Finite Group Databasebeta

← #6494442

Subgroups — #6494442

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33310C211yesmaximal, characteristic, complemented
5#33263C211yesmaximal, characteristic, complemented
6#33258C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#30305C211yesmaximal, characteristic, complemented
9#30305C211yesmaximal, characteristic, complemented
10#30304C211yesmaximal, characteristic, complemented
11#28837C214yesmaximal, complemented
12#28819C214yesmaximal, complemented
13#26078C214yesmaximal
14#26072C214yesmaximal
15#21984C211yesmaximal, characteristic
16#21981C211yesmaximal, characteristic
17#20824C211yesmaximal, characteristic
18#20800C211yesmaximal, characteristic
19#20472C211yesmaximal, characteristic
20C2×#544—21no—
21C2×#501C2214yes—
22C2×#498C2214yes—
23C2×#431C2211yescharacteristic
24C2×#428C2214yes—
25C2×C4×M16C2211yescharacteristic
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2859C2211yescharacteristic
34#2660—21no—
35#2660—21no—
36#2629—21no—
37#2628—21no—
38#2570—21no—
39#2570C2212yes—
40#2552—21no—
41#2552C2212yes—
42#2551—21no—
43#2551C2212yes—
44#2550—21no—
45#2550C2212yes—
46#2441—21no—
47#2441C2212yes—
48#2440—21no—
49#2440—21no—
50#2440C2212yes—
51#2440C2212yes—
52#2439—21no—
53#2439C2212yes—
54#2331—21no—
55#2330—21no—
56#2317—21no—
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C4×#117C2211yescharacteristic
62#2085C2214yes—
63#2084C2214yes—
64#2084C2214yes—
65#2083C2214yes—
66#2059C2214yes—
67#2058C2214yes—
68#2058C2214yes—
69#2057C2214yes—
70#1949C2214yes—
71#1948C2214yes—
72#1944C2214yes—
73#1943C2214yes—
74#1942C2214yes—
75#1942C2214yes—
76#1854C2214yes—
77#1853C2214yes—
78#1847C2214yes—
79#1847C2214yes—
80#1845C2211yescharacteristic
81#1845C2211yescharacteristic
82#1845C2211yescharacteristic
83#1845C2211yescharacteristic
84#1844C2211yescharacteristic
85#1844C2211yescharacteristic
86#1844C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2214yes—
89#1844C2211yescharacteristic
90#1844C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1809C2214yes—
98#1776C2214yes—
99#1771C2212yes—
100#1771C2212yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×M16C2312yes—
108#544—48no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2312yes—
119C2×#125—22no—
120C2×#125—21no—
121C2×#125—21no—
122C2×#125—21no—
123C2×#125—21no—
124C2×#125C2312yes—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—22no—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C2×#117D411yescharacteristic
131C2×#117C2312yes—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136#501—48no—
137#498—48no—
138#431Q818yescomplemented
139#428—48no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C4×M16Q818yescomplemented
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162#400—22no—
163#399—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#396—22no—
171#396—22no—
172#395—22no—
173#395—22no—
174#395—22no—
175#395—22no—
176#395—22no—
177#393—22no—
178#391—22no—
179#391—22no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—22no—
195#389—22no—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#385—22no—
209#385—22no—
210#384—21no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384—21no—
215#384—21no—
216#384—21no—
217#384—21no—
218#384C2312yes—
219#384C2312yes—
220#384C2312yes—
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2312yes—
230#382—21no—
231#382—21no—
232#382—21no—
233#382—21no—
234#382—21no—
235#382—21no—
236#382C2312yes—
237#382C2312yes—
238#380—21no—
239#380—21no—
240#380—21no—
241#380—21no—
242#380C2312yes—
243#380C2312yes—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257C4×C22⋊C4C2311yescharacteristic
258#376D411yescharacteristic
259#376D411yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—22no—
273C22×Q8—22no—
274C22×Q8—22no—
