Finite Group Databasebeta

← #6578237

Subgroups — #6578237

Order 512.

2175 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3116C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic, complemented
3C4×#550C211yesmaximal, characteristic
4#33163C211yesmaximal, characteristic, complemented
5#32907C212yesmaximal, complemented
6#28438C211yesmaximal, characteristic, complemented
7#28280C211yesmaximal, characteristic, complemented
8#26642C214yesmaximal, complemented
9#26231C212yesmaximal, complemented
10#25755C212yesmaximal
11#25171C214yesmaximal, complemented
12#24659C212yesmaximal, complemented
13#24151C212yesmaximal, complemented
14#23190C212yesmaximal, complemented
15#23004C212yesmaximal, complemented
16#22799C211yesmaximal, characteristic, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#550C2211yescharacteristic
19C2×#533C2212yes—
20C2×C4×#41C2212yes—
21C2×#492C2212yes—
22#3116—216no—
23C2×#485C2214yes—
24C2×#477C2211yescharacteristic
25C2×#476C2212yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#462C2212yes—
29C2×#461C2212yes—
30C2×#461C2212yes—
31C2×#452C2214yes—
32C2×#452C2214yes—
33C2×#441C2214yes—
34C2×#440C2214yes—
35C2×#432C2212yes—
36#2868C2211yescharacteristic
37#2857C2212yes—
38#2552—21no—
39#2542C2212yes—
40#2538—21no—
41#2441—22no—
42#2429C2214yes—
43#2423—22no—
44#2366—21no—
45#2366C2211yescharacteristic
46#2365C2212yes—
47#2363—22no—
48#2363C2212yes—
49#2354—21no—
50#2354—21no—
51#2354C2211yescharacteristic
52Q8×C4⋊C4—21no—
53Q8×C4⋊C4C2211yescharacteristic
54#2341C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127—21no—
57C4×#127C2211yescharacteristic
58C4×#121—22no—
59C4×#121C2212yes—
60C42×Q8C2211yescharacteristic
61C42×D4—21no—
62C42×D4C2211yescharacteristic
63#2065C2214yes—
64#2036C2214yes—
65#2035C2214yes—
66#2013C2214yes—
67#2000C2214yes—
68#2000C2214yes—
69#1987C2211yescharacteristic
70#1982C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1973C2214yes—
75#1956C2212yes—
76#1954C2212yes—
77#1948C2212yes—
78#1947C2211yescharacteristic
79#1935C2214yes—
80#1924C2214yes—
81#1921C2212yes—
82#1880C2214yes—
83#1878C2214yes—
84#1877C2214yes—
85#1873C2214yes—
86#1865C2214yes—
87#1863C2212yes—
88#1842C2211yescharacteristic
89#1803C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93#1770C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C22×#41C2312yes—
97C22×Q16C2314yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100#550—24no—
101#550—24no—
102#533—28no—
103#533—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127—21no—
109C2×#127C2311yescharacteristic
110C2×#125—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C2×#121—22no—
115C2×#121C2312yes—
116C4×#41—28no—
117C4×#41—44no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2314yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4D411yescharacteristic
133C2×C4×D4D411yescharacteristic
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2311yescharacteristic
137C2×#117—22no—
138C2×#117—22no—
139C2×#117C2312yes—
140C22×C42—22no—
141C22×C42C2312yes—
142#492—28no—
143#485—216no—
144#477D418yescomplemented
145#476—28no—
146#462—28no—
147#462—28no—
148#462—44no—
149#461—28no—
150#461—44no—
151#452—216no—
152#452—48no—
153#441—216no—
154#440—216no—
155#432—28no—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106C2314yes—
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2311yescharacteristic
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2312yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×C22⋊C8C2314yes—
174C2×C22⋊C8C2314yes—
175C2×#97C2314yes—
176C2×C4×C8C2312yes—
177#399—24no—
178#398—24no—
179#396—24no—
180#391—21no—
181#391C2314yes—
182#390—21no—
183#390—22no—
184#390—21no—
