Finite Group Databasebeta

← #7538152

Subgroups — #7538152

Order 512.

1585 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#32659C214yesmaximal, complemented
4#28835C212yesmaximal, complemented
5#28687C211yesmaximal, characteristic, complemented
6#28541C212yesmaximal, complemented
7#28508C211yesmaximal, characteristic, complemented
8#26530C212yesmaximal, complemented
9#26087C212yesmaximal, complemented
10#25452C212yesmaximal, complemented
11#25068C214yesmaximal, complemented
12#24811C212yesmaximal, complemented
13#23250C211yesmaximal, characteristic, complemented
14#23099C212yesmaximal
15#22968C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#533—21no—
19C2×#498C2212yes—
20C2×#476C2211yescharacteristic
21C2×#474C2212yes—
22C2×#474C2212yes—
23C2×#459C2211yescharacteristic
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2857—21no—
27#2804—22no—
28#2580—22no—
29#2580C2214yes—
30#2571C2212yes—
31#2563—21no—
32#2563C2212yes—
33#2552C2212yes—
34#2542C2212yes—
35#2538C2212yes—
36#2487—22no—
37#2441—22no—
38#2429—22no—
39#2423—22no—
40#2366C2212yes—
41#2365C2212yes—
42#2354C2212yes—
43Q8×C4⋊C4C2212yes—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C42×D4C2211yescharacteristic
50#2094C2214yes—
51#2093C2214yes—
52#2092C2214yes—
53#2085C2214yes—
54#2084C2214yes—
55#2083C2214yes—
56#2067C2214yes—
57#2065C2214yes—
58#1990C2212yes—
59#1987C2211yescharacteristic
60#1985C2214yes—
61#1982C2211yescharacteristic
62#1972C2214yes—
63#1957C2212yes—
64#1949C2214yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1947C2211yescharacteristic
69#1943C2214yes—
70#1939C2212yes—
71#1939C2212yes—
72#1926C2214yes—
73#1925C2212yes—
74#1921C2214yes—
75#1920C2212yes—
76#1879C2214yes—
77#1879C2212yes—
78#1864C2212yes—
79#1864C2214yes—
80#1847C2214yes—
81#1845C2212yes—
82#1802C2211yescharacteristic
83#1802C2214yes—
84#1802C2211yescharacteristic
85C4×#106C2212yes—
86#1770C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#533—48no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127C2311yescharacteristic
101C2×#125—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2314yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C22×C4⋊C4—22no—
119C22×C42—21no—
120#498—44no—
121#476—24no—
122#474—28no—
123#474—44no—
124#459Q818yescomplemented
125C4×D8—44no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149#400—24no—
150#399—24no—
151#399—24no—
152#398—24no—
153#396—24no—
154#393—24no—
155#393—24no—
156#391—22no—
157#391—21no—
158#391D412yes—
159#390—22no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2314yes—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389—21no—
170#389D412yes—
171#389D412yes—
172#389C2312yes—
173#389C2314yes—
174#389C2314yes—
175#388—24no—
176#387—24no—
177#386—24no—
178#385—24no—
179#385—24no—
180#385—24no—
181#384—21no—
182#384—21no—
183#384D411yescharacteristic
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2314yes—
187#384C2312yes—
188#384C2312yes—
189#382—22no—
190#382C2314yes—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C43C2311yescharacteristic
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2312yes—
218#336C2314yes—
219#336C2312yes—
220C22×Q8—22no—
221C22×Q8—21no—
222C22×Q8C2×D412yes—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C22×D4C2411yescharacteristic
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C2×D8—42no—
233C2×D8—42no—
234C2×D8C2×Q818yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241C4⋊Q8—22no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—44no—
245C4⋊Q8—82no—
246#127—22no—
247#127C2×Q814yes—
248#125—48no—
249C22⋊Q8—48no—
250C22⋊Q8—48no—
251C22⋊Q8—84no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—48no—
255C4×Q8—48no—
256C4×Q8—82no—
257C4×D4—24no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—41no—
263C4×D4—41no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—82no—
268C4×D4—84no—
269C4×D4C2×Q814yes—
