Finite Group Databasebeta

← #6606251

Subgroups — #6606251

Order 512.

2109 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3328C212yesmaximal
2D4×C4⋊Q8C211yesmaximal, characteristic
3C4×#556C211yesmaximal, characteristic
4#33163C211yesmaximal, characteristic, complemented
5#33114C212yesmaximal, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28283C211yesmaximal, characteristic, complemented
8#25276C212yesmaximal, complemented
9#25095C214yesmaximal, complemented
10#24760C212yesmaximal, complemented
11#24632C212yesmaximal, complemented
12#24496C214yesmaximal, complemented
13#24212C212yesmaximal, complemented
14#22650C211yesmaximal, characteristic
15#22430C212yesmaximal
16C4×#462C211yesmaximal, characteristic, complemented
17#21259C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×D4×Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21#3328—216no—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#475C2212yes—
25C2×#470C2214yes—
26C2×#462C2212yes—
27C2×#462C2211yescharacteristic
28C2×#462C2212yes—
29C2×#462C2212yes—
30C2×#451C2214yes—
31C2×C4×Q16C2212yes—
32C2×#436C2212yes—
33C2×#423C2212yes—
34C2×#419C2214yes—
35C2×#419C2214yes—
36#2868C2211yescharacteristic
37#2857C2212yes—
38#2551—21no—
39#2551—21no—
40#2546C2212yes—
41#2440—22no—
42#2440—22no—
43#2434C2214yes—
44#2373—22no—
45#2367—21no—
46#2365C2211yescharacteristic
47#2362C2212yes—
48#2356—21no—
49#2355C2211yescharacteristic
50#2354C2211yescharacteristic
51D4×C4⋊C4—21no—
52D4×C4⋊C4—21no—
53#2347C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#126—22no—
57C4×#124—21no—
58C4×C22⋊Q8C2212yes—
59C42×Q8—21no—
60C42×Q8C2211yescharacteristic
61C42×D4C2211yescharacteristic
62C4×#117C2212yes—
63#2066C2214yes—
64#2066C2214yes—
65#2056C2214yes—
66#2043C2214yes—
67#2043C2214yes—
68#2016C2214yes—
69#1981C2214yes—
70#1981C2211yescharacteristic
71#1981C2214yes—
72#1981C2212yes—
73#1981C2212yes—
74#1981C2212yes—
75#1981C2212yes—
76#1981C2212yes—
77#1981C2212yes—
78#1956C2212yes—
79#1948C2212yes—
80#1947C2211yescharacteristic
81#1939C2212yes—
82#1928C2214yes—
83#1928C2212yes—
84#1927C2214yes—
85#1894C2214yes—
86#1878C2212yes—
87#1873C2214yes—
88#1851C2214yes—
89#1842C2211yescharacteristic
90#1803C2212yes—
91#1803C2211yescharacteristic
92#1797C2212yes—
93#1787C2214yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C23×Q8C2314yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C22×M16C2314yes—
100#556—24no—
101#556D418yes—
102D4×Q8—28no—
103D4×Q8—28no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—28no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—44no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2311yescharacteristic
115C2×#126—22no—
116C2×#124—21no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8C2314yes—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—22no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117C2312yes—
132C2×#117C2312yes—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—22no—
135C22×C42C2312yes—
136#487—28no—
137#485—216no—
138#475—28no—
139#470—216no—
140#462—24no—
141#462—28no—
142#462—44no—
143#462—44no—
144#451—216no—
145C4×Q16—28no—
146#436—28no—
147C2×#107C2312yes—
148C2×#106C2314yes—
149#423—44no—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2312yes—
156#419—216no—
157#419—48no—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×C22⋊C8C2314yes—
172C2×C4×C8C2312yes—
173#398—24no—
174#395—24no—
175#395—24no—
176#391—21no—
177#391—21no—
178#391C2312yes—
179#391C2312yes—
180#391C2314yes—
181#390—21no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#385—22no—
201#385—24no—
202#385—24no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#383—22no—
210#382—21no—
211#380—22no—
212#380—22no—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2312yes—
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—
240#376—22no—
241C43—22no—
242C43—21no—
243C43C2311yescharacteristic
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2312yes—
249C22×Q8—21no—
250C22×Q8—24no—
251C22×Q8—28no—
252C22×Q8—28no—
253C22×Q8—28no—
254C22×Q8—44no—
255C22×Q8—44no—
256C22×Q8—44no—
257C22×Q8C2414yes—
258C22×Q8C2411yescharacteristic
259C22×Q8C2412yes—
260C22×D4C2×D411yescharacteristic
261C23×C4—24no—
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2412yes—
265C2×Q16—24no—
266C2×Q16—24no—
267C2×Q16—28no—
268C2×Q16—42no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×M16—28no—
272C2×M16—44no—
273C2×M16—44no—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C4⋊Q8—28no—
278C4⋊Q8—24no—
279C4⋊Q8—22no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—24no—
284C4⋊Q8—48no—
285C4⋊Q8—48no—
