Finite Group Databasebeta

← #7538151

Subgroups — #7538151

Order 512.

1577 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59823C211yesmaximal, characteristic, complemented
2#36194C211yesmaximal, characteristic, complemented
3#32666C214yesmaximal, complemented
4#28836C212yesmaximal, complemented
5#28687C211yesmaximal, characteristic, complemented
6#28541C212yesmaximal, complemented
7#28509C211yesmaximal, characteristic, complemented
8#26533C212yesmaximal, complemented
9#26090C212yesmaximal, complemented
10#25451C212yesmaximal, complemented
11#25073C214yesmaximal, complemented
12#24812C212yesmaximal, complemented
13#23250C211yesmaximal, characteristic, complemented
14#23099C212yesmaximal
15#22969C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554—21no—
19C2×#498C2212yes—
20C2×#476C2212yes—
21C2×#476C2212yes—
22C2×#474C2211yescharacteristic
23C2×#459C2211yescharacteristic
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2857—21no—
27#2732—22no—
28#2571C2212yes—
29#2563C2212yes—
30#2552C2212yes—
31#2542C2212yes—
32#2538—21no—
33#2538C2212yes—
34#2538C2214yes—
35#2497—22no—
36#2492—22no—
37#2454—22no—
38#2439—22no—
39#2417—22no—
40#2369C2212yes—
41#2366C2212yes—
42#2355C2212yes—
43#2354C2212yes—
44D4×C4⋊C4C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48#2285—21no—
49C42×D4C2211yescharacteristic
50#2094C2214yes—
51#2094C2214yes—
52#2092C2214yes—
53#2085C2214yes—
54#2084C2214yes—
55#2084C2214yes—
56#2065C2214yes—
57#2065C2214yes—
58#1990C2212yes—
59#1988C2211yescharacteristic
60#1985C2214yes—
61#1982C2211yescharacteristic
62#1972C2214yes—
63#1957C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1948C2214yes—
67#1947C2211yescharacteristic
68#1947C2211yescharacteristic
69#1944C2214yes—
70#1939C2212yes—
71#1939C2212yes—
72#1926C2214yes—
73#1925C2212yes—
74#1920C2214yes—
75#1918C2212yes—
76#1879C2214yes—
77#1879C2212yes—
78#1862C2212yes—
79#1862C2214yes—
80#1847C2214yes—
81#1844C2212yes—
82#1804C2214yes—
83#1802C2211yescharacteristic
84#1802C2211yescharacteristic
85C4×#107C2212yes—
86#1770C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#554—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—24no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2314yes—
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C22×C4⋊C4—22no—
118C22×C42—21no—
119#498—44no—
120#476—28no—
121#476—44no—
122#474—24no—
123#459Q818yescomplemented
124C4×D8—44no—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#399—24no—
150#398—24no—
151#396—24no—
152#396—24no—
153#396—24no—
154#393—24no—
155#391—22no—
156#391—21no—
157#391—24no—
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—21no—
169#389D412yes—
170#389D412yes—
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#388—24no—
175#387—24no—
176#386—24no—
177#385—24no—
178#384—21no—
179#384—21no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#384C2314yes—
186#382—22no—
187#382C2314yes—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—24no—
193#380—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—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#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#336C2312yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220C22×Q8—21no—
221C22×Q8—22no—
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—22no—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—82no—
245C4⋊Q8—82no—
246#127—22no—
247#127C2×Q814yes—
248#125—48no—
249C22⋊Q8—48no—
250C22⋊Q8—416no—
251C22⋊Q8—84no—
252C22⋊Q8—84no—
253C4×Q8—42no—
254C4×Q8—42no—
255C4×Q8—44no—
256C4×D4—22no—
257C4×D4—24no—
258C4×D4—22no—
259C4×D4—24no—
260C4×D4—22no—
261C4×D4—41no—
262C4×D4—41no—
263C4×D4—44no—
264C4×D4—44no—
265C4×D4—48no—
266C4×D4—44no—
267C4×D4—82no—
268C4×D4C2×Q814yes—
269#117—84no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2414yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—24no—
