Finite Group Databasebeta

← #6594108

Subgroups — #6594108

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C211yesmaximal, characteristic
2#36181C211yesmaximal, characteristic
3#33291C211yesmaximal, characteristic, complemented
4#33043C211yesmaximal, characteristic, complemented
5#33043C211yesmaximal, characteristic, complemented
6#29220C211yesmaximal, characteristic, complemented
7#29144C211yesmaximal, characteristic, complemented
8#27975C211yesmaximal, characteristic, complemented
9#27972C211yesmaximal, characteristic, complemented
10#27962C211yesmaximal, characteristic, complemented
11#27960C211yesmaximal, characteristic, complemented
12#27732C214yesmaximal, complemented
13#27700C214yesmaximal, complemented
14#27699C214yesmaximal, complemented
15#22486C214yesmaximal
16C4×#487C211yesmaximal, characteristic, complemented
17#20760C211yesmaximal, characteristic, complemented
18C2×#1834C211yesmaximal, characteristic, complemented
19C4×C8×D4C211yesmaximal, characteristic
20C2×#552C2212yes—
21C2×#487C2211yescharacteristic
22C2×C8×D4C2212yes—
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2570C2212yes—
32#2565C2212yes—
33#2511—21no—
34#2507—21no—
35#2506—21no—
36#2506—21no—
37#2503C2212yes—
38#2499C2212yes—
39#2497—21no—
40#2496—21no—
41#2496—21no—
42#2487—21no—
43#2486—21no—
44#2486—21no—
45#2442C2212yes—
46#2441C2212yes—
47#2423—21no—
48#2417—21no—
49#2360—21no—
50#2352C2212yes—
51#2345—21no—
52#2344—21no—
53#2344—21no—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C42×D4C2211yescharacteristic
61#2101C2214yes—
62#2101C2214yes—
63#2099C2214yes—
64#2074C2214yes—
65#2072C2214yes—
66#2026C2214yes—
67#1991C2214yes—
68#1991C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1989C2214yes—
73#1956C2211yescharacteristic
74#1956C2211yescharacteristic
75#1956C2211yescharacteristic
76#1933C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1873C2214yes—
82#1872C2214yes—
83#1835C2214yes—
84#1834C2211yescharacteristic
85#1834C2211yescharacteristic
86#1834C2211yescharacteristic
87#1834C2211yescharacteristic
88#1834C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2214yes—
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1825C2214yes—
95#1823C2214yes—
96#1805C2214yes—
97C8×C4⋊C4C2212yes—
98C4×C4⋊C8C2211yescharacteristic
99C4×#103C2214yes—
100C8×C22⋊C4C2212yes—
101C8×C22⋊C4C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1735C2214yes—
105C2×#340C2214yes—
106C42×C8C2211yescharacteristic
107C23×C8C2312yes—
108C23×C8C2312yes—
109#552—216no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#126—22no—
113C2×#125—22no—
114C2×#125C2312yes—
115C2×#125C2312yes—
116C2×#123—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117C2312yes—
128C2×#117C2312yes—
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#487—24no—
138C8×D4—28no—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8—24no—
147C2×C4×C8—24no—
148C2×C4×C8C2312yes—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2312yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2311yescharacteristic
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#397—22no—
166#397—22no—
167#396—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#389—21no—
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—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—22no—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#386—22no—
205#385—22no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384—21no—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2312yes—
213#384C2312yes—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#382C2312yes—
219#382C2312yes—
220#382C2312yes—
221#382C2312yes—
222#381—22no—
223#381—22no—
224#380—21no—
225#380—21no—
226#380—21no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—22no—
235#380C2312yes—
236#380C2312yes—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
248C4×C22⋊C4—22no—
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4C2311yescharacteristic
254C4×C22⋊C4C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—21no—
272C22×Q8C2412yes—
273C22×D4C2×D411yescharacteristic
274C23×C4—22no—
275C23×C4—22no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C23×C4C2411yescharacteristic
