Finite Group Databasebeta

← #6497746

Subgroups — #6497746

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#33278C211yesmaximal, characteristic, complemented
5#33207C211yesmaximal, characteristic
6#33043C211yesmaximal, characteristic
7#30305C211yesmaximal, characteristic, complemented
8#30280C211yesmaximal, characteristic, complemented
9#29877C211yesmaximal, characteristic
10#29845C211yesmaximal, characteristic
11#28947C214yesmaximal, complemented
12#28847C214yesmaximal
13#28066C214yesmaximal
14#28030C214yesmaximal, complemented
15#22246C211yesmaximal, characteristic, complemented
16#22218C211yesmaximal, characteristic, complemented
17C4×#498C211yesmaximal, characteristic, complemented
18C4×#487C211yesmaximal, characteristic, complemented
19C8×#117C211yesmaximal, characteristic
20C2×#541—21no—
21C2×#498C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2647—21no—
36#2642—21no—
37#2628—21no—
38#2621—21no—
39#2511—21no—
40#2511C2212yes—
41#2506—21no—
42#2506—21no—
43#2506C2212yes—
44#2506C2212yes—
45#2496—21no—
46#2496—21no—
47#2496C2212yes—
48#2496C2212yes—
49#2492—21no—
50#2492C2212yes—
51#2451—21no—
52#2451C2212yes—
53#2423—21no—
54#2423C2212yes—
55#2331—21no—
56#2327—21no—
57#2314—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2096C2214yes—
64#2096C2214yes—
65#2088C2214yes—
66#2084C2214yes—
67#2077C2214yes—
68#2066C2214yes—
69#2058C2214yes—
70#2035C2214yes—
71#1990C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1941C2214yes—
81#1941C2214yes—
82#1933C2214yes—
83#1933C2214yes—
84#1933C2214yes—
85#1933C2214yes—
86#1927C2214yes—
87#1920C2214yes—
88#1919C2214yes—
89#1892C2214yes—
90#1865C2214yes—
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1834C2214yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1807C2214yes—
97C4×#106C2214yes—
98C8×C4⋊C4C2212yes—
99C8×C4⋊C4C2212yes—
100C4×#103C2214yes—
101C8×C22⋊C4C2212yes—
102C8×C22⋊C4C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×C8C2312yes—
108#541—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×#126—22no—
119C2×#126—22no—
120C2×#125—22no—
121C2×#125—22no—
122C2×#125—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C2312yes—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8C2312yes—
131C2×#117D411yescharacteristic
132C2×#117D411yescharacteristic
133C22×C42C2311yescharacteristic
134#498—24no—
135#498—24no—
136#487—24no—
137#487—24no—
138#431—24no—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C4×C8—24no—
150C2×C4×C8C2312yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2312yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#400—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#400—22no—
166#399—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#395—22no—
174#393—22no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—22no—
184#391—22no—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#388—22no—
210#388—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#384—21no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2312yes—
228#384C2312yes—
229#384C2312yes—
230#382—21no—
231#382—21no—
232#382—21no—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380C2312yes—
239#380C2312yes—
240#380C2312yes—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257#376D411yescharacteristic
258#376D411yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8C2412yes—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C22×C8—24no—
282C22×C8—24no—
283C22×C8—42no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—44no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—82no—
301#126—48no—
302#126—48no—
303#125—48no—
304#125—48no—
305#125—44no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309C22⋊Q8—82no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—82no—
314#117—24no—
315#117SD1618yes—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
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—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⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
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×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×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#106—28no—
388#106—28no—
389#106—44no—
390#106—44no—
391C4⋊C8—24no—
392#103—28no—
393#103—28no—
394#103—28no—
395#103—44no—
396C22⋊C8—24no—
397C4×C8—24no—
398C4×C8—24no—
399C4×C8—24no—
400C4×C8—22no—
401C4×C8—22no—
402C4×C8—42no—
403C4×C8—42no—
404C4×C8#4114yes—
405C4×C8#4114yes—
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#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
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436C24C22×D411yescharacteristic
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—82no—
452C2×Q8—82no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
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×C4C2×SD1614yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2511yescharacteristic
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—48no—
495C2×C8C2×#4114yes—
496C2×C8C2×#4114yes—
497C2×C82−1+414yes—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4C2×SD1614yes—
541C4⋊C4C2×SD1614yes—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4C2×SD1614yes—
549C22⋊C4C2×SD1614yes—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—21no—
558C42—21no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—44no—
564C42—44no—
565C42—82no—
566C42—82no—
567C42C2×SD1614yes—
568C42C2×SD1614yes—
569C42#13514yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23C22×SD1612yes—
579C23C23×D411yescharacteristic
580Q8—44no—
581Q8—44no—
582Q8—84no—
583Q8—84no—
584Q8—84no—
585Q8—164no—
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—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
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—21no—
613C2×C4—44no—
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—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4#53212yes—
641C2×C4#53212yes—
642C2×C4C22×SD1612yes—
643C2×C4C22×SD1612yes—
644C2×C4C22×SD1612yes—
645C2×C4C22×SD1612yes—
646C2×C4C22×SD1612yes—
647C2×C4C22×SD1612yes—
648C2×C4C2×#13512yes—
649C2×C4C2×#13512yes—
650C2×C4C2×#13512yes—
651C2×C4#57712yes—
652C2×C4#57812yes—
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×2−1+412yes—
656C2×C4#58512yes—
657C8—24no—
658C8—24no—
659C8—24no—
660C8—42no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22C2×#53211yescharacteristic
669C22#349911yescharacteristic
670C22#350011yescharacteristic
671C22C23×SD1611yescharacteristic
672C22C22×#13511yescharacteristic
673C22C2×#57711yescharacteristic
674C22C2×#57811yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#316714yes—
695C4#317814yes—
696C4#346112yes—
697C4#349512yes—
698C2—42no—
699C2#3531811yescharacteristic
700C2C2×#316711yescharacteristic
701C2C2×#317811yescharacteristic
702C2#6246911yescharacteristic
703C2#6247311yescharacteristic
704C2#6252611yescharacteristic
705C2#6253111yescharacteristic
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. #33289C2semidirect
4. #33278C2semidirect
7. #30305C2semidirect
8. #30280C2semidirect
11. #28947C2semidirect
14. #28030C2semidirect
15. #22246C2semidirect
16. #22218C2semidirect
17. C4×#498C2semidirect
18. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#352832
#332892
#332782
#332072
#330432
#303052
#302802
#298772
#298452
#289472
#288472
#280662
#280302
#222462
#222182
C4×#4982
C4×#4872
C8×#1172