Finite Group Databasebeta

← #6591600

Subgroups — #6591600

Order 512.

2183 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3303C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic, complemented
3C4×#547C211yesmaximal, characteristic, complemented
4#33197C211yesmaximal, characteristic, complemented
5#32904C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28252C211yesmaximal, characteristic, complemented
8#27649C212yesmaximal, complemented
9#25319C212yesmaximal, complemented
10#25213C214yesmaximal, complemented
11#24812C212yesmaximal, complemented
12#24753C212yesmaximal, complemented
13#24537C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic, complemented
15#22489C212yesmaximal, complemented
16C4×#459C211yesmaximal, characteristic, complemented
17#20645C212yesmaximal, complemented
18C2×#555C2212yes—
19C2×#547C2211yescharacteristic
20C2×C4×#41C2212yes—
21#3303—216no—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#476C2212yes—
25C2×#465C2214yes—
26C2×#460C2212yes—
27C2×#461C2212yes—
28C2×#459C2211yescharacteristic
29C2×#449C2214yes—
30C2×C4×D8C2212yes—
31C2×C8×D4C2212yes—
32C22×C4⋊C8C2212yes—
33C2×#416C2214yes—
34#2868C2211yescharacteristic
35#2856C2212yes—
36#2563C2212yes—
37#2537—21no—
38#2537—21no—
39#2500C2214yes—
40#2491—22no—
41#2491—22no—
42#2365C2211yescharacteristic
43#2362C2212yes—
44#2361—21no—
45#2359—22no—
46#2354—21no—
47#2354—21no—
48#2354C2211yescharacteristic
49D4×C4⋊C4C2211yescharacteristic
50#2344—21no—
51#2339C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127C2211yescharacteristic
54C4×C22⋊Q8C2212yes—
55C4×#123—22no—
56C4×#121—21no—
57#2285C2212yes—
58C42×D4C2211yescharacteristic
59C42×D4C2211yescharacteristic
60C2×C4×C4⋊C4—21no—
61#2085C2214yes—
62#2074C2214yes—
63#2072C2214yes—
64#2065C2214yes—
65#2055C2214yes—
66#2055C2214yes—
67#1990C2212yes—
68#1987C2212yes—
69#1982C2211yescharacteristic
70#1948C2212yes—
71#1948C2212yes—
72#1947C2211yescharacteristic
73#1939C2212yes—
74#1925C2212yes—
75#1893C2214yes—
76#1880C2214yes—
77#1879C2214yes—
78#1879C2212yes—
79#1875C2214yes—
80#1850C2214yes—
81#1842C2211yescharacteristic
82#1825C2214yes—
83#1805C2214yes—
84#1804C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#1735C2214yes—
96C22×#41C2312yes—
97C22×SD16C2314yes—
98C22×D8C2312yes—
99C23×C8C2314yes—
100#555—28no—
101#555—28no—
102#547—24no—
103#547D418yescomplemented
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#124—24no—
110C2×#123—22no—
111C2×C22⋊Q8C2312yes—
112C2×#121—21no—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C4×#41—28no—
118C4×#41—44no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×#117—22no—
134C2×#117—22no—
135C2×#117C2314yes—
136C2×#117C2312yes—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C42—22no—
140C22×C42—21no—
141C22×C42C2312yes—
142#487—28no—
143#485—216no—
144#476—28no—
145#465—216no—
146#461—28no—
147#460—44no—
148#459—24no—
149#449—216no—
150C4×D8—28no—
151C8×D4—28no—
152C2×#107C2312yes—
153C2×#106C2314yes—
154C2×C4⋊C8—44no—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2314yes—
161#416—216no—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×C22⋊C8C2314yes—
172C2×C4×C8C2312yes—
173#394—24no—
174#394—24no—
175#393—24no—
176#391C2312yes—
177#391C2312yes—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—22no—
182#390—22no—
183#390C2312yes—
184#390C2314yes—
185#389—21no—
186#389—21no—
187#389—21no—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—24no—
193#388—24no—
194#387—22no—
195#387—22no—
196#387—22no—
197#386—22no—
198#386—22no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2312yes—
207#382C2314yes—
208#381—22no—
209#380—22no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—21no—
214#380C2312yes—
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⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—22no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4D412yes—
235C4×C22⋊C4C2312yes—
236C2×#95—22no—
237C43C2311yescharacteristic
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#336C2314yes—
246#336C2312yes—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41—42no—
251C2×#41—42no—
252C2×#41C2×D418yes—
253C22×Q8—22no—
254C22×Q8C2412yes—
255C22×D4—22no—
256C22×D4—24no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4C2×D411yescharacteristic
260C22×D4C2412yes—
261C22×D4C2411yescharacteristic
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2412yes—
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C2×D8—24no—
275C2×D8—24no—
276C2×D8—24no—
277C22×C8—28no—
278C22×C8—44no—
279C22×C8—44no—
280C22×C8C2412yes—
281C22×C8C2414yes—
282C22×C8C2412yes—
283C4⋊Q8—24no—
284C4⋊Q8—22no—
285C4⋊Q8—24no—
286C4⋊Q8—82no—
287C4⋊Q8—82no—
288C4⋊Q8C2×D414yes—
289#127—22no—
290#127—22no—
291#124—416no—
292#123—216no—
293C22⋊Q8—24no—
294C22⋊Q8C2×D418yes—
295#121—28no—
296#121—28no—
297#121—28no—
298#121—48no—
299#121—84no—
300#121—84no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—42no—
305C4×D4—24no—
306C4×D4—22no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—22no—
310C4×D4—24no—
311C4×D4—22no—
312C4×D4—42no—
313C4×D4—41no—
314C4×D4—42no—
315C4×D4—44no—
316C4×D4—48no—
317C4×D4—48no—
