Finite Group Databasebeta

← #6493482

Subgroups — #6493482

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic, complemented
3#33161C211yesmaximal, characteristic, complemented
4#33069C211yesmaximal, characteristic, complemented
5#32903C212yesmaximal, complemented
6#29999C211yesmaximal, characteristic, complemented
7#29920C212yesmaximal, complemented
8#29796C211yesmaximal, characteristic, complemented
9#27407C214yesmaximal, complemented
10#27207C214yesmaximal, complemented
11#26179C214yesmaximal, complemented
12#25165C214yesmaximal, complemented
13#22153C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic, complemented
15#21505C212yesmaximal, complemented
16C2×#540—21no—
17C22×C4×Q8C2211yescharacteristic
18C2×#460C2212yes—
19C2×#460C2212yes—
20C2×#460C2214yes—
21C2×#448C2214yes—
22C2×#440C2214yes—
23#2867C2211yescharacteristic
24#2859C2211yescharacteristic
25#2856C2212yes—
26#2641—21no—
27#2619—21no—
28#2445—22no—
29#2445C2214yes—
30#2428—22no—
31#2428C2214yes—
32#2355—22no—
33#2355C2212yes—
34Q8×C4⋊C4C2212yes—
35#2345—22no—
36#2345C2212yes—
37Q8×C22⋊C4C2212yes—
38#2328—21no—
39C4×#127C2212yes—
40#2295—21no—
41#2295—21no—
42#2295C2211yescharacteristic
43#2295C2211yescharacteristic
44#2285—21no—
45#2285C2211yescharacteristic
46C42×Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48C4×#117—21no—
49C4×#117C2211yescharacteristic
50#2093C2214yes—
51#2087C2214yes—
52#2071C2214yes—
53#2068C2214yes—
54#2041C2214yes—
55#2033C2214yes—
56#2033C2214yes—
57#2014C2214yes—
58#1982C2212yes—
59#1982C2214yes—
60#1982C2212yes—
61#1973C2214yes—
62#1947C2212yes—
63#1947C2212yes—
64#1936C2214yes—
65#1925C2214yes—
66#1924C2214yes—
67#1921C2214yes—
68#1917C2214yes—
69#1880C2214yes—
70#1880C2214yes—
71#1880C2214yes—
72#1877C2214yes—
73#1874C2214yes—
74#1863C2214yes—
75#1862C2214yes—
76#1842C2211yescharacteristic
77#1831C2211yescharacteristic
78#1803C2212yes—
79#1803C2212yes—
80#1797C2212yes—
81#1767C2212yes—
82#1762C2214yes—
83C23×Q8C2311yescharacteristic
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86#540—48no—
87C2×#127—21no—
88C2×#127—21no—
89C2×#127—22no—
90C2×#127C2312yes—
91C2×#127C2314yes—
92C2×#125—21no—
93C2×#123—24no—
94C2×#121—22no—
95C2×#121—22no—
96C2×#121—24no—
97C2×#121C2314yes—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4C2314yes—
111C2×#117—21no—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118#460—28no—
119#460—28no—
120#460—48no—
121#448—48no—
122#440—48no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2314yes—
138#394—24no—
139#394—24no—
140#394—24no—
141#392—24no—
142#391—22no—
143#391—22no—
144#391C2312yes—
145#391C2312yes—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2312yes—
157#389C2312yes—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#386—24no—
165#386—24no—
166#384—22no—
167#384C2312yes—
168#384C2312yes—
169#382—22no—
170#382—21no—
171#382—21no—
172#380—21no—
173#380—22no—
174#380—22no—
175#380C2312yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2311yescharacteristic
189C4×C4⋊C4C2311yescharacteristic
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4D412yes—
194C4×C22⋊C4C2312yes—
195#376—21no—
196#376—21no—
197#376D411yescharacteristic
198#376D411yescharacteristic
199C43—21no—
200C43—21no—
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—21no—
213C22×Q8—24no—
214C22×Q8—42no—
215C22×Q8C2411yescharacteristic
216C22×Q8C2411yescharacteristic
217C22×D4—24no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4C2414yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C2×SD16—28no—
230C2×SD16—28no—
231C2×SD16—28no—
232C2×SD16—28no—
