Finite Group Databasebeta

← #6574538

Subgroups — #6574538

Order 512.

2093 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic, complemented
3C4×#547C211yesmaximal, characteristic
4#33197C211yesmaximal, characteristic, complemented
5#32958C212yesmaximal, complemented
6#28511C211yesmaximal, characteristic, complemented
7#28283C211yesmaximal, characteristic, complemented
8#26461C212yesmaximal, complemented
9#26349C214yesmaximal, complemented
10#26041C212yesmaximal, complemented
11#25956C214yesmaximal, complemented
12#24837C212yesmaximal, complemented
13#24303C212yesmaximal, complemented
14#22658C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#22417C212yesmaximal
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#547C2211yescharacteristic
19C2×#533C2212yes—
20C2×C4×#41C2212yes—
21#3076—216no—
22C2×#479C2214yes—
23C2×#477C2212yes—
24C2×#477C2212yes—
25C2×#475C2212yes—
26C2×#478C2214yes—
27C2×#476C2212yes—
28C2×#474C2211yescharacteristic
29C2×C8×D4C2212yes—
30C2×#427C2214yes—
31C22×C4⋊C8C2212yes—
32C2×#416C2214yes—
33#2868C2211yescharacteristic
34#2857C2212yes—
35#2552C2212yes—
36#2542—21no—
37#2538—21no—
38#2441C2214yes—
39#2429—22no—
40#2423—22no—
41#2365—21no—
42#2365C2211yescharacteristic
43#2362C2212yes—
44#2361—21no—
45#2359—22no—
46#2354C2212yes—
47#2354C2211yescharacteristic
48Q8×C4⋊C4—21no—
49D4×C4⋊C4C2211yescharacteristic
50#2344—21no—
51#2339C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C22⋊Q8C2212yes—
55C4×#123—22no—
56C4×#121—21no—
57C42×D4C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C2×C4×C4⋊C4—21no—
60#2095C2214yes—
61#2087C2214yes—
62#2068C2214yes—
63#2066C2214yes—
64#2059C2214yes—
65#2026C2214yes—
66#1988C2211yescharacteristic
67#1987C2212yes—
68#1981C2212yes—
69#1956C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1918C2212yes—
76#1892C2214yes—
77#1873C2214yes—
78#1865C2214yes—
79#1862C2212yes—
80#1862C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1850C2214yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1808C2214yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92#1788C2214yes—
93C4×#107C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#102C2212yes—
96C22×#41C2312yes—
97C23×C8C2314yes—
98#547—24no—
99#547Q818yescomplemented
100#533—28no—
101#533—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125C2314yes—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121—21no—
114C4×#41—28no—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C2×#117C2312yes—
132C2×#117C2314yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C42—22no—
136C22×C42—21no—
137C22×C42C2312yes—
138#479—216no—
139#478—216no—
140#477—28no—
141#477—44no—
142#476—28no—
143#475—28no—
144#474D418yescomplemented
145C8×D4—28no—
146#427—216no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8—28no—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2311yescharacteristic
160#416—216no—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×C22⋊C8C2314yes—
169C2×C4×C8C2312yes—
170#399—24no—
171#398—24no—
172#396—24no—
173#391—21no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#390—21no—
178#390—22no—
179#389—21no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—24no—
186#389C2312yes—
187#389C2312yes—
188#389C2314yes—
189#388—22no—
190#388—24no—
191#388—22no—
192#388—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—24no—
197#386—22no—
198#386—22no—
199#385—22no—
200#385—22no—
201#384—22no—
202#384—21no—
203#384—21no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#381—22no—
209#380—22no—
210#380—21no—
211#380C2312yes—
212#380C2314yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4C2312yes—
235C2×#95—22no—
236C43C2311yescharacteristic
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2312yes—
242#340C2312yes—
243#340C2312yes—
244#336C2314yes—
245#336C2312yes—
246C2×#41—24no—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—42no—
250C2×#41—42no—
251C2×#41C2×Q818yes—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8C2412yes—
255C22×D4—22no—
256C22×D4C2×D411yescharacteristic
257C22×D4C2411yescharacteristic
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—21no—
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2412yes—
266C22×C8—28no—
267C22×C8—28no—
268C22×C8—44no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—24no—
276C4⋊Q8—44no—
277C4⋊Q8—82no—
278C4⋊Q8C2×D414yes—
279C4⋊Q8C2×Q814yes—
280#125—28no—
281#125—44no—
282#123—216no—
283C22⋊Q8—24no—
284C22⋊Q8—48no—
285C22⋊Q8—48no—
286C22⋊Q8—84no—
287C22⋊Q8C2×Q818yes—
288#121—28no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—42no—