318C4×D4—44no—
319C4×D4C2×D414yes—
320C4×D4C2×D414yes—
321C4×D4#4114yes—
322C4×D4#4114yes—
323#117—28no—
324#117—28no—
325#117—24no—
326#117—48no—
327#117—42no—
328#117—48no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—48no—
382C2×C42—44no—
383C2×C42C2×D412yes—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2412yes—
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—24no—
394#106—28no—
395C4⋊C8—24no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398C4⋊C8—24no—
399C4⋊C8—44no—
400C4⋊C8—42no—
401#103—28no—
402#103—28no—
403#103—28no—
404#102—24no—
405#102—28no—
406#102—24no—
407#102—24no—
408#102—42no—
409#102—44no—
410C22⋊C8—28no—
411C4×C8—24no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—22no—
419#95—22no—
420#95—24no—
421#95—24no—
422#95—22no—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426C24—21no—
427C24—22no—
428C24C22×D412yes—
429#41—216no—
430#41—48no—
431#41—48no—
432#41—48no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8C22×D414yes—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—48no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—84no—
464C2×D4—84no—
465C2×D4#12114yes—
466C2×D4C22×D412yes—
467C2×D4C2×#4112yes—
468C2×D42−1+412yes—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—81no—
503C22×C4—82no—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D414yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C2×#4114yes—
515C22×C4C2511yescharacteristic
516SD16—88no—
517D8—48no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—24no—
521C2×C8—22no—
522C2×C8—24no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—24no—
527C2×C8—88no—
528C4⋊C4—24no—
529C4⋊C4—28no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—81no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4#12114yes—
556C4⋊C4C22×D412yes—
557C4⋊C4C2×#4112yes—
558C22⋊C4—28no—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—42no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—42no—
566C22⋊C4—44no—
567C22⋊C4—48no—
568C22⋊C4—48no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—84no—
572C22⋊C4—84no—
573C22⋊C4—82no—
574C22⋊C4#12118yes—
575C42—24no—
576C42—24no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—21no—
582C42—21no—
583C42—41no—
584C42—44no—
585C42—44no—
586C42—48no—
587C42—44no—
588C42—42no—
589C42—42no—
590C42—42no—
591C42#13514yes—
592C42C22×D412yes—
593C42C2×#4112yes—
594C42C2×#4112yes—
595C42H(5,𝔽2)12yes—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23C2×#12114yes—
617C23C23×D411yescharacteristic
618Q8—24no—
619Q8—24no—
620Q8—42no—
621Q8—42no—
622Q8—164no—
623D4—22no—
624D4—22no—
625D4—22no—
626D4—28no—
627D4—416no—
628D4—42no—
629D4—42no—
630D4—42no—
631D4—42no—
632D4—84no—
633D4—84no—
634D4—168no—
635D4#54714yes—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—22no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—416no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—48no—
670C2×C4—41no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4—88no—
685C2×C4—88no—
686C2×C4C2×#12114yes—
687C2×C4C2×#12112yes—
688C2×C4C2×#12112yes—
689C2×C4#54612yes—
690C2×C4#54612yes—
691C2×C4C2×#13512yes—
692C2×C4C2×#13512yes—
693C2×C4C2×#13512yes—
694C2×C4#57512yes—
695C2×C4#57712yes—
696C2×C4C23×D411yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C2×C4C22×#4111yescharacteristic
699C2×C4C2×H(5,𝔽2)11yescharacteristic
700C2×C4C2×2−1+411yescharacteristic
701C2×C4#58511yescharacteristic
702C8—42no—
703C8—42no—
704C8—44no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—24no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22C22×#12111yescharacteristic
727C22C2×#54611yescharacteristic
728C22#348911yescharacteristic
729C22C22×#13511yescharacteristic
730C22C2×#57511yescharacteristic
731C22C2×#57711yescharacteristic
732C22#358511yescharacteristic
733C4—22no—
734C4—24no—
735C4—22no—
736C4—22no—
737C4—44no—
738C4—41no—
739C4—42no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4—84no—
744C4—81no—
745C4—82no—
746C4—82no—
747C4#303514yes—
748C4#330314yescomplemented
749C4C2×#54711yescharacteristic
750C4#347511yescharacteristic
751C4#349611yescharacteristic
752C4#352011yescharacteristic
753C2—44no—
754C2—44no—
755C2—162no—
756C2#3615311yescharacteristic
757C2C2×#303511yescharacteristic
758C2#6043111yescharacteristic
759C2C2×#330311yescharacteristic
760C2#6116611yescharacteristic
761C2#6209811yescharacteristic
762C2#6213411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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×#3303C2semidirect
2. #36187C2semidirect
3. C4×#547C2semidirect
4. #33197C2semidirect
5. #32904C2semidirect
6. #28717C2semidirect
7. #28252C2semidirect
8. #27649C2semidirect
9. #25319C2semidirect
10. #25213C2semidirect
11. #24812C2semidirect
12. #24753C2semidirect
13. #24537C2semidirect
14. D4×C4⋊C8C2semidirect
15. #22489C2semidirect
16. C4×#459C2semidirect
17. #20645C2semidirect
103. #547D4semidirect
748. C4#3303semidirect

Maximal subgroups

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

subgroupindex
C2×#33032
#361872
C4×#5472
#331972
#329042
#287172
#282522
#276492
#253192
#252132
#248122
#247532
#245372
D4×C4⋊C82
#224892
C4×#4592
#206452