Finite Group Databasebeta

← #6498956

Subgroups — #6498956

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#33289C211yesmaximal, characteristic, complemented
5#33263C211yesmaximal, characteristic, complemented
6#33256C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#30305C211yesmaximal, characteristic, complemented
9#30305C211yesmaximal, characteristic, complemented
10#30304C211yesmaximal, characteristic, complemented
11#28803C214yesmaximal, complemented
12#28802C214yesmaximal, complemented
13#28783C214yesmaximal
14#28782C214yesmaximal
15#21984C211yesmaximal, characteristic
16#21981C211yesmaximal, characteristic
17#20824C211yesmaximal, characteristic
18#20764C211yesmaximal, characteristic
19#20472C211yesmaximal, characteristic
20C2×#544—21no—
21C2×#431C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2660—21no—
32#2660—21no—
33#2629—21no—
34#2628—21no—
35#2570—21no—
36#2570C2212yes—
37#2552—21no—
38#2552C2212yes—
39#2551—21no—
40#2551C2212yes—
41#2550—21no—
42#2550C2212yes—
43#2441—21no—
44#2441C2212yes—
45#2440—21no—
46#2440—21no—
47#2440C2212yes—
48#2440C2212yes—
49#2439—21no—
50#2439C2212yes—
51#2331—21no—
52#2330—21no—
53#2317—21no—
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2085C2214yes—
60#2084C2214yes—
61#2084C2214yes—
62#2083C2214yes—
63#2059C2214yes—
64#2058C2214yes—
65#2058C2214yes—
66#2057C2214yes—
67#1993C2214yes—
68#1992C2214yes—
69#1990C2214yes—
70#1990C2214yes—
71#1949C2214yes—
72#1948C2214yes—
73#1944C2214yes—
74#1943C2214yes—
75#1941C2214yes—
76#1941C2214yes—
77#1854C2214yes—
78#1853C2214yes—
79#1845C2211yescharacteristic
80#1845C2211yescharacteristic
81#1844C2211yescharacteristic
82#1844C2211yescharacteristic
83#1844C2211yescharacteristic
84#1844C2211yescharacteristic
85#1836C2214yes—
86#1834C2211yescharacteristic
87#1834C2211yescharacteristic
88#1834C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1809C2214yes—
97#1776C2214yes—
98#1771C2212yes—
99#1771C2212yes—
100#1753C2212yes—
101#1753C2212yes—
102#1744C2211yescharacteristic
103#1744C2211yescharacteristic
104#1736C2214yes—
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#431Q818yescomplemented
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C4×M16Q818yescomplemented
146C2×#97C2312yes—
147C2×#97C2312yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155#400—22no—
156#399—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#396—22no—
164#396—22no—
165#395—22no—
166#395—22no—
167#395—22no—
168#395—22no—
169#395—22no—
170#393—22no—
171#391—22no—
172#391—22no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—22no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384—21no—
210#384—21no—
211#384C2312yes—
212#384C2312yes—
213#384C2312yes—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2312yes—
223#382—21no—
224#382—21no—
225#382—21no—
226#382—21no—
227#382—21no—
228#382—21no—
229#382C2312yes—
230#382C2312yes—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380C2312yes—
236#380C2312yes—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C22⋊C4D411yescharacteristic
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4C2311yescharacteristic
250C4×C22⋊C4C2311yescharacteristic
251#376D411yescharacteristic
252#376D411yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8—22no—
272C23×C4—22no—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C2×M16—42no—
277C2×M16—42no—
278C2×M16C2×Q818yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—48no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—82no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293#125—44no—
294#125—44no—
295#125—48no—
296#125—44no—
297#125—44no—
298#125—82no—
299C22⋊Q8—48no—
300C22⋊Q8—48no—
301#117—24no—
302#117—24no—
303#117—44no—
304#117—44no—
305#117—82no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
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—44no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—82no—
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C22⋊C4C2×D411yescharacteristic
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×Q814yes—
380C2×C42C2×Q814yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—44no—
388#107—44no—
389#106—44no—
390#106—44no—
391#106—44no—
392#106—44no—
393C4⋊C8C2×Q818yes—
394C22⋊C8C2×Q818yes—
395#97—42no—
396#97—42no—
397#97C2×Q818yes—
398C4×C8—42no—
399C4×C8—42no—
400C4×C8C2×Q814yes—
401C4×C8C2×Q814yes—
402C4×C8C2×Q814yes—
403C4×C8C2×Q814yes—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
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#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428C24C22×D411yescharacteristic
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—82no—
433C2×Q8—82no—
434C2×Q8—82no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×Q8—82no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4C4⋊Q814yes—
456C22×C4C4⋊Q814yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C2511yescharacteristic
473M16—48no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8C22×Q814yes—
483C2×C8C22×Q814yes—
484C2×C8C22×Q814yes—
485C2×C8C22×Q814yes—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—44no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—44no—
554C42—82no—
555C42—82no—
556C42C4⋊Q814yes—
557C42C4⋊Q814yes—
558C42AGL(1,ℤ/8)14yes—
559C42#13714yes—
560C42C22×Q812yes—
561C42C22×Q812yes—
562C42C22×Q812yes—
563C42C22×Q812yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23C2×C4⋊Q812yes—
571C23C23×D411yescharacteristic
572Q8—164no—
573Q8—164no—
574Q8—164no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
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—164no—
630C2×C4C2×C4⋊Q812yes—
631C2×C4C2×C4⋊Q812yes—
632C2×C4C2×C4⋊Q812yes—
633C2×C4C2×C4⋊Q812yes—
634C2×C4C2×C4⋊Q812yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×#13712yes—
639C2×C4C2×#13712yes—
640C2×C4C2×#13712yes—
641C2×C4#57712yes—
642C2×C4#57712yes—
643C2×C4C23×Q811yescharacteristic
644C2×C4C23×Q811yescharacteristic
645C2×C4#58512yes—
646C2×C4#58512yes—
647C8—24no—
648C8—24no—
649C8—24no—
650C8—24no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22C22×C4⋊Q811yescharacteristic
659C22#350011yescharacteristic
660C22#350011yescharacteristic
661C22C22×AGL(1,ℤ/8)11yescharacteristic
662C22C22×#13711yescharacteristic
663C22C2×#57711yescharacteristic
664C22C2×#57711yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#316314yes—
685C4#316514yes—
686C4#345812yes—
687C4#350412yes—
688C2—42no—
689C2#3533011yescharacteristic
690C2C2×#316311yescharacteristic
691C2C2×#316511yescharacteristic
692C2#6253911yescharacteristic
693C2#6254011yescharacteristic
694C2#6254111yescharacteristic
695C2#6254211yescharacteristic
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. #33297C2semidirect
4. #33289C2semidirect
5. #33263C2semidirect
6. #33256C2semidirect
7. #30307C2semidirect
8. #30305C2semidirect
9. #30305C2semidirect
10. #30304C2semidirect
11. #28803C2semidirect
12. #28802C2semidirect
136. #431Q8semidirect
145. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#352752
#332972
#332892
#332632
#332562
#303072
#303052
#303042
#288032
#288022
#287832
#287822
#219842
#219812
#208242
#207642
#204722