293C4×Q8—48no—
294C4×Q8—48no—
295C4×Q8—44no—
296C4×D4—24no—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—42no—
302C4×D4—41no—
303C4×D4—42no—
304C4×D4—48no—
305C4×D4—48no—
306C4×D4C2×D414yes—
307C4×D4C2×Q814yes—
308C4×D4C2×Q814yes—
309C4×D4#4114yes—
310#117—24no—
311#117—28no—
312#117—48no—
313#117—42no—
314#117—44no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—48no—
370C2×C42—44no—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D412yes—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—28no—
382#107—42no—
383#107—44no—
384#107C2×D418yes—
385#106—28no—
386#106—28no—
387C4⋊C8—28no—
388C4⋊C8—24no—
389C4⋊C8—24no—
390C4⋊C8—22no—
391C4⋊C8—42no—
392C4⋊C8C2×D414yes—
393#103—28no—
394#103—28no—
395#103—44no—
396#103—44no—
397#102—28no—
398#102—24no—
399#102C2×D418yes—
400C22⋊C8—28no—
401C4×C8—24no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418C24—21no—
419C24C22×D412yes—
420#41—216no—
421#41—48no—
422#41—48no—
423#41—48no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8C22×Q814yes—
436C2×D4—28no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4C22⋊Q814yes—
445C2×D4C22×D412yes—
446C2×D4C22×Q812yes—
447C2×D42−1+412yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4—82no—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×Q814yes—
491C22×C4C2×#4114yes—
492C22×C4C2511yescharacteristic
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—416no—
502C2×C8C22×D414yes—
503C4⋊C4—24no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—81no—
533C4⋊C4—84no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4C22⋊Q814yes—
537C4⋊C4C22×D414yes—
538C4⋊C4C22×D412yes—
539C4⋊C4C22×Q812yes—
540C22⋊C4—28no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—82no—
550C22⋊C4—84no—
551C22⋊C4C22⋊Q818yes—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—21no—
559C42—21no—
560C42—41no—
561C42—44no—
562C42—44no—
563C42—48no—
564C42—44no—
565C42—42no—
566C42—42no—
567C42—42no—
568C42—44no—
569C42#13514yes—
570C42C22×D412yes—
571C42C22×Q812yes—
572C42C2×#4112yes—
573C42H(5,𝔽2)12yes—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23C2×C22⋊Q814yes—
588C23C23×D411yescharacteristic
589Q8—24no—
590Q8—24no—
591Q8—42no—
592Q8—42no—
593Q8—48no—
594Q8—88no—
595D4—28no—
596D4—22no—
597D4—22no—
598D4—22no—
599D4—416no—
600D4D4×Q814yes—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
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—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—416no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—48no—
636C2×C4—48no—
637C2×C4—48no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—88no—
653C2×C4C2×C22⋊Q812yes—
654C2×C4C2×C22⋊Q814yes—
655C2×C4C2×C22⋊Q812yes—
656C2×C4D4212yes—
657C2×C4D4212yes—
658C2×C4C2×#13512yes—
659C2×C4C2×#13512yes—
660C2×C4C2×#13512yes—
661C2×C4#57512yes—
662C2×C4#57812yes—
663C2×C4C23×D411yescharacteristic
664C2×C4C23×Q811yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C2×C4#58511yescharacteristic
669C8—24no—
670C8—44no—
671C8—42no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22C22×C22⋊Q811yescharacteristic
687C22C2×D4211yescharacteristic
688C22#348911yescharacteristic
689C22C22×#13511yescharacteristic
690C22C2×#57511yescharacteristic
691C22C2×#57811yescharacteristic
692C22#358511yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—24no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—41no—
703C4—44no—
704C4—82no—
705C4—84no—
706C4—82no—
707C4—81no—
708C4#307614yescomplemented
709C4#328314yes—
710C4C2×D4×Q811yescharacteristic
711C4D4×#4111yescharacteristic
712C4#350111yescharacteristic
713C4#352011yescharacteristic
714C2—44no—
715C2—44no—
716C2D4×C22⋊Q811yescharacteristic
717C2C2×#307611yescharacteristic
718C2#6049811yescharacteristic
719C2C2×#328311yescharacteristic
720C2#6107011yescharacteristic
721C2#6209811yescharacteristic
722C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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×#3076C2semidirect
2. #36134C2semidirect
4. #33197C2semidirect
5. #32958C2semidirect
6. #28511C2semidirect
7. #28283C2semidirect
8. #26461C2semidirect
9. #26349C2semidirect
10. #26041C2semidirect
11. #25956C2semidirect
12. #24837C2semidirect
13. #24303C2semidirect
14. #22658C2semidirect
15. D4×C4⋊C8C2semidirect
17. C4×#474C2semidirect
99. #547Q8semidirect
144. #474D4semidirect
708. C4#3076semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
#361342
C4×#5472
#331972
#329582
#285112
#282832
#264612
#263492
#260412
#259562
#248372
#243032
#226582
D4×C4⋊C82
#224172
C4×#4742