Finite Group Databasebeta

← #6497110

Subgroups — #6497110

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#33067C211yesmaximal, characteristic, complemented
5#32954C212yesmaximal
6#29996C211yesmaximal, characteristic, complemented
7#29937C212yesmaximal
8#29866C211yesmaximal, characteristic, complemented
9#27522C214yesmaximal
10#27458C214yesmaximal
11#27362C214yesmaximal, complemented
12#27298C214yesmaximal, complemented
13#22112C212yesmaximal, complemented
14#21922C211yesmaximal, characteristic
15C4×#475C212yesmaximal, complemented
16C2×#541—21no—
17C22×C4×Q8C2211yescharacteristic
18C2×#475C2212yes—
19C2×#475C2212yes—
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2642—21no—
24#2621—21no—
25#2506—22no—
26#2506C2214yes—
27#2496—22no—
28#2496C2214yes—
29#2355—22no—
30#2355C2212yes—
31Q8×C4⋊C4C2212yes—
32#2345—22no—
33#2345C2212yes—
34Q8×C22⋊C4C2212yes—
35#2327—21no—
36C4×C4⋊Q8C2212yes—
37#2295—21no—
38#2295—21no—
39#2295C2211yescharacteristic
40#2295C2211yescharacteristic
41#2285—21no—
42#2285C2211yescharacteristic
43C42×Q8C2211yescharacteristic
44C42×Q8C2211yescharacteristic
45C4×#117—21no—
46C4×#117C2211yescharacteristic
47#2100C2214yes—
48#2088C2214yes—
49#2084C2214yes—
50#2084C2214yes—
51#2077C2214yes—
52#2077C2214yes—
53#2072C2214yes—
54#2066C2214yes—
55#1988C2212yes—
56#1988C2212yes—
57#1948C2212yes—
58#1948C2212yes—
59#1948C2214yes—
60#1941C2214yes—
61#1933C2214yes—
62#1927C2214yes—
63#1923C2214yes—
64#1920C2214yes—
65#1919C2214yes—
66#1919C2214yes—
67#1916C2214yes—
68#1865C2214yes—
69#1865C2214yes—
70#1865C2214yes—
71#1865C2214yes—
72#1853C2214yes—
73#1842C2211yescharacteristic
74#1831C2211yescharacteristic
75#1803C2212yes—
76#1803C2212yes—
77#1803C2214yes—
78#1797C2212yes—
79#1787C2214yes—
80C4×#106C2214yes—
81C4×C4⋊C8C2212yes—
82C4×#103C2214yes—
83C23×Q8C2311yescharacteristic
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—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8—22no—
97C2×C4×Q8C2311yescharacteristic
98C2×C4×Q8C2311yescharacteristic
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×Q8C2311yescharacteristic
103C2×#117—21no—
104C2×#117D411yescharacteristic
105C2×#117D411yescharacteristic
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110#475—28no—
111#475—28no—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C4×C8C2314yes—
125#400—24no—
126#400—24no—
127#398—24no—
128#397—24no—
129#391—22no—
130#391—22no—
131#391—22no—
132#391—22no—
133#391C2312yes—
134#391C2312yes—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389C2314yes—
141#389C2314yes—
142#389C2312yes—
143#389C2314yes—
144#389C2312yes—
145#387—22no—
146#387—22no—
147#387—24no—
148#387—24no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#384—22no—
154#384—22no—
155#384C2312yes—
156#384C2314yes—
157#384C2312yes—
158#382—22no—
159#382—21no—
160#382—21no—
161#380—21no—
162#380—22no—
163#380—22no—
164#380—22no—
165#380C2312yes—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2311yescharacteristic
182C4×C4⋊C4C2311yescharacteristic
183C4×C4⋊C4C2312yes—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4C2312yes—
188#376—21no—
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—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—21no—
210C22×Q8—22no—
211C22×Q8—24no—
212C22×Q8—42no—
213C22×Q8C2411yescharacteristic
214C22×Q8C2411yescharacteristic
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2411yescharacteristic
220C22×C8C2414yes—
221C22×C8C2414yes—
