Finite Group Databasebeta

← #6577854

Subgroups — #6577854

Order 512.

2069 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic
3C4×D4×Q8C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic, complemented
5#32957C212yesmaximal, complemented
6#28511C211yesmaximal, characteristic, complemented
7#28255C211yesmaximal, characteristic, complemented
8#26559C214yesmaximal, complemented
9#25016C212yesmaximal
10#24906C214yesmaximal, complemented
11#24702C212yesmaximal, complemented
12#24243C212yesmaximal
13#24155C212yesmaximal, complemented
14#22998C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic
16C4⋊C4×Q16C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#533C2212yes—
20C22×C4×Q8C2212yes—
21#3104—216no—
22C2×#484C2214yes—
23C2×#477C2212yes—
24C2×#477C2212yes—
25C2×#477C2212yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#462C2211yescharacteristic
29C2×#451C2214yes—
30C2×C4×Q16C2214yes—
31C2×C4×Q16C2214yes—
32C2×C4×Q16C2212yes—
33C2×#431C2212yes—
34#2868C2211yescharacteristic
35#2857C2212yes—
36#2552—21no—
37#2542—21no—
38#2538C2212yes—
39#2441—22no—
40#2429—22no—
41#2423C2214yes—
42#2365—21no—
43#2365—21no—
44#2365C2211yescharacteristic
45#2360—22no—
46#2360C2212yes—
47#2354—21no—
48Q8×C4⋊C4C2212yes—
49Q8×C4⋊C4C2211yescharacteristic
50D4×C4⋊C4—21no—
51D4×C4⋊C4C2211yescharacteristic
52Q8×C22⋊C4C2212yes—
53C4×C4⋊Q8—21no—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C22⋊Q8—22no—
57C4×C22⋊Q8C2212yes—
58C42×Q8C2211yescharacteristic
59C42×D4—21no—
60C42×D4C2211yescharacteristic
61#2095C2214yes—
62#2076C2214yes—
63#2056C2214yes—
64#2036C2214yes—
65#2036C2214yes—
66#2001C2214yes—
67#1987C2212yes—
68#1981C2211yescharacteristic
69#1956C2212yes—
70#1954C2212yes—
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1942C2214yes—
75#1933C2214yes—
76#1928C2212yes—
77#1923C2214yes—
78#1878C2212yes—
79#1878C2214yes—
80#1863C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1852C2214yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2211yescharacteristic
91#1797C2212yes—
92#1787C2214yes—
93C8×C4⋊C4C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2312yes—
99D4×Q8—24no—
100D4×Q8—24no—
101#533—28no—
102#533—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#125—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2314yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—28no—
119C2×C4×Q8—44no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2314yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4D411yescharacteristic
131C2×C4×D4D411yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×#117—22no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2312yes—
136C22×C42—22no—
137C22×C42C2312yes—
138#484—216no—
139#477—28no—
140#477—28no—
141#477—44no—
142#462—28no—
143#462—28no—
144#462D418yescomplemented
145#451—216no—
146C4×Q16—216no—
147C4×Q16—28no—
148C4×Q16—216no—
149#431—28no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#107C2312yes—
153C2×#107C2314yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2311yescharacteristic
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×C22⋊C8C2314yes—
169C2×C22⋊C8C2314yes—
170C2×C4×C8C2312yes—
171C2×C4×C8C2314yes—
172#399—24no—
173#398—24no—
174#396—24no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#390—22no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—24no—
193#389C2314yes—
194#389C2312yes—
195#388—24no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—24no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—21no—
204#384—22no—
205#384—21no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2312yes—
210#384C2312yes—
211#380—22no—
212#380—22no—
213#380C2312yes—
214#380C2314yes—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4D412yes—
236C4×C22⋊C4D412yes—
237C4×C22⋊C4C2312yes—
238C43—21no—
239C43C2311yescharacteristic
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#340C2314yes—
244#340C2312yes—
245#336C2314yes—
246#336C2312yes—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—42no—
254C22×Q8—42no—
255C22×Q8C2412yes—
256C22×Q8C2412yes—
257C22×Q8C2411yescharacteristic
258C22×D4C2×D411yescharacteristic
259C23×C4—22no—
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2412yes—
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16C2×D418yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—28no—
274C4⋊Q8—24no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—82no—
280C4⋊Q8C2×D414yes—
281C4⋊Q8#4114yes—
282#125—48no—
283C22⋊Q8—216no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—28no—