282C22×C8—24no—
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8—42no—
287C22×C8—42no—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296#126—48no—
297#125—28no—
298#125—44no—
299#125—48no—
300#123—48no—
301C22⋊Q8—28no—
302C22⋊Q8—28no—
303C22⋊Q8—44no—
304C22⋊Q8—82no—
305C4×Q8—28no—
306C4×Q8—82no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4#4118yes—
310#117—28no—
311#117—48no—
312#117—44no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
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—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
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—44no—
346C2×C4⋊C4—44no—
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⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C4⋊C8—24no—
394#103—28no—
395#103—44no—
396#103—44no—
397#103—44no—
398C22⋊C8—24no—
399C22⋊C8—24no—
400C4×C8—24no—
401C4×C8—24no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—24no—
405C4×C8—22no—
406C4×C8—42no—
407C4×C8#4114yes—
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#95—22no—
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434C24C22×D411yescharacteristic
435C24C22×D411yescharacteristic
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4#12414yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
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—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4#13514yes—
477C22×C4#13514yes—
478C22×C4C22×D411yescharacteristic
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×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2511yescharacteristic
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×C8—22no—
504C2×C8—48no—
505C2×C8—48no—
506C2×C8C2×#4114yes—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4#12414yes—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4#12414yes—
560C22⋊C4#12414yes—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—22no—
574C42—44no—
575C42—44no—
576C42—82no—
577C42—82no—
578C42C2×#4114yes—
579C42C2×#4112yes—
580C422−1+412yes—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23C2×#12412yes—
592C23C2×#12412yes—
593C23C2×#13512yes—
594C23C2×#13512yes—
595C23C23×D411yescharacteristic
596Q8—44no—
597Q8—44no—
598Q8—84no—
599D4—28no—
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—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—21no—
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—44no—
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—88no—
657C2×C4—164no—
658C2×C4C2×#12412yes—
659C2×C4C2×#12412yes—
660C2×C4C2×#12412yes—
661C2×C4C2×#12412yes—
662C2×C4#54712yes—
663C2×C4#54712yes—
664C2×C4C2×#13512yes—
665C2×C4C2×#13512yes—
666C2×C4C2×#13512yes—
667C2×C4C2×#13512yes—
668C2×C4#57712yes—
669C2×C4#57812yes—
670C2×C4C22×#4111yescharacteristic
671C2×C4C22×#4112yes—
672C2×C4C2×2−1+411yescharacteristic
673C2×C4#58512yes—
674C8—24no—
675C8—42no—
676C8—42no—
677C8—42no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22#313114yes—
692C22C22×#12411yescharacteristic
693C22C2×#54711yescharacteristic
694C22#350011yescharacteristic
695C22C22×#13511yescharacteristic
696C22C22×#13511yescharacteristic
697C22C2×#57711yescharacteristic
698C22C2×#57811yescharacteristic
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4#333614yes—
718C4#333714yes—
719C4#347512yes—
720C4#351512yes—
721C2—24no—
722C2—42no—
723C2#3616411yescharacteristic
724C2C2×#313111yescharacteristic
725C2#6062311yescharacteristic
726C2C2×#333611yescharacteristic
727C2C2×#333711yescharacteristic
728C2#6252811yescharacteristic
729C2#6253111yescharacteristic
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. #33291C2semidirect
4. #33043C2semidirect
5. #33043C2semidirect
6. #29220C2semidirect
7. #29144C2semidirect
8. #27975C2semidirect
9. #27972C2semidirect
10. #27962C2semidirect
11. #27960C2semidirect
12. #27732C2semidirect
13. #27700C2semidirect
14. #27699C2semidirect
16. C4×#487C2semidirect
17. #20760C2semidirect
18. C2×#1834C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#361812
#332912
#330432
#292202
#291442
#279752
#279722
#279622
#279602
#277322
#277002
#276992
#224862
C4×#4872
#207602
C2×#18342
C4×C8×D42