Finite Group Databasebeta

← #6497115

Subgroups — #6497115

Order 512.

1559 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic, complemented
2#34125C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#33069C211yesmaximal, characteristic, complemented
5#32955C212yesmaximal
6#29998C211yesmaximal, characteristic, complemented
7#29941C212yesmaximal, complemented
8#29864C211yesmaximal, characteristic, complemented
9#27527C214yesmaximal
10#27463C214yesmaximal, complemented
11#27367C214yesmaximal, complemented
12#27303C214yesmaximal, complemented
13#22110C212yesmaximal, complemented
14#21922C211yesmaximal, characteristic, complemented
15C4×#475C212yesmaximal, complemented
16C2×#541—21no—
17C2×C4×#41C2211yescharacteristic
18C2×#475C2212yes—
19C2×#474C2212yes—
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2642—21no—
24#2621—21no—
25#2506—22no—
26#2506C2214yes—
27#2496—22no—
28#2496C2214yes—
29#2357—22no—
30#2357C2212yes—
31#2351C2212yes—
32#2347—22no—
33#2347C2212yes—
34#2341C2212yes—
35#2327—21no—
36C4×C4⋊Q8C2212yes—
37#2293—21no—
38#2293—21no—
39#2293C2211yescharacteristic
40#2293C2211yescharacteristic
41#2283—21no—
42#2283C2211yescharacteristic
43C42×Q8C2211yescharacteristic
44C42×Q8C2211yescharacteristic
45C4×#117—21no—
46C4×#117C2211yescharacteristic
47#2101C2214yes—
48#2088C2214yes—
49#2085C2214yes—
50#2083C2214yes—
51#2077C2214yes—
52#2076C2214yes—
53#2071C2214yes—
54#2065C2214yes—
55#1988C2212yes—
56#1988C2212yes—
57#1948C2212yes—
58#1948C2214yes—
59#1947C2212yes—
60#1942C2214yes—
61#1934C2214yes—
62#1925C2214yes—
63#1923C2214yes—
64#1921C2214yes—
65#1919C2214yes—
66#1918C2214yes—
67#1916C2214yes—
68#1865C2214yes—
69#1864C2214yes—
70#1863C2214yes—
71#1863C2214yes—
72#1854C2214yes—
73#1842C2211yescharacteristic
74#1831C2211yescharacteristic
75#1803C2212yes—
76#1803C2214yes—
77#1802C2212yes—
78#1797C2212yes—
79#1788C2214yes—
80C4×#106C2214yes—
81C4×C4⋊C8C2212yes—
82C4×#103C2214yes—
83C22×#41C2311yescharacteristic
84#541—48no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×#126—24no—
89C2×#125—24no—
90C2×#125—21no—
91C2×C22⋊Q8—22no—
92C4×#41—24no—
93C4×#41—24no—
94C2×C4×Q8—22no—
95C2×C4×Q8C2311yescharacteristic
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—21no—
106C2×#117D411yescharacteristic
107C2×#117D411yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111#475—28no—
112#474—28no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8C2314yes—
125C2×C4×C8C2314yes—
126#400—24no—
127#400—24no—
128#398—24no—
129#397—24no—
130#391—22no—
131#391—22no—
132#390—22no—
133#390—22no—
134#390C2312yes—
135#390C2312yes—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389C2314yes—
142#389C2314yes—
143#389C2312yes—
144#389C2314yes—
145#389C2312yes—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#387—24no—
153#387—24no—
154#384—22no—
155#384C2312yes—
156#384C2314yes—
157#383—21no—
158#383—22no—
159#382—21no—
160#382—22no—
161#382C2312yes—
162#381—21no—
163#381—22no—
164#380—22no—
165#380—22no—
166#380C2312yes—
167#380C2314yes—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2311yescharacteristic
181C4×C4⋊C4C2312yes—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4D412yes—
188C4×C22⋊C4C2312yes—
189#376—21no—
190#376D411yescharacteristic
191#376D411yescharacteristic
192C43—21no—
193C43—21no—
194C43C2311yescharacteristic
195C43C2311yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206#336C2314yes—
207C2×#41—22no—
208C2×#41—22no—
209C2×#41—24no—
210C2×#41—42no—
211C22×Q8—22no—
212C22×Q8C2411yescharacteristic
213C22×D4—21no—
214C22×D4C2411yescharacteristic
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2411yescharacteristic