287C22⋊Q8—44no—
288C22⋊Q8—48no—
289C22⋊Q8—84no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—24no—
293C4×Q8—28no—
294C4×Q8—22no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—48no—
301C4×Q8—42no—
302C4×Q8—42no—
303C4×Q8—41no—
304C4×Q8—44no—
305C4×Q8—48no—
306C4×Q8—48no—
307C4×Q8C2×D414yes—
308C4×Q8#4114yes—
309C4×D4—28no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—42no—
313C4×D4—44no—
314C4×D4Q1618yescomplemented
315#117—48no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—42no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—48no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—28no—
387#107—24no—
388#107—44no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391C4⋊C8—22no—
392C4⋊C8—42no—
393C4⋊C8C2×D414yes—
394#103—28no—
395#103—28no—
396#103—24no—
397#103—28no—
398#103—28no—
399#103—24no—
400#103—42no—
401#103—44no—
402#103C2×D418yes—
403C22⋊C8—28no—
404C22⋊C8—44no—
405C4×C8—28no—
406C4×C8—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—24no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—24no—
420#95C2×D412yes—
421#95C2×D412yes—
422#95C2×D412yes—
423C24C22×D412yes—
424C2×Q8—28no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—48no—
444C2×Q8—48no—
445C2×Q8—48no—
446C2×Q8—44no—
447C2×Q8C22×D414yes—
448C2×Q8C22×D412yes—
449C2×Q8C2×#4114yes—
450C2×Q8C2×#4112yes—
451C2×Q82−1+412yes—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4C2×Q1614yes—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4C2×Q1618yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2511yescharacteristic
492Q16—216no—
493Q16—416no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8C22×D414yes—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—81no—
537C4⋊C4—82no—
538C4⋊C4—84no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4C2×Q1614yes—
542C4⋊C4C22×D412yes—
543C4⋊C4C2×#4112yes—
544C22⋊C4—28no—
545C22⋊C4—24no—
546C22⋊C4—42no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—84no—
550C22⋊C4C2×Q1618yes—
551C42—24no—
552C42—24no—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—41no—
562C42—44no—
563C42—44no—
564C42—48no—
565C42—42no—
566C42—44no—
567C42—44no—
568C42—42no—
569C42—42no—
570C42—81no—
571C42C2×Q1614yes—
572C42C22×D412yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C42H(5,𝔽2)12yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23C22×Q1614yes—
583C23C23×D411yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8—24no—
588Q8—24no—
589Q8—24no—
590Q8—24no—
591Q8—416no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—88no—
597Q8#54714yes—
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—22no—
610C2×C4—216no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—416no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—84no—
650C2×C4—88no—
651C2×C4#52512yes—
652C2×C4#52512yes—
653C2×C4D4212yes—
654C2×C4D4212yes—
655C2×C4C22×Q1612yes—
656C2×C4C22×Q1614yes—
657C2×C4C22×Q1612yes—
658C2×C4C22×Q1612yes—
659C2×C4#57512yes—
660C2×C4#57812yes—
661C2×C4C23×D411yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C2×C4#58511yescharacteristic
667C8—24no—
668C8—44no—
669C8—42no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22C2×#52511yescharacteristic
678C22C2×D4211yescharacteristic
679C22#348911yescharacteristic
680C22C23×Q1611yescharacteristic
681C22C2×#57511yescharacteristic
682C22C2×#57811yescharacteristic
683C22#358511yescharacteristic
684C4—24no—
685C4—24no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—41no—
696C4—82no—
697C4—84no—
698C4—82no—
699C4—81no—
700C4#310414yescomplemented
701C4D4×Q1614yes—
702C4C2×#54711yescharacteristic
703C4D4×#4111yescharacteristic
704C4#350611yescharacteristic
705C4#352011yescharacteristic
706C2—44no—
707C2#3610011yescharacteristic
708C2C2×#310411yescharacteristic
709C2#6060011yescharacteristic
710C2C2×D4×Q1611yescharacteristic
711C2#6107011yescharacteristic
712C2#6210011yescharacteristic
713C2#6214611yescharacteristic
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
1. C2×#3104C2semidirect
4. #33198C2semidirect
5. #32957C2semidirect
6. #28511C2semidirect
7. #28255C2semidirect
8. #26559C2semidirect
10. #24906C2semidirect
11. #24702C2semidirect
13. #24155C2semidirect
14. #22998C2semidirect
16. C4⋊C4×Q16C2semidirect
17. C4×#462C2semidirect
144. #462D4semidirect
314. C4×D4Q16semidirect
700. C4#3104semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
#361342
C4×D4×Q82
#331982
#329572
#285112
#282552
#265592
#250162
#249062
#247022
#242432
#241552
#229982
#227952
C4⋊C4×Q162
C4×#4622