Finite Group Databasebeta

← #6594408

Subgroups — #6594408

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 733 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#33207C211yesmaximal, characteristic, complemented
5#33207C211yesmaximal, characteristic, complemented
6#29177C211yesmaximal, characteristic, complemented
7#29176C211yesmaximal, characteristic, complemented
8#27990C214yesmaximal, complemented
9#27961C211yesmaximal, characteristic, complemented
10#27960C211yesmaximal, characteristic, complemented
11#27876C211yesmaximal, characteristic, complemented
12#27875C211yesmaximal, characteristic, complemented
13#27835C214yesmaximal, complemented
14#27658C214yesmaximal
15#22609C214yesmaximal, complemented
16C4×#498C211yesmaximal, characteristic, complemented
17#20816C211yesmaximal, characteristic, complemented
18C2×#1844C211yesmaximal, characteristic, complemented
19C4×C8×D4C211yesmaximal, characteristic
20C2×#552—21no—
21C2×#498C2211yescharacteristic
22C2×C8×D4C2212yes—
23C22×#106C2214yes—
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2570—21no—
33#2565—21no—
34#2511—21no—
35#2507C2212yes—
36#2506—21no—
37#2506C2212yes—
38#2503—21no—
39#2499—21no—
40#2497C2212yes—
41#2496—21no—
42#2496C2212yes—
43#2487C2212yes—
44#2486—21no—
45#2486C2212yes—
46#2442—21no—
47#2441—21no—
48#2423—21no—
49#2417—21no—
50#2360—21no—
51#2352—21no—
52#2345C2212yes—
53#2344—21no—
54#2344C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2100C2214yes—
63#2099C2214yes—
64#2088C2214yes—
65#2088C2214yes—
66#2077C2214yes—
67#2077C2214yes—
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1941C2214yes—
75#1941C2214yes—
76#1941C2214yes—
77#1941C2214yes—
78#1941C2214yes—
79#1920C2214yes—
80#1920C2214yes—
81#1919C2214yes—
82#1919C2214yes—
83#1844C2211yescharacteristic
84#1844C2214yes—
85#1844C2211yescharacteristic
86#1844C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1824C2214yes—
94#1823C2214yes—
95#1807C2214yes—
96#1807C2214yes—
97#1807C2214yes—
98C4×#106C2214yes—
99C4×#106C2214yes—
100C8×C4⋊C4C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102C8×C22⋊C4C2212yes—
103C8×C22⋊C4C2212yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×C8C2312yes—
108C23×C8C2312yes—
109#552—48no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#126—22no—
113C2×#125—22no—
114C2×#125—21no—
115C2×#125—21no—
116C2×#123—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—21no—
127C2×#117—21no—
128C2×#117—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#498—24no—
138C8×D4—28no—
139C2×#106—216no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C4×C8—24no—
150C2×C4×C8—24no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2312yes—
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#391—21no—
175#391—21no—
176#391—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#389C2312yes—
189#389C2312yes—
190#389C2312yes—
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#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2312yes—
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—22no—
234#380C2312yes—
235#380C2312yes—
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⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4C2312yes—
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#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×D4C2×D411yescharacteristic
275C23×C4—22no—
276C23×C4—22no—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282C23×C4C2411yescharacteristic
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8—24no—
287C22×C8—24no—
288C22×C8#4118yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297#126—48no—
298#125—48no—
299#125—44no—
300#125—82no—
301#123—48no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C22⋊Q8—82no—
306C4×Q8—44no—
307C4×Q8—82no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4SD1618yes—
311#117—44no—
312#117—48no—
313#117—82no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
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—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—82no—
349C2×C4⋊C4—82no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394#106—28no—
395#106—28no—
396#106—28no—
397#106—28no—
398#106—28no—
399#106—28no—
400C4⋊C8—24no—
401C22⋊C8—24no—
402C22⋊C8—24no—
403C4×C8—24no—
404C4×C8—24no—
405C4×C8—24no—
406C4×C8—24no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8#4114yes—
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#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437C24C22×D411yescharacteristic
438C24C22×D411yescharacteristic
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4C2×SD1614yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
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—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4C2×SD1614yes—
479C22×C4C2×SD1614yes—
480C22×C4#13514yes—
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×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4114yes—
494C22×C4C2511yescharacteristic
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—216no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—48no—
505C2×C8C2×#4114yes—
506C2×C8C2×#4114yes—
507C2×C8C2×#4114yes—
508C2×C82−1+414yes—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4C2×SD1614yes—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4C2×SD1614yes—
563C22⋊C4C2×SD1614yes—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—21no—
572C42—21no—
573C42—22no—
574C42—22no—
575C42—21no—
576C42—21no—
577C42—44no—
578C42—44no—
579C42—82no—
580C42—82no—
581C42C2×SD1614yes—
582C42C2×#4112yes—
583C422−1+412yes—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23#53212yes—
595C23C22×SD1612yes—
596C23C22×SD1612yes—
597C23C2×#13512yes—
598C23C23×D411yescharacteristic
599Q8—84no—
600Q8—84no—
601D4—28no—
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—21no—
613C2×C4—21no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—44no—
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—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—88no—
659C2×C4—164no—
660C2×C4#53212yes—
661C2×C4#54712yes—
662C2×C4#54712yes—
663C2×C4C22×SD1612yes—
664C2×C4C22×SD1612yes—
665C2×C4C22×SD1612yes—
666C2×C4C22×SD1612yes—
667C2×C4C22×SD1612yes—
668C2×C4C2×#13512yes—
669C2×C4C2×#13512yes—
670C2×C4#57712yes—
671C2×C4#57712yes—
672C2×C4C22×#4111yescharacteristic
673C2×C4C22×#4112yes—
674C2×C4C2×2−1+412yes—
675C2×C4C2×2−1+411yescharacteristic
676C8—24no—
677C8—42no—
678C8—42no—
679C8#55218yescomplemented
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22#316714yes—
694C22C2×#53211yescharacteristic
695C22C2×#54711yescharacteristic
696C22#349911yescharacteristic
697C22C23×SD1611yescharacteristic
698C22C22×#13511yescharacteristic
699C22C2×#57711yescharacteristic
700C22C2×#57711yescharacteristic
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4#333514yes—
720C4#333714yes—
721C4C2×#55212yes—
722C4#350912yes—
723C2—24no—
724C2—42no—
725C2#3618111yescharacteristic
726C2C2×#316711yescharacteristic
727C2#6071211yescharacteristic
728C2C2×#333511yescharacteristic
729C2C2×#333711yescharacteristic
730C2#6246911yescharacteristic
731C2#6247011yescharacteristic
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. #33207C2semidirect
5. #33207C2semidirect
6. #29177C2semidirect
7. #29176C2semidirect
8. #27990C2semidirect
9. #27961C2semidirect
10. #27960C2semidirect
11. #27876C2semidirect
12. #27875C2semidirect
13. #27835C2semidirect
15. #22609C2semidirect
16. C4×#498C2semidirect
17. #20816C2semidirect
18. C2×#1844C2semidirect
679. C8#552semidirect

Maximal subgroups

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

subgroupindex
#361892
#361812
#332562
#332072
#291772
#291762
#279902
#279612
#279602
#278762
#278752
#278352
#276582
#226092
C4×#4982
#208162
C2×#18442
C4×C8×D42