185#390—22no—
186#390—22no—
187#390—22no—
188#390—21no—
189#390—21no—
190#390C2312yes—
191#390C2312yes—
192#389—22no—
193#389—21no—
194#389—22no—
195#389—22no—
196#389—24no—
197#389—22no—
198#389C2312yes—
199#389C2312yes—
200#388—24no—
201#388—22no—
202#388—22no—
203#388—22no—
204#387—24no—
205#386—22no—
206#386—22no—
207#386—22no—
208#384—21no—
209#384—22no—
210#384—21no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#380—22no—
217#380—22no—
218#380—22no—
219#380C2312yes—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4D412yes—
238C4×C22⋊C4D412yes—
239C4×C22⋊C4C2312yes—
240C43—21no—
241C43C2311yescharacteristic
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2312yes—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—24no—
253C2×#41—42no—
254C2×#41—42no—
255C22×Q8—21no—
256C22×Q8C2414yes—
257C22×Q8C2411yescharacteristic
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4C2×D411yescharacteristic
263C22×D4C2412yes—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4C2×D412yes—
268C23×C4C2×D412yes—
269C23×C4C2412yes—
270C2×Q16—28no—
271C2×Q16—28no—
272C2×Q16—28no—
273C2×SD16—28no—
274C2×SD16—28no—
275C2×SD16—28no—
276C2×SD16—44no—
277C2×SD16—44no—
278C2×SD16—44no—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C22×C8C2412yes—
282C4⋊Q8—24no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—22no—
286C4⋊Q8—24no—
287C4⋊Q8—42no—
288C4⋊Q8—42no—
289C4⋊Q8C2×D414yes—
290#127—28no—
291#127—22no—
292#127#4114yes—
293#125—48no—
294C22⋊Q8—28no—
295C22⋊Q8—44no—
296C22⋊Q8—48no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299#121—216no—
300#121—24no—
301#121—24no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—24no—
305C4×Q8—22no—
306C4×Q8—28no—
307C4×Q8—41no—
308C4×Q8—42no—
309C4×Q8—44no—
310C4×Q8—48no—
311C4×Q8—44no—
312C4×Q8C2×D414yes—
313C4×Q8#4114yes—
314C4×D4—28no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—24no—
321C4×D4—42no—
322C4×D4—42no—
323C4×D4—48no—
324C4×D4—48no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—82no—
328C4×D4—84no—
329#117—24no—
330#117—48no—
331#117—42no—
332#117—48no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
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—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—24no—
379C2×C42—24no—
380C2×C42—22no—
381C2×C42—24no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42—22no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—48no—
389C2×C42C2×D412yes—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D412yes—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2412yes—
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—42no—
400#107C2×D418yes—
401#106—28no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—22no—
405C4⋊C8—42no—
406C4⋊C8C2×D414yes—
407#103—24no—
408#103—28no—
409#103—28no—
410#103—28no—
411#103—44no—
412#103—44no—
413#103C2×D418yes—
414#102—28no—
415#102—44no—
416C22⋊C8—28no—
417C22⋊C8—44no—
418#97—28no—
419C4×C8—24no—
420#95—22no—
421#95—22no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—24no—
426#95—22no—
427#95—22no—
428#95—24no—
429#95C2×D412yes—
430#95C2×D412yes—
431#95C2×D412yes—
432C24—22no—
433C24C22×D412yes—
434#41—48no—
435#41—48no—
436#41—48no—
437#41—48no—
438C2×Q8—24no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8C22×D412yes—
451C2×Q8C2×#4112yes—
452C2×Q8H(5,𝔽2)12yes—
453C2×D4—28no—
454C2×D4—28no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—81no—
470C2×D4—81no—
471C2×D4—82no—
472C2×D4—82no—
473C2×D4C2×#4114yes—
474C22×C4—21no—
475C22×C4—28no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—82no—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C2511yescharacteristic
513Q16—416no—
514SD16—416no—
