Finite Group Databasebeta

← #6497111

Subgroups — #6497111

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#33069C211yesmaximal, characteristic, complemented
5#32957C212yesmaximal
6#29999C211yesmaximal, characteristic, complemented
7#29939C212yesmaximal
8#29869C211yesmaximal, characteristic, complemented
9#27524C214yesmaximal
10#27460C214yesmaximal
11#27364C214yesmaximal, complemented
12#27300C214yesmaximal, complemented
13#22113C212yesmaximal, complemented
14#21922C211yesmaximal, characteristic
15C4×#477C212yesmaximal, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#477C2212yes—
18C2×#477C2212yes—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21#2857C2212yes—
22#2647—21no—
23#2629—21no—
24#2507C2214yes—
25#2497C2214yes—
26#2446—22no—
27#2429—22no—
28#2355—22no—
29#2355C2212yes—
30Q8×C4⋊C4C2212yes—
31#2345—22no—
32#2345C2212yes—
33Q8×C22⋊C4C2212yes—
34#2330—21no—
35#2314—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#2087C2214yes—
49#2083C2214yes—
50#2083C2214yes—
51#2076C2214yes—
52#2076C2214yes—
53#2072C2214yes—
54#2068C2214yes—
55#1987C2212yes—
56#1987C2212yes—
57#1947C2212yes—
58#1947C2212yes—
59#1947C2214yes—
60#1942C2214yes—
61#1933C2214yes—
62#1928C2214yes—
63#1923C2214yes—
64#1921C2214yes—
65#1921C2214yes—
66#1921C2214yes—
67#1916C2214yes—
68#1863C2214yes—
69#1863C2214yes—
70#1863C2214yes—
71#1863C2214yes—
72#1852C2214yes—
73#1842C2211yescharacteristic
74#1831C2211yescharacteristic
75#1803C2212yes—
76#1803C2212yes—
77#1803C2214yes—
78#1797C2212yes—
79#1787C2214yes—
80C4×#107C2214yes—
81C4×C4⋊C8C2212yes—
82C4×#103C2214yes—
83C23×Q8C2311yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×#125—21no—
87C2×C22⋊Q8—22no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×C4×Q8—22no—
93C2×C4×Q8C2311yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2311yescharacteristic
98C2×C4×Q8C2311yescharacteristic
99C2×#117—21no—
100C2×#117D411yescharacteristic
101C2×#117D411yescharacteristic
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106#477—28no—
107#477—28no—
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×C22⋊C8C2314yes—
120C2×C4×C8C2314yes—
121#400—24no—
122#398—24no—
123#398—24no—
124#396—24no—
125#391—22no—
126#391—22no—
127#391—22no—
128#391—22no—
129#391C2312yes—
130#391C2312yes—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—24no—
135#389—22no—
136#389—22no—
137#389C2314yes—
138#389C2314yes—
139#389C2312yes—
140#389C2314yes—
141#389C2312yes—
142#388—24no—
143#387—24no—
144#387—22no—
145#387—22no—
146#387—24no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#384—22no—
152#384—21no—
153#384—21no—
154#384C2312yes—
155#384C2314yes—
156#384C2312yes—
157#382—22no—
158#382—21no—
159#382—21no—
160#380—22no—
161#380—21no—
162#380—22no—
163#380—22no—
164#380C2312yes—
165#380C2314yes—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2311yescharacteristic
181C4×C4⋊C4C2311yescharacteristic
182C4×C4⋊C4C2312yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4C2312yes—
187#376—21no—
188#376—21no—
189#376D411yescharacteristic
190#376D411yescharacteristic
191C43—21no—
192C43—21no—
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2314yes—
205#336C2314yes—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8C2411yescharacteristic
213C22×Q8C2411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
219C22×C8C2414yes—
220C22×C8C2414yes—
221C4⋊Q8—24no—
222C4⋊Q8—24no—
223C4⋊Q8—84no—
224#125—44no—
225C22⋊Q8—84no—
226C4×Q8—28no—
227C4×Q8—28no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—42no—
237C4×Q8—42no—
238C4×Q8—84no—
239C4×Q8#4114yes—
240C4×Q8#4114yes—
241#117—42no—
242#117—44no—
243#117Q1618yes—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2411yescharacteristic
273C2×C4⋊C4C2411yescharacteristic
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D412yes—
278C2×C42—28no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2412yes—
302C2×C42C2412yes—
303C2×C42C2411yescharacteristic
304#107—28no—
305#107—28no—
306#107—44no—
307#107—44no—
308C4⋊C8—24no—
309C4⋊C8—24no—
310C4⋊C8—44no—
311#103—28no—
312#103—28no—
313#103—28no—
314#103—44no—
315C22⋊C8—44no—
316C4×C8—44no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—21no—
324#95—21no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337C24C22×D411yescharacteristic
338C2×Q8—24no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—48no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8C2×#4112yes—
352C2×Q8C2×#4112yes—
353C2×Q82−1+414yes—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—82no—
374C22×C4C2×Q1614yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2511yescharacteristic
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4C2×Q1618yes—
422C4⋊C4C2×#4112yes—
423C4⋊C4C2×#4112yes—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—84no—
427C22⋊C4C2×Q1618yes—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—21no—
437C42—21no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—42no—
443C42—42no—
444C42—44no—
445C42—44no—
446C42—42no—
447C42—42no—
448C42—82no—
449C42—82no—
450C42—84no—
451C42C2×Q1614yes—
452C42C2×Q1614yes—
453C42C2×#4112yes—
454C42C2×#4112yes—
455C23—22no—
456C23—21no—
457C23—22no—
458C23—21no—
459C23C22×Q1612yes—
460C23C23×D411yescharacteristic
461Q8—22no—
462Q8—24no—
463Q8—24no—
464Q8—48no—
465Q8—41no—
466Q8—41no—
467Q8—88no—
468Q8#54114yes—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—28no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4#52512yes—
514C2×C4#52512yes—
515C2×C4C22×Q1614yes—
516C2×C4C22×Q1614yes—
517C2×C4C22×Q1612yes—
518C2×C4C22×Q1612yes—
519C2×C4#57512yes—
520C2×C4C22×#4111yescharacteristic
521C2×C4C22×#4111yescharacteristic
522C2×C4C2×2−1+412yes—
523C2×C4#58512yes—
524C8—44no—
525C8—44no—
526C22—22no—
527C22—24no—
528C22—41no—
529C22—42no—
530C22—41no—
531C22C2×#52511yescharacteristic
532C22#348912yes—
533C22C23×Q1611yescharacteristic
534C22C2×#57511yescharacteristic
535C22#358512yes—
536C4—22no—
537C4—24no—
538C4—24no—
539C4—22no—
540C4—24no—
541C4—41no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—41no—
547C4—42no—
548C4—84no—
549C4—84no—
550C4—82no—
551C4—82no—
552C4#310414yes—
553C4C2×#54111yescharacteristic
554C4#345511yescharacteristic
555C4#349512yes—
556C2—42no—
557C2#3531011yescharacteristic
558C2C2×#310411yescharacteristic
559C2#6056811yescharacteristic
560C2#6210012yes—
561C2#6211712yes—
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. #33198C2semidirect
4. #33069C2semidirect
6. #29999C2semidirect
8. #29869C2semidirect
11. #27364C2semidirect
12. #27300C2semidirect
13. #22113C2semidirect
15. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#352842
Q8×#1172
#331982
#330692
#329572
#299992
#299392
#298692
#275242
#274602
#273642
#273002
#221132
#219222
C4×#4772