Finite Group Databasebeta

← #6849811

Subgroups — #6849811

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C211yesmaximal, characteristic
2#36181C211yesmaximal, characteristic
3#33256C211yesmaximal, characteristic, complemented
4#32476C214yesmaximal, complemented
5#31772C214yesmaximal
6#29845C211yesmaximal, characteristic, complemented
7#29845C211yesmaximal, characteristic, complemented
8#29191C211yesmaximal, characteristic, complemented
9#29176C211yesmaximal, characteristic, complemented
10#27972C211yesmaximal, characteristic, complemented
11#27961C211yesmaximal, characteristic, complemented
12#27889C211yesmaximal, characteristic, complemented
13#27876C211yesmaximal, characteristic, complemented
14#27160C214yesmaximal, complemented
15#27034C214yesmaximal, complemented
16#21981C211yesmaximal, characteristic, complemented
17#21981C211yesmaximal, characteristic, complemented
18C4×#487C211yesmaximal, characteristic, complemented
19#21264C211yesmaximal, characteristic
20C2×#552—21no—
21C2×#487C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2570—21no—
31#2565—21no—
32#2511C2212yes—
33#2507—21no—
34#2506—21no—
35#2506C2212yes—
36#2503—21no—
37#2499—21no—
38#2497—21no—
39#2496—21no—
40#2496C2212yes—
41#2487—21no—
42#2486—21no—
43#2486C2212yes—
44#2442—21no—
45#2441—21no—
46#2423C2212yes—
47#2417C2212yes—
48#2360C2212yes—
49#2352—21no—
50#2345—21no—
51#2344—21no—
52#2344C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C42×D4C2211yescharacteristic
60#2100C2214yes—
61#2088C2214yes—
62#2077C2214yes—
63#2072C2214yes—
64#2066C2214yes—
65#2066C2214yes—
66#2066C2214yes—
67#2035C2214yes—
68#2035C2214yes—
69#2035C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1956C2211yescharacteristic
74#1956C2211yescharacteristic
75#1956C2211yescharacteristic
76#1954C2212yes—
77#1948C2214yes—
78#1948C2214yes—
79#1941C2214yes—
80#1941C2214yes—
81#1939C2212yes—
82#1933C2214yes—
83#1927C2214yes—
84#1923C2212yes—
85#1923C2212yes—
86#1920C2214yes—
87#1919C2214yes—
88#1872C2214yes—
89#1865C2214yes—
90#1853C2214yes—
91#1853C2214yes—
92#1844C2211yescharacteristic
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1834C2211yescharacteristic
101#1824C2214yes—
102C4×C4⋊C8C2211yescharacteristic
103C4×#103C2214yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#552—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#126—22no—
111C2×#125—22no—
112C2×#125—21no—
113C2×#125—21no—
114C2×#123—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#487—24no—
136#431—24no—
137#431—24no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2311yescharacteristic
159#400—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#399—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#397—22no—
169#397—22no—
170#396—22no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
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—21no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#386—22no—
208#385—22no—
209#384—21no—
210#384—21no—
211#384—21no—
212#384—21no—
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2312yes—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2312yes—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#381—22no—
226#381—22no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—22no—
236#380C2312yes—
237#380C2312yes—
238#380C2312yes—
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2311yescharacteristic
251C4×C22⋊C4—22no—
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257C4×C22⋊C4C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8C2412yes—
272C22×D4C2×D411yescharacteristic
273C23×C4—22no—
274C23×C4—22no—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289#126—48no—
290#125—48no—
291#125—44no—
292#125—82no—
293#123—48no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4SD1618yes—
303#117—44no—
304#117—48no—
305#117—82no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
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—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4—82no—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
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×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#106—44no—
387#106—44no—
388#106—44no—
389#106—44no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392C4⋊C8—42no—
393#103—28no—
394#103—44no—
395#103—44no—
396#103—44no—
397C22⋊C8—24no—
398C22⋊C8—24no—
399C22⋊C8—42no—
400C22⋊C8—42no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—42no—
405C4×C8—42no—
406C4×C8#4114yes—
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#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
433C24C22×D411yescharacteristic
434C24C22×D411yescharacteristic
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4C2×SD1614yes—
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—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4C2×SD1614yes—
478C22×C4C2×SD1614yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2511yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8C2×#4114yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4C2×SD1614yes—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4C2×SD1614yes—
557C22⋊C4C2×SD1614yes—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—22no—
570C42—22no—
571C42—44no—
572C42—44no—
573C42—82no—
574C42—82no—
575C42C2×SD1614yes—
576C42C2×#4112yes—
577C422−1+412yes—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23C22×SD1612yes—
591C23C22×SD1612yes—
592C23C23×D411yescharacteristic
593Q8—44no—
594Q8—44no—
595Q8—84no—
596D4—44no—
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—28no—
606C2×C4—28no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—44no—
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—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—88no—
656C2×C4—164no—
657C2×C4#54712yes—
658C2×C4#54712yes—
659C2×C4C22×SD1612yes—
660C2×C4C22×SD1612yes—
661C2×C4C22×SD1612yes—
662C2×C4C22×SD1612yes—
663C2×C4C22×SD1612yes—
664C2×C4#57512yes—
665C2×C4#57712yes—
666C2×C4#57812yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C2×C4#58512yes—
670C2×C4#58512yes—
671C8—24no—
672C8—42no—
673C8—42no—
674C8—42no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22#344911yescharacteristic
690C22C2×#54711yescharacteristic
691C22#350011yescharacteristic
692C22C23×SD1611yescharacteristic
693C22C2×#57511yescharacteristic
694C22C2×#57711yescharacteristic
695C22C2×#57811yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4#333514yes—
715C4#335214yes—
716C4#351212yes—
717C4#356512yes—
718C2—42no—
719C2—42no—
720C2#3948511yescharacteristic
721C2#6094111yescharacteristic
722C2#6100811yescharacteristic
723C2C2×#333511yescharacteristic
724C2C2×#335211yescharacteristic
725C2#6252611yescharacteristic
726C2#6254411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33256C2semidirect
4. #32476C2semidirect
6. #29845C2semidirect
7. #29845C2semidirect
8. #29191C2semidirect
9. #29176C2semidirect
10. #27972C2semidirect
11. #27961C2semidirect
12. #27889C2semidirect
13. #27876C2semidirect
14. #27160C2semidirect
15. #27034C2semidirect
16. #21981C2semidirect
17. #21981C2semidirect
18. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#361812
#332562
#324762
#317722
#298452
#291912
#291762
#279722
#279612
#278892
#278762
#271602
#270342
#219812
C4×#4872
#212642