Finite Group Databasebeta

← #6566083

Subgroups — #6566083

Order 512.

2161 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3335C212yesmaximal, complemented
2#36179C211yesmaximal, characteristic
3#34332C211yesmaximal, characteristic, complemented
4#29866C211yesmaximal, characteristic, complemented
5#29755C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28574C212yesmaximal, complemented
8#28202C211yesmaximal, characteristic, complemented
9#27647C212yesmaximal, complemented
10#25451C212yesmaximal, complemented
11#25204C214yesmaximal, complemented
12#24813C212yesmaximal, complemented
13#24578C214yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15C2×#2084C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22485C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#547—21no—
20C2×#547—21no—
21C2×#547C2212yes—
22#3335—216no—
23C2×#498C2212yes—
24C2×#481C2214yes—
25C2×#483C2214yes—
26C2×#475C2212yes—
27C2×#476C2212yes—
28C2×#465C2214yes—
29C2×C4×SD16C2212yes—
30C2×C8×D4C2212yes—
31C22×#106C2214yes—
32#2868C2211yescharacteristic
33#2856C2212yes—
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2501—22no—
38#2501—22no—
39#2501C2214yes—
40#2370—21no—
41#2369C2211yescharacteristic
42#2360—22no—
43#2360C2212yes—
44Q8×C4⋊C4C2211yescharacteristic
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2211yescharacteristic
47#2345C2212yes—
48#2330—21no—
49#2329—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2295C2211yescharacteristic
54C2×#389C2212yes—
55C2×#384C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2214yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2212yes—
63#2084C2212yes—
64#2084C2211yescharacteristic
65#2084C2214yes—
66#2072C2212yes—
67#2072C2211yescharacteristic
68#2072C2212yes—
69#2065C2212yes—
70#2065C2214yes—
71#2065C2214yes—
72#2055C2212yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2014C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1948C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1926C2214yes—
82#1926C2214yes—
83#1893C2214yes—
84#1880C2214yes—
85#1874C2214yes—
86#1850C2214yes—
87#1831C2211yescharacteristic
88#1823C2214yes—
89#1803C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94C2×#340C2214yes—
95C2×#336C2212yes—
96C22×#41—22no—
97C22×SD16C2312yes—
98C23×C8C2314yes—
99#554—24no—
100#554D418yescomplemented
101#547—28no—
102#547—28no—
103#547—48no—
104#547—48no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#125—21no—
111C2×#123—24no—
112C2×#123—24no—
113C2×#123—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8C2312yes—
119C2×#121—22no—
120C2×#121—22no—
121C2×#121C2314yes—
122C2×C4×Q8—21no—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2312yes—
132C2×C4×D4C2311yescharacteristic
133C22×C4⋊C4—22no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C42C2312yes—
140#498—28no—
141#483—216no—
142#481—216no—
143#476—28no—
144#475—28no—
145#465—216no—
146C4×SD16—28no—
147C8×D4—28no—
148C2×#106—216no—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2312yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2314yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C4×C8C2312yes—
166#399—24no—
167#399—24no—
168#399—24no—
169#391—21no—
170#391—22no—
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#390—21no—
174#390—21no—
175#390—22no—
176#390—22no—
177#390C2312yes—
178#390C2314yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2312yes—
190#389C2311yescharacteristic
191#387—21no—
192#387—22no—
193#387—21no—
194#387—22no—
195#387—22no—
196#387—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384—22no—
204#384—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2314yes—
208#384C2312yes—
209#384C2312yes—
210#380—22no—
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229#376—21no—
230C2×#95—22no—
231C43C2311yescharacteristic
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2314yes—
244C2×#41—48no—
245C2×#41—48no—
246C2×#41—48no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8C2411yescharacteristic
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—24no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—42no—
265C22×C8—28no—
266C22×C8—28no—