233C2×SD16—44no—
234C2×SD16—44no—
235C22×C8C2414yes—
236C22×C8C2414yes—
237#127—24no—
238#127—24no—
239#127—44no—
240#127—44no—
241#127—44no—
242#127—44no—
243#127—84no—
244#125—44no—
245#123—416no—
246#121—416no—
247#121—48no—
248#121—44no—
249#121—44no—
250#121—84no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8#4114yes—
264C4×Q8#4114yes—
265C4×D4—48no—
266C4×D4—44no—
267C4×D4—44no—
268C4×D4—84no—
269#117—24no—
270#117—24no—
271#117—44no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C42—28no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330#107—44no—
331#107—44no—
332C4⋊C8—24no—
333C4⋊C8—24no—
334C4⋊C8—44no—
335#103—28no—
336#103—44no—
337#102—28no—
338#102—28no—
339#102—44no—
340#102—44no—
341#102—44no—
342#102—44no—
343C22⋊C8—44no—
344#97—44no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—21no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361C24—24no—
362C24C22×D411yescharacteristic
363C2×Q8—24no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—44no—
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8C2×#4112yes—
374C2×Q8C2×#4112yes—
375C2×Q8H(5,𝔽2)14yes—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—82no—
419C22×C4AGL(1,ℤ/8)14yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C2511yescharacteristic
430SD16—416no—
431SD16—88no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4C2×#4112yes—
463C4⋊C4C2×#4112yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—42no—
489C42—42no—
490C42—44no—
491C42—44no—
492C42—42no—
493C42—42no—
494C42—82no—
495C42—82no—
496C42#13714yes—
497C42C2×#4112yes—
498C42C2×#4112yes—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—21no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23C2×AGL(1,ℤ/8)12yes—
511C23C23×D411yescharacteristic
512Q8—22no—
513Q8—24no—
514Q8—24no—
515Q8—48no—
516Q8—41no—
517Q8—41no—
518Q8#54014yescomplemented
519D4—44no—
520D4—44no—
521D4—44no—
522D4—44no—
523D4—88no—
524D4—88no—
525D4—168no—
526D4—168no—
527D4—168no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—82no—
569C2×C4—84no—
570C2×C4—168no—
571C2×C4#52512yes—
572C2×C4#52512yes—
573C2×C4C2×AGL(1,ℤ/8)12yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4C2×#13714yes—
576C2×C4C2×#13712yes—
577C2×C4C22×#4111yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C2×H(5,𝔽2)12yes—
580C2×C4#58512yes—
581C8—44no—
582C8—44no—
583C22—24no—
584C22—22no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22C2×#52511yescharacteristic
596C22#348812yes—
597C22C22×AGL(1,ℤ/8)11yescharacteristic
598C22C22×#13711yescharacteristic
599C22#358512yes—
600C4—24no—
601C4—22no—
602C4—22no—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—84no—
615C4#311214yes—
616C4C2×#54011yescharacteristic
617C4#345511yescharacteristic
618C4#349412yes—
619C2—42no—
620C2—164no—
621C2#3526111yescharacteristic
622C2C2×#311211yescharacteristic
623C2#6056811yescharacteristic
624C2#6196312yes—
625C2#6196512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35278C2
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
2. Q8×#117C2semidirect
3. #33161C2semidirect
4. #33069C2semidirect
5. #32903C2semidirect
6. #29999C2semidirect
7. #29920C2semidirect
8. #29796C2semidirect
9. #27407C2semidirect
10. #27207C2semidirect
11. #26179C2semidirect
12. #25165C2semidirect
13. #22153C2semidirect
14. #21926C2semidirect
15. #21505C2semidirect
518. Q8#540semidirect

Maximal subgroups

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

subgroupindex
#352782
Q8×#1172
#331612
#330692
#329032
#299992
#299202
#297962
#274072
#272072
#261792
#251652
#221532
#219262
#215052