275C23×C4—22no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×M16—24no—
280C2×M16—24no—
281C2×M16C2×Q818yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C4⋊Q8—44no—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—48no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—82no—
294C4⋊Q8—82no—
295C4⋊Q8—82no—
296#125—44no—
297#125—44no—
298#125—48no—
299#125—44no—
300#125—44no—
301#125—82no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304#117—24no—
305#117—24no—
306#117—44no—
307#117—44no—
308#117—82no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—82no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4C2×D411yescharacteristic
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×Q814yes—
383C2×C42C2×Q814yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—28no—
391#107—28no—
392#107—44no—
393#107—44no—
394#106—28no—
395#106—28no—
396#106—44no—
397#106—44no—
398#106—44no—
399#106—44no—
400C4⋊C8C2×Q818yes—
401C22⋊C8C2×Q818yes—
402#97—24no—
403#97—24no—
404#97C2×Q818yes—
405C4×C8—24no—
406C4×C8—24no—
407C4×C8C2×Q814yes—
408C4×C8C2×Q814yes—
409C4×C8C2×Q814yes—
410C4×C8C2×Q814yes—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435C24C22×D411yescharacteristic
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8—82no—
442C2×Q8—82no—
443C2×Q8—82no—
444C2×Q8—82no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4C4⋊Q814yes—
463C22×C4C4⋊Q814yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q812yes—
478C22×C4C22×Q812yes—
479C22×C4C2511yescharacteristic
480M16—216no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8C22×Q814yes—
488C2×C8C22×Q814yes—
489C2×C8C22×Q814yes—
490C2×C8C22×Q814yes—
491C2×C8H(5,𝔽2)14yes—
492C2×C82−1+414yes—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—44no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—44no—
561C42—82no—
562C42—82no—
563C42C4⋊Q814yes—
564C42C4⋊Q814yes—
565C42AGL(1,ℤ/8)14yes—
566C42#13714yes—
567C42C22×Q812yes—
568C42C22×Q812yes—
569C42C22×Q812yes—
570C42C22×Q812yes—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23C2×C4⋊Q812yes—
578C23C23×D411yescharacteristic
579Q8—164no—
580Q8—164no—
581Q8—164no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—164no—
637C2×C4C2×C4⋊Q812yes—
638C2×C4C2×C4⋊Q812yes—
639C2×C4C2×C4⋊Q812yes—
640C2×C4C2×C4⋊Q812yes—
641C2×C4C2×C4⋊Q812yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×#13712yes—
646C2×C4C2×#13712yes—
647C2×C4C2×#13712yes—
648C2×C4#57712yes—
649C2×C4#57712yes—
650C2×C4C23×Q811yescharacteristic
651C2×C4C23×Q811yescharacteristic
652C2×C4C2×H(5,𝔽2)12yes—
653C2×C4C2×2−1+412yes—
654C8—24no—
655C8—24no—
656C8—24no—
657C8#54418yescomplemented
658C22—22no—
659C22—22no—
660C22—22no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22C22×C4⋊Q811yescharacteristic
666C22#349811yescharacteristic
667C22#349911yescharacteristic
668C22C22×AGL(1,ℤ/8)11yescharacteristic
669C22C22×#13711yescharacteristic
670C22C2×#57711yescharacteristic
671C22C2×#57711yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#316314yes—
692C4#316514yes—
693C4C2×#54412yes—
694C4#350312yes—
695C2—42no—
696C2#3527511yescharacteristic
697C2C2×#316311yescharacteristic
698C2C2×#316511yescharacteristic
699C2#6242111yescharacteristic
700C2#6242211yescharacteristic
701C2#6247711yescharacteristic
702C2#6247811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33316C2semidirect
4. #33310C2semidirect
5. #33263C2semidirect
6. #33258C2semidirect
7. #30307C2semidirect
8. #30305C2semidirect
9. #30305C2semidirect
10. #30304C2semidirect
11. #28837C2semidirect
12. #28819C2semidirect
138. #431Q8semidirect
152. C4×M16Q8semidirect
657. C8#544semidirect

Maximal subgroups

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

subgroupindex
#352942
#352752
#333162
#333102
#332632
#332582
#303072
#303052
#303042
#288372
#288192
#260782
#260722
#219842
#219812
#208242
#208002
#204722