270#117—48no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—84no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—44no—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2414yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#107—28no—
326#107—42no—
327#107—44no—
328#106—24no—
329#106—42no—
330#106—44no—
331#106—44no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—44no—
335C4⋊C8C2×Q814yes—
336C4⋊C8#4114yes—
337#102—28no—
338#102—24no—
339#102—42no—
340#102—42no—
341#102—42no—
342#102—42no—
343#102—44no—
344#102C2×Q818yes—
345#97—44no—
346C4×C8—24no—
347C4×C8—24no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—24no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—21no—
361C24—22no—
362C2×Q8—24no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—81no—
370C2×Q8—81no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4C22×Q814yes—
388C2×D4C22×Q812yes—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2511yescharacteristic
420D8—48no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8C22×Q814yes—
432C2×C8C2×#4114yes—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—84no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—84no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4C22×Q812yes—
468C22⋊C4—24no—
469C22⋊C4—48no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—82no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C42—22no—
480C42—22no—
481C42—24no—
482C42—24no—
483C42—21no—
484C42—21no—
485C42—41no—
486C42—48no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—44no—
491C42AGL(1,ℤ/8)14yes—
492C42C22×Q812yes—
493C42C2×#4112yes—
494C42H(5,𝔽2)12yes—
495C422−1+412yes—
496C23—22no—
497C23—24no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23C23×D411yescharacteristic
511Q8—48no—
512Q8—84no—
513Q8—88no—
514D4—22no—
515D4—24no—
516D4—22no—
517D4—24no—
518D4—48no—
519D4—41no—
520D4—42no—
521D4—42no—
522D4—41no—
523D4—84no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—88no—
564C2×C4—168no—
565C2×C4D4×Q812yes—
566C2×C4D4×Q812yes—
567C2×C4#55012yes—
568C2×C4#55012yes—
569C2×C4C2×AGL(1,ℤ/8)12yes—
570C2×C4C2×AGL(1,ℤ/8)12yes—
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4#57612yes—
573C2×C4#57712yes—
574C2×C4C23×Q811yescharacteristic
575C2×C4C22×#4111yescharacteristic
576C2×C4C2×H(5,𝔽2)11yescharacteristic
577C2×C4C2×2−1+411yescharacteristic
578C2×C4#58511yescharacteristic
579C2×C4#58511yescharacteristic
580C8—24no—
581C8—24no—
582C8—42no—
583C8—42no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—81no—
591C22—82no—
592C22—82no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22—82no—
597C22—81no—
598C22C2×D4×Q811yescharacteristic
599C22C2×#55011yescharacteristic
600C22#348511yescharacteristic
601C22C22×AGL(1,ℤ/8)11yescharacteristic
602C22C2×#57611yescharacteristic
603C22C2×#57711yescharacteristic
604C22#358611yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—44no—
609C4—41no—
610C4—44no—
611C4—42no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4—84no—
617C4—82no—
618C4#338414yes—
619C4#339914yes—
620C4Q8×#4111yescharacteristic
621C4#350111yescharacteristic
622C4#351511yescharacteristic
623C4#354711yescharacteristic
624C2—82no—
625C2—84no—
626C2#5905211yescharacteristic
627C2C2×#338411yescharacteristic
628C2#6138111yescharacteristic
629C2C2×#339911yescharacteristic
630C2#6142011yescharacteristic
631C2#6171711yescharacteristic
632C2#6178811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59864C2semidirect
2. #36134C2semidirect
3. #32659C2semidirect
4. #28835C2semidirect
5. #28687C2semidirect
6. #28541C2semidirect
7. #28508C2semidirect
8. #26530C2semidirect
9. #26087C2semidirect
10. #25452C2semidirect
11. #25068C2semidirect
12. #24811C2semidirect
13. #23250C2semidirect
15. #22968C2semidirect
16. C4×#476C2semidirect
124. #459Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
#361342
#326592
#288352
#286872
#285412
#285082
#265302
#260872
#254522
#250682
#248112
#232502
#230992
#229682
C4×#4762
C4×#4592