Finite Group Databasebeta

← #6482071

Subgroups — #6482071

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37640C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic
3#32989C214yesmaximal, complemented
4#29901C214yesmaximal, complemented
5#26301C212yesmaximal, complemented
6#26296C212yesmaximal, complemented
7#26236C212yesmaximal, complemented
8#26233C212yesmaximal, complemented
9#25017C212yesmaximal, complemented
10#25016C212yesmaximal, complemented
11#24970C212yesmaximal, complemented
12#24969C212yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#21787C211yesmaximal, characteristic
16#21389C211yesmaximal, characteristic, complemented
17#21362C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#520C2212yes—
21C2×#484C2214yes—
22C2×#485C2214yes—
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2820—22no—
26#2624—22no—
27#2620—22no—
28#2616C2214yes—
29#2567—21no—
30#2548—21no—
31#2547C2212yes—
32#2542C2212yes—
33#2538C2212yes—
34#2538C2212yes—
35#2361—22no—
36#2361—22no—
37#2354C2212yes—
38#2354C2212yes—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41#2326—22no—
42#2311C2212yes—
43C4×C4⋊Q8—21no—
44C4×#127—21no—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C4×#117C2211yescharacteristic
48#2077C2214yes—
49#2076C2214yes—
50#2075C2214yes—
51#2075C2214yes—
52#2036C2214yes—
53#2036C2214yes—
54#2036C2214yes—
55#2035C2214yes—
56#1990C2212yes—
57#1988C2211yescharacteristic
58#1988C2211yescharacteristic
59#1981C2211yescharacteristic
60#1981C2211yescharacteristic
61#1956C2212yes—
62#1954C2212yes—
63#1954C2212yes—
64#1954C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1948C2211yescharacteristic
68#1947C2211yescharacteristic
69#1947C2211yescharacteristic
70#1947C2212yes—
71#1947C2212yes—
72#1873C2214yes—
73#1873C2212yes—
74#1873C2212yes—
75#1872C2214yes—
76#1872C2214yes—
77#1854C2212yes—
78#1854C2212yes—
79#1854C2214yes—
80#1852C2214yes—
81#1852C2214yes—
82#1845C2212yes—
83#1834C2212yes—
84#1825C2214yes—
85#1823C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1780C2214yes—
89C4×#107C2212yes—
90C4×#106C2212yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C22×#41C2312yes—
94#533—24no—
95#533—24no—
96#532—24no—
97#532—24no—
98C2×C4⋊Q8D411yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127D411yescharacteristic
102C2×#125—21no—
103C2×#124—22no—
104C2×#124D412yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2314yes—
109C2×#121D412yes—
110C2×#121D412yes—
111#520—44no—
112#520—44no—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4—24no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C2×#117—21no—
124C2×#117C2311yescharacteristic
125C2×#117C2314yes—
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132#485—48no—
133#484—48no—
134#431—24no—
135#431—24no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2312yes—
159C2×C4×C8C2312yes—
160#399—24no—
161#397—24no—
162#396—24no—
163#396—24no—
164#395—24no—
165#394—24no—
166#394—24no—
167#394—24no—
168#391—22no—
169#391—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#388—24no—
181#388—24no—
182#386—24no—
183#385—24no—
184#385—24no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#382—21no—
193#380—21no—
194#380C2314yes—
195#380C2314yes—
196#380C2314yes—
197#380C2314yes—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209#376—21no—
210C43—21no—
211#340C2314yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2314yes—
216#340C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223C2×#41—24no—
224C2×#41—24no—
225C2×#41—42no—
226C2×#41—42no—
227C2×#41—42no—
228C2×#41—42no—
229C22×Q8—21no—
230C22×Q8C2×D412yes—
231C22×Q8C2412yes—
232C22×D4C2×D412yes—
233C22×D4C2×D412yes—
234C22×D4C2×D414yes—
235C22×D4C2×D412yes—
236C23×C4—22no—
237C23×C4—24no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251#127—24no—
252#125—44no—
253#124—28no—
254#124—84no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262#121—28no—
263#121—28no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—48no—
267C4×Q8—44no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—48no—
273C4×D4—416no—
274C4×D4—84no—
275C4×D4—84no—
276#117—22no—
277#117—22no—
278#117—28no—
279#117—41no—
280#117—41no—
281#117—44no—
282#117—44no—
283#117—82no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4C2×D414yes—
329C2×C22⋊C4C2×D414yes—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42#4114yes—
342C2×C42#4114yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—42no—
347#107—44no—
348#107—44no—
349#106—24no—
350#106—42no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355#103—24no—
356#103—24no—
357#103—42no—
358#103—42no—
359#103—44no—
360#103—44no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363C22⋊C8—44no—
364C22⋊C8—44no—
365C22⋊C8—44no—
366C22⋊C8—44no—
367C4×C8—24no—
368C4×C8—24no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—24no—
379C24C22×D414yes—
380C24C22×D411yescharacteristic
381#41—48no—
382#41—48no—
383#41—48no—
384#41—48no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—44no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4#12314yes—
439C22×C4#12314yes—
440C22×C4C22×D414yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2511yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4H(5,𝔽2)12yes—
495C4⋊C4H(5,𝔽2)12yes—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C42—24no—
510C42—48no—
511C42—42no—
512C42—44no—
513C42—44no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42#12314yes—
519C42#12314yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23C2×#12312yes—
538C23C23×D411yescharacteristic
539Q8—48no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—84no—
545D4—48no—
546D4—416no—
547D4—48no—
548D4—48no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—168no—
596C2×C4—164no—
597C2×C4C2×#12312yes—
598C2×C4C2×#12312yes—
599C2×C4C2×#12312yes—
600C2×C4C2×#12312yes—
601C2×C4C2×#12312yes—
602C2×C4#57612yes—
603C2×C4#57712yes—
604C2×C4#57812yes—
605C2×C4#57812yes—
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4#58511yescharacteristic
611C2×C4#58511yescharacteristic
612C8—42no—
613C8—42no—
614C8—42no—
615C8—42no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22C22×#12311yescharacteristic
631C22#348411yescharacteristic
632C22#348511yescharacteristic
633C22C2×#57611yescharacteristic
634C22C2×#57711yescharacteristic
635C22C2×#57811yescharacteristic
636C22C2×#57811yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—81no—
645C4—81no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4#309714yes—
651C4#309814yes—
652C4#346011yescharacteristic
653C4#346011yescharacteristic
654C4#350611yescharacteristic
655C4#350611yescharacteristic
656C2—42no—
657C2—84no—
658C2#3515211yescharacteristic
659C2C2×#309711yescharacteristic
660C2C2×#309811yescharacteristic
661C2#6166511yescharacteristic
662C2#6166811yescharacteristic
663C2#6178911yescharacteristic
664C2#6179111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #32989C2semidirect
4. #29901C2semidirect
5. #26301C2semidirect
6. #26296C2semidirect
7. #26236C2semidirect
8. #26233C2semidirect
9. #25017C2semidirect
10. #25016C2semidirect
11. #24970C2semidirect
12. #24969C2semidirect
13. #21982C2semidirect
14. #21860C2semidirect
16. #21389C2semidirect
17. #21362C2semidirect

Maximal subgroups

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

subgroupindex
#376402
#342072
#329892
#299012
#263012
#262962
#262362
#262332
#250172
#250162
#249702
#249692
#219822
#218602
#217872
#213892
#213622