222C4⋊Q8—24no—
223C4⋊Q8—24no—
224C4⋊Q8—44no—
225C4⋊Q8—44no—
226#126—416no—
227#125—416no—
228#125—44no—
229C22⋊Q8—48no—
230C4×Q8—28no—
231C4×Q8—28no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—42no—
241C4×Q8—42no—
242C4×Q8—48no—
243C4×Q8#4114yes—
244C4×Q8#4114yes—
245#117—42no—
246#117—44no—
247#117SD1618yes—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
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⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2411yescharacteristic
277C2×C4⋊C4C2411yescharacteristic
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C42—28no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2412yes—
306C2×C42C2412yes—
307C2×C42C2411yescharacteristic
308#106—28no—
309#106—28no—
310#106—44no—
311#106—44no—
312C4⋊C8—24no—
313C4⋊C8—24no—
314C4⋊C8—44no—
315#103—28no—
316#103—28no—
317#103—28no—
318#103—44no—
319C22⋊C8—44no—
320C4×C8—44no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—21no—
328#95—21no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341C24C22×D411yescharacteristic
342C2×Q8—24no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—44no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8C2×#4112yes—
356C2×Q8C2×#4112yes—
357C2×Q82−1+414yes—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—82no—
378C22×C4C2×SD1614yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2511yescharacteristic
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4C2×SD1618yes—
426C4⋊C4C2×#4112yes—
427C4⋊C4C2×#4112yes—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—48no—
431C22⋊C4C2×SD1618yes—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—42no—
447C42—42no—
448C42—44no—
449C42—44no—
450C42—48no—
451C42—42no—
452C42—42no—
453C42—82no—
454C42—82no—
455C42C2×SD1614yes—
456C42C2×SD1614yes—
457C42C2×#4112yes—
458C42C2×#4112yes—
459C23—22no—
460C23—21no—
461C23—22no—
462C23—21no—
463C23C22×SD1612yes—
464C23C23×D411yescharacteristic
465Q8—22no—
466Q8—24no—
467Q8—24no—
468Q8—48no—
469Q8—41no—
470Q8—41no—
471Q8—88no—
472Q8#54114yescomplemented
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4#52512yes—
518C2×C4#52512yes—
519C2×C4C22×SD1614yes—
520C2×C4C22×SD1614yes—
521C2×C4C22×SD1612yes—
522C2×C4C22×SD1612yes—
523C2×C4#57512yes—
524C2×C4C22×#4111yescharacteristic
525C2×C4C22×#4111yescharacteristic
526C2×C4C2×2−1+412yes—
527C2×C4#58512yes—
528C8—44no—
529C8—44no—
530C22—22no—
531C22—24no—
532C22—41no—
533C22—42no—
534C22—41no—
535C22C2×#52511yescharacteristic
536C22#348912yes—
537C22C23×SD1611yescharacteristic
538C22C2×#57511yescharacteristic
539C22#358612yes—
540C4—22no—
541C4—24no—
542C4—24no—
543C4—22no—
544C4—24no—
545C4—41no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—41no—
551C4—42no—
552C4—84no—
553C4—84no—
554C4—82no—
555C4—82no—
556C4#310314yes—
557C4C2×#54111yescharacteristic
558C4#345511yescharacteristic
559C4#349512yes—
560C2—42no—
561C2#3531011yescharacteristic
562C2C2×#310311yescharacteristic
563C2#6056911yescharacteristic
564C2#6209912yes—
565C2#6211812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #33196C2semidirect
4. #33067C2semidirect
6. #29996C2semidirect
8. #29866C2semidirect
11. #27362C2semidirect
12. #27298C2semidirect
13. #22112C2semidirect
15. C4×#475C2semidirect
472. Q8#541semidirect

Maximal subgroups

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

subgroupindex
#353192
Q8×#1172
#331962
#330672
#329542
#299962
#299372
#298662
#275222
#274582
#273622
#272982
#221122
#219222
C4×#4752