286C4⋊Q8—44no—
287C4⋊Q8—42no—
288C4⋊Q8—42no—
289C4⋊Q8—48no—
290C4⋊Q8C2×D414yes—
291C4⋊Q8#4114yes—
292#126—216no—
293#124—28no—
294C22⋊Q8—24no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—416no—
298C22⋊Q8C2×D418yes—
299C4×Q8—28no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×Q8—41no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8C2×D414yes—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—48no—
313C4×D4C2×D414yes—
314C4×D4#4114yes—
315#117—24no—
316#117—24no—
317#117—42no—
318#117C2×D418yes—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—48no—
343C2×C4⋊C4—48no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—24no—
386#106—28no—
387C4⋊C8—24no—
388C4⋊C8—22no—
389C4⋊C8—22no—
390C4⋊C8—24no—
391C4⋊C8—44no—
392C4⋊C8—42no—
393#103—24no—
394#103—24no—
395#103—24no—
396#103—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—24no—
401#103—28no—
402#103—44no—
403#103—44no—
404#103—44no—
405#103—44no—
406C22⋊C8—28no—
407C4×C8—24no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—21no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—24no—
422#95—24no—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426C24C22×D412yes—
427C2×Q8—24no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—48no—
445C2×Q8—416no—
446C2×Q8—416no—
447C2×Q8—48no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—84no—
451C2×Q8—88no—
452C2×Q8—88no—
453C2×Q8C22×D412yes—
454C2×Q8C2×#4114yes—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4#12114yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—42no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—84no—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D414yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4114yes—
491C22×C4C2511yescharacteristic
492Q16—48no—
493Q16—88no—
494M16—88no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4#12114yes—
538C4⋊C4#13714yes—
539C4⋊C4C22×D414yes—
540C4⋊C4C22×D412yes—
541C4⋊C4H(5,𝔽2)12yes—
542C22⋊C4—28no—
543C22⋊C4—28no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—82no—
549C22⋊C4#12118yes—
550C42—28no—
551C42—24no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—21no—
559C42—21no—
560C42—41no—
561C42—44no—
562C42—42no—
563C42—44no—
564C42—42no—
565C42—44no—
566C42—42no—
567C42—44no—
568C42—42no—
569C42—81no—
570C42—82no—
571C42#13714yes—
572C42C22×D412yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C42H(5,𝔽2)12yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23C2×#12114yes—
583C23C23×D411yescharacteristic
584Q8—22no—
585Q8—24no—
586Q8—24no—
587Q8—22no—
588Q8—41no—
589Q8—42no—
590Q8—41no—
591Q8—42no—
592Q8—48no—
593Q8—44no—
594Q8—44no—
595Q8—44no—
596Q8—44no—
597Q8—816no—
598D4—28no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—48no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—48no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—816no—
649C2×C4C2×#12114yes—
650C2×C4C2×#12112yes—
651C2×C4C2×#12112yes—
652C2×C4#54612yes—
653C2×C4#54612yes—
654C2×C4C2×#13714yes—
655C2×C4C2×#13712yes—
656C2×C4C2×#13712yes—
657C2×C4C2×#13712yes—
658C2×C4C2×#13712yes—
659C2×C4#57812yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4#58511yescharacteristic
666C8—42no—
667C8—42no—
668C8—44no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22C22×#12111yescharacteristic
677C22C2×#54611yescharacteristic
678C22#348811yescharacteristic
679C22C22×#13711yescharacteristic
680C22C22×#13711yescharacteristic
681C22C2×#57811yescharacteristic
682C22#358611yescharacteristic
683C4—22no—
684C4—22no—
685C4—24no—
686C4—22no—
687C4—24no—
688C4—42no—
689C4—41no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—82no—
696C4—82no—
697C4—84no—
698C4—81no—
699C4#303914yes—
700C4#332814yescomplemented
701C4D4×#4111yescharacteristic
702C4#348111yescharacteristic
703C4#349811yescharacteristic
704C4#351411yescharacteristic
705C2—44no—
706C2#3625811yescharacteristic
707C2C2×#303911yescharacteristic
708C2#6043211yescharacteristic
709C2C2×#332811yescharacteristic
710C2#6115711yescharacteristic
711C2#6196411yescharacteristic
712C2#6200011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
4. #33163C2semidirect
5. #33114C2semidirect
6. #28748C2semidirect
7. #28283C2semidirect
8. #25276C2semidirect
9. #25095C2semidirect
10. #24760C2semidirect
11. #24632C2semidirect
12. #24496C2semidirect
13. #24212C2semidirect
16. C4×#462C2semidirect
17. #21259C2semidirect
700. C4#3328semidirect

Maximal subgroups

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

subgroupindex
C2×#33282
D4×C4⋊Q82
C4×#5562
#331632
#331142
#287482
#282832
#252762
#250952
#247602
#246322
#244962
#242122
#226502
#224302
C4×#4622
#212592