515SD16—88no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—24no—
519C2×C8—22no—
520C2×C8—24no—
521C2×C8—22no—
522C2×C8—24no—
523C2×C8—22no—
524C2×C8C22×D414yes—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—22no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—48no—
543C4⋊C4—48no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—81no—
553C4⋊C4—82no—
554C4⋊C4—84no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4C22×D414yes—
558C4⋊C4C22×D412yes—
559C4⋊C4C2×#4112yes—
560C22⋊C4—28no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—42no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—48no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—84no—
574C42—24no—
575C42—24no—
576C42—24no—
577C42—24no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—21no—
582C42—22no—
583C42—21no—
584C42—41no—
585C42—44no—
586C42—44no—
587C42—48no—
588C42—42no—
589C42—42no—
590C42—44no—
591C42—42no—
592C42—81no—
593C42#13714yes—
594C42C22×D412yes—
595C42C2×#4112yes—
596C42C2×#4112yes—
597C42H(5,𝔽2)12yes—
598C23—22no—
599C23—22no—
600C23—24no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23C23×D411yescharacteristic
613Q8—22no—
614Q8—22no—
615Q8—22no—
616Q8—48no—
617Q8—44no—
618Q8—44no—
619Q8—44no—
620Q8—44no—
621Q8#55014yes—
622D4—24no—
623D4—24no—
624D4—416no—
625D4—44no—
626D4—48no—
627D4—42no—
628D4—42no—
629D4—84no—
630D4—88no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—21no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—416no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—48no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—84no—
680C2×C4—88no—
681C2×C4—88no—
682C2×C4#52512yes—
683C2×C4#52512yes—
684C2×C4D4212yes—
685C2×C4D4212yes—
686C2×C4C2×#13712yes—
687C2×C4C2×#13712yes—
688C2×C4C2×#13712yes—
689C2×C4#57512yes—
690C2×C4#57712yes—
691C2×C4C23×D411yescharacteristic
692C2×C4C22×#4111yescharacteristic
693C2×C4C22×#4111yescharacteristic
694C2×C4C2×H(5,𝔽2)11yescharacteristic
695C2×C4C2×H(5,𝔽2)11yescharacteristic
696C2×C4#58511yescharacteristic
697C8—24no—
698C8—44no—
699C8—42no—
700C22—44no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—44no—
707C22—42no—
708C22—42no—
709C22—44no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22C2×#52511yescharacteristic
715C22C2×D4211yescharacteristic
716C22#348811yescharacteristic
717C22C22×#13711yescharacteristic
718C22C2×#57511yescharacteristic
719C22C2×#57711yescharacteristic
720C22#358611yescharacteristic
721C4—24no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—42no—
726C4—41no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—44no—
731C4—42no—
732C4—84no—
733C4—81no—
734C4—82no—
735C4—82no—
736C4#311614yescomplemented
737C4#327814yes—
738C4C2×#55011yescharacteristic
739C4D4×#4111yescharacteristic
740C4#350611yescharacteristic
741C4#351911yescharacteristic
742C2—82no—
743C2—84no—
744C2#3610111yescharacteristic
745C2C2×#311611yescharacteristic
746C2#6059111yescharacteristic
747C2C2×#327811yescharacteristic
748C2#6106711yescharacteristic
749C2#6196411yescharacteristic
750C2#6198711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3116C2semidirect
2. #36134C2semidirect
4. #33163C2semidirect
5. #32907C2semidirect
6. #28438C2semidirect
7. #28280C2semidirect
8. #26642C2semidirect
9. #26231C2semidirect
11. #25171C2semidirect
12. #24659C2semidirect
13. #24151C2semidirect
14. #23190C2semidirect
15. #23004C2semidirect
16. #22799C2semidirect
17. C4×#477C2semidirect
144. #477D4semidirect
736. C4#3116semidirect

Maximal subgroups

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

subgroupindex
C2×#31162
#361342
C4×#5502
#331632
#329072
#284382
#282802
#266422
#262312
#257552
#251712
#246592
#241512
#231902
#230042
#227992
C4×#4772