221C22×C8C2414yes—
222C22×C8C2414yes—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—44no—
226C4⋊Q8—44no—
227#126—416no—
228#125—44no—
229#125—416no—
230C22⋊Q8—48no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—42no—
237C4×Q8—42no—
238C4×Q8—48no—
239C4×Q8#4114yes—
240C4×D4—28no—
241C4×D4—28no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4#4114yes—
246#117—28no—
247#117—22no—
248#117—22no—
249#117—24no—
250#117—24no—
251#117—44no—
252#117—44no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2411yescharacteristic
274C2×C4⋊C4C2411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D412yes—
286C2×C42—28no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2412yes—
309C2×C42C2412yes—
310C2×C42C2411yescharacteristic
311#107—28no—
312#107—44no—
313#106—28no—
314#106—44no—
315C4⋊C8—24no—
316C4⋊C8—24no—
317C4⋊C8—44no—
318#103—28no—
319#103—28no—
320#102—28no—
321#102—44no—
322C22⋊C8—44no—
323C4×C8—44no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—21no—
332#95—21no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341#95C2×D412yes—
342C24—21no—
343C24C22×D411yescharacteristic
344#41—48no—
345#41—44no—
346#41—44no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8C2×#4112yes—
353C2×Q82−1+414yes—
354C2×D4—24no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—44no—
359C2×D4C2×#4112yes—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—82no—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C2511yescharacteristic
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4C2×#4112yes—
426C4⋊C4C2×#4112yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C42—24no—
439C42—24no—
440C42—22no—
441C42—24no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—42no—
452C42—42no—
453C42—44no—
454C42—44no—
455C42—42no—
456C42—42no—
457C42—48no—
458C42—82no—
459C42—82no—
460C42#13514yes—
461C42AGL(1,ℤ/8)14yes—
462C42C2×#4112yes—
463C42C2×#4112yes—
464C23—22no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23C23×D411yescharacteristic
475Q8—22no—
476Q8—24no—
477Q8—88no—
478Q8#54114yescomplemented
479D4—24no—
480D4—48no—
481D4—41no—
482D4—41no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—84no—
528C2×C4#52512yes—
529C2×C4#52512yes—
530C2×C4C2×#13514yes—
531C2×C4C2×#13512yes—
532C2×C4C2×AGL(1,ℤ/8)14yes—
533C2×C4C2×AGL(1,ℤ/8)12yes—
534C2×C4C22×#4111yescharacteristic
535C2×C4C22×#4111yescharacteristic
536C2×C4C2×2−1+412yes—
537C2×C4#58512yes—
538C8—44no—
539C8—44no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—82no—
549C22—82no—
550C22C2×#52511yescharacteristic
551C22#348912yes—
552C22C22×#13511yescharacteristic
553C22C22×AGL(1,ℤ/8)11yescharacteristic
554C22#358612yes—
555C4—24no—
556C4—22no—
557C4—24no—
558C4—24no—
559C4—41no—
560C4—42no—
561C4—42no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—84no—
567C4—82no—
568C4—82no—
569C4—84no—
570C4#310614yes—
571C4C2×#54111yescharacteristic
572C4#345511yescharacteristic
573C4#349512yes—
574C2—81no—
575C2—82no—
576C2#3531011yescharacteristic
577C2C2×#310611yescharacteristic
578C2#6056911yescharacteristic
579C2#6210112yes—
580C2#6211112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
1. #35319C2semidirect
3. #33196C2semidirect
4. #33069C2semidirect
6. #29998C2semidirect
7. #29941C2semidirect
8. #29864C2semidirect
10. #27463C2semidirect
11. #27367C2semidirect
12. #27303C2semidirect
13. #22110C2semidirect
14. #21922C2semidirect
15. C4×#475C2semidirect
478. Q8#541semidirect

Maximal subgroups

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

subgroupindex
#353192
#341252
#331962
#330692
#329552
#299982
#299412
#298642
#275272
#274632
#273672
#273032
#221102
#219222
C4×#4752