267C22×C8—44no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—24no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8C2×D414yes—
277#127—42no—
278#125—28no—
279#123—416no—
280#123—416no—
281#123—416no—
282C22⋊Q8—24no—
283C22⋊Q8—216no—
284C22⋊Q8—48no—
285C22⋊Q8—48no—
286C22⋊Q8—48no—
287C22⋊Q8C2×D418yes—
288#121—28no—
289#121—28no—
290#121—48no—
291#121—48no—
292C4×Q8—22no—
293C4×Q8—44no—
294C4×Q8—82no—
295C4×Q8C2×D414yes—
296C4×D4—28no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4SD1618yescomplemented
304C4×D4C2×D414yes—
305C4×D4#4114yes—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
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—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—48no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D418yes—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#106—24no—
375#106—28no—
376#106—28no—
377#106—28no—
378#106—28no—
379C4⋊C8—24no—
380C4⋊C8—24no—
381C4⋊C8—44no—
382#103—24no—
383#103—28no—
384#102—24no—
385#102—24no—
386#102—28no—
387#102—42no—
388#102—44no—
389C22⋊C8—28no—
390C4×C8—24no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24—22no—
408C24C22×D412yes—
409#41—832no—
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—81no—
418C2×Q8—81no—
419C2×Q8C22×D412yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4#12114yes—
436C2×D4C2×SD1614yes—
437C2×D4AGL(1,ℤ/8)14yes—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C2×SD1618yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4114yes—
477C22×C4C2511yescharacteristic
478SD16—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—416no—
488C2×C82−1+414yes—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—81no—
522C4⋊C4#12114yes—
523C4⋊C4C2×SD1614yes—
524C4⋊C4C22×D412yes—
525C4⋊C4C22×D414yes—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4#12118yes—
536C22⋊C4C2×SD1618yes—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—22no—
543C42—42no—
544C42—42no—
545C42—41no—
546C42—42no—
547C42—82no—
548C42—82no—
549C42—82no—
550C42C2×SD1614yes—
551C42C22×D412yes—
552C42C2×#4112yes—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23C2×#12114yes—
565C23C22×SD1614yes—
566C23C2×AGL(1,ℤ/8)14yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—42no—
570Q8—84no—
571Q8—88no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—84no—
577D4—84no—
578D4—816no—
579D4#460116yes—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—88no—
627C2×C4—816no—
628C2×C4—88no—
629C2×C4—164no—
630C2×C4C2×#12112yes—
631C2×C4C2×#12114yes—
632C2×C4C2×#12112yes—
633C2×C4C22×SD1612yes—
634C2×C4C22×SD1612yes—
635C2×C4C22×SD1614yes—
636C2×C4C22×SD1612yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4#57712yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×2−1+412yes—
642C2×C4#58512yes—
643C8—24no—
644C8—42no—
645C8—44no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22C2×#46018yes—
660C22C22×#12111yescharacteristic
661C22#348511yescharacteristic
662C22#349911yescharacteristic
663C22C23×SD1611yescharacteristic
664C22C22×AGL(1,ℤ/8)11yescharacteristic
665C22C2×#57711yescharacteristic
666C22#358611yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—24no—
671C4—22no—
672C4—24no—
673C4—42no—
674C4—44no—
675C4—42no—
676C4—41no—
677C4—42no—
678C4—82no—
679C4—81no—
680C4—81no—
681C4—82no—
682C4—84no—
683C4C2×#46014yes—
684C4#348912yes—
685C4#351512yes—
686C2—28no—
687C2—162no—
688C2#3602011yescharacteristic
689C2C22×#46011yescharacteristic
690C2#6042911yescharacteristic
691C2#6169511yescharacteristic
692C2#6172811yescharacteristic
693C2#6246911yescharacteristic
694C2#6250311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
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×#3335C2semidirect
3. #34332C2semidirect
4. #29866C2semidirect
5. #29755C2semidirect
6. #28687C2semidirect
7. #28574C2semidirect
8. #28202C2semidirect
9. #27647C2semidirect
10. #25451C2semidirect
11. #25204C2semidirect
12. #24813C2semidirect
13. #24578C2semidirect
14. #23678C2semidirect
15. C2×#2084C2semidirect
16. #22696C2semidirect
17. #22485C2semidirect
100. #554D4semidirect
303. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#33352
#361792
#343322
#298662
#297552
#286872
#285742
#282022
#276472
#254512
#252042
#248132
#245782
#236782
C2×#20842
#226962
#224852