325#107—42no—
326#107—42no—
327#106—28no—
328#106—44no—
329#106—44no—
330#106—44no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—44no—
334C4⋊C8C2×Q814yes—
335C4⋊C8#4114yes—
336#102—24no—
337#102—28no—
338#102—42no—
339#102—42no—
340#102—42no—
341#102—42no—
342#102—44no—
343#102C2×Q818yes—
344#97—44no—
345C4×C8—24no—
346C4×C8—24no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95C2×D414yes—
358#95C2×D414yes—
359C24—21no—
360C24—22no—
361C2×Q8—24no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—44no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4C22×Q814yes—
387C2×D4C22×Q812yes—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—48no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4C22×D414yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D414yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2511yescharacteristic
419D8—48no—
420C2×C8—22no—
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×C8C22×Q814yes—
431C2×C8C2×#4114yes—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4C22×Q812yes—
467C22⋊C4—24no—
468C22⋊C4—42no—
469C22⋊C4—48no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—82no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—24no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—44no—
489C42—84no—
490C42AGL(1,ℤ/8)14yes—
491C42C22×Q812yes—
492C42C2×#4112yes—
493C42H(5,𝔽2)12yes—
494C422−1+412yes—
495C23—22no—
496C23—24no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23C23×D411yescharacteristic
510Q8—84no—
511Q8—84no—
512Q8—164no—
513D4—22no—
514D4—24no—
515D4—22no—
516D4—24no—
517D4—48no—
518D4—41no—
519D4—42no—
520D4—42no—
521D4—41no—
522D4—84no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—88no—
563C2×C4—168no—
564C2×C4D4×Q812yes—
565C2×C4D4×Q812yes—
566C2×C4#55012yes—
567C2×C4#55012yes—
568C2×C4C2×AGL(1,ℤ/8)12yes—
569C2×C4C2×AGL(1,ℤ/8)12yes—
570C2×C4C2×AGL(1,ℤ/8)12yes—
571C2×C4#57712yes—
572C2×C4#57712yes—
573C2×C4C23×Q811yescharacteristic
574C2×C4C22×#4111yescharacteristic
575C2×C4C2×H(5,𝔽2)11yescharacteristic
576C2×C4C2×2−1+411yescharacteristic
577C2×C4#58511yescharacteristic
578C2×C4#58511yescharacteristic
579C8—24no—
580C8—24no—
581C8—42no—
582C8—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—81no—
590C22—82no—
591C22—81no—
592C22—82no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—82no—
597C22C2×D4×Q811yescharacteristic
598C22C2×#55011yescharacteristic
599C22#348511yescharacteristic
600C22C22×AGL(1,ℤ/8)11yescharacteristic
601C22C2×#57711yescharacteristic
602C22C2×#57711yescharacteristic
603C22#358511yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—44no—
608C4—41no—
609C4—44no—
610C4—42no—
611C4—81no—
612C4—82no—
613C4—81no—
614C4—82no—
615C4—82no—
616C4—84no—
617C4#338414yes—
618C4#339714yes—
619C4Q8×#4111yescharacteristic
620C4#350111yescharacteristic
621C4#351511yescharacteristic
622C4#354711yescharacteristic
623C2—82no—
624C2—84no—
625C2#5905211yescharacteristic
626C2C2×#338411yescharacteristic
627C2#6138211yescharacteristic
628C2C2×#339711yescharacteristic
629C2#6142111yescharacteristic
630C2#6172011yescharacteristic
631C2#6179211yescharacteristic
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. #59823C2semidirect
2. #36194C2semidirect
3. #32666C2semidirect
4. #28836C2semidirect
5. #28687C2semidirect
6. #28541C2semidirect
7. #28509C2semidirect
8. #26533C2semidirect
9. #26090C2semidirect
10. #25451C2semidirect
11. #25073C2semidirect
12. #24812C2semidirect
13. #23250C2semidirect
15. #22969C2semidirect
16. C4×#474C2semidirect
123. #459Q8semidirect

Maximal subgroups

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

subgroupindex
#598232
#361942
#326662
#288362
#286872
#285412
#285092
#265332
#260902
#254512
#250732
#248122
#232502
#230992
#229692
C4×#4742
C4×#4592