidirect
13. #26360C2semidirect
14. #25836C2semidirect
16. #25647C2semidirect
19. #21119C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33742
C2×#33562
#397442
#365622
#331962
#306522
#306202
#302802
#298302
#286842
#286062
#265742
#263602
#258362
#257022
#256472
#255702
#222142
#211192