Finite Group Databasebeta

← #6487848

Subgroups — #6487848

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#32959C214yesmaximal
4#29972C214yesmaximal, complemented
5#26111C214yesmaximal, complemented
6#26107C214yesmaximal, complemented
7#26045C214yesmaximal, complemented
8#26041C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic, complemented
10#21981C211yesmaximal, characteristic, complemented
11#21951C211yesmaximal, characteristic, complemented
12#20732C211yesmaximal, characteristic
13#20732C211yesmaximal, characteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2212yes—
16C23×C4⋊C4—21no—
17C2×#478C2218yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2851C2214yes—
21#2646—22no—
22#2571C2212yescomplemented
23#2571C2212yescomplemented
24#2570—22no—
25#2570—22no—
26#2552C2212yescomplemented
27#2552C2212yescomplemented
28D4×C4⋊C4—22no—
29D4×C4⋊C4—22no—
30D4×C4⋊C4C2214yes—
31D4×C4⋊C4C2214yes—
32#2317—21no—
33#2317—21no—
34C2×#389C2212yes—
35C2×#385—22no—
36#2285C2211yescharacteristic
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#380—21no—
40C2×C4×C4⋊C4C2211yescharacteristic
41#2101C2218yescomplemented
42#2088C2218yescomplemented
43#2087C2218yescomplemented
44#2059C2218yescomplemented
45#1948C2214yescomplemented
46#1947C2214yescomplemented
47#1939C2214yescomplemented
48#1939C2214yescomplemented
49#1864C2218yes—
50#1862C2218yes—
51#1853C2214yes—
52#1852C2214yes—
53#1845C2212yescomplemented
54#1844C2212yescomplemented
55#1834C2212yescomplemented
56#1834C2212yescomplemented
57#1823C2212yes—
58#1823C2212yes—
59#1804C2218yes—
60#1802C2212yes—
61#1802C2212yes—
62#1802C2212yes—
63#1802C2212yes—
64#1797C2214yes—
65#1786C2212yes—
66#1786C2212yes—
67C4×#102C2214yes—
68C4×#102C2214yes—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C24×C4—22no—
72C2×#125C2314yes—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—48no—
78C2×C4×D4Q818yes—
79C2×C4×D4C2318yes—
80C2×C4×D4C2312yes—
81C2×C4×D4C2312yes—
82C2×#117—21no—
83C2×#117—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—48no—
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—22no—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98#478—416no—
99#431—24no—
100#431—24no—
101C2×#107C2318yes—
102C2×#106C2318yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2318yes—
106C2×#102C2314yes—
107C2×#102C2314yes—
108C2×#102C2314yes—
109C2×#102C2314yes—
110C2×C22⋊C8C2312yes—
111C2×C22⋊C8C2312yes—
112C2×C4×C8C2312yes—
113C2×C4×C8C2312yes—
114#400—24no—
115#400—24no—
116#398—24no—
117#398—24no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389—24no—
123#389C2318yes—
124#389C2314yes—
125#389C2314yes—
126#387—24no—
127#387—24no—
128#385—24no—
129#385—24no—
130#385—28no—
131#385—28no—
132#384—22no—
133#384—22no—
134#384—21no—
135#384—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2312yes—
141#384C2312yes—
142#382—22no—
143#382—22no—
144#382—21no—
145#382—21no—
146#382C2314yes—
147#380—21no—
148#380—21no—
149#380—24no—
150#380—24no—
151#380C2312yes—
152#380C2312yes—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4D412yes—
156C4×C4⋊C4D412yes—
157C4×C4⋊C4C2312yes—
158C4×C4⋊C4C2312yes—
159C4×C4⋊C4C2312yes—
160C4×C4⋊C4C2312yes—
161C2×#95—22no—
162C2×#95D412yes—
163#340C2314yes—
164#340C2314yes—
165#340C2314yes—
166#340C2314yes—
167#340C2318yes—
168#340C2318yes—
169#336C2318yes—
170C25—21no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4C2×Q814yes—
176C22×D4C2412yes—
177C23×C4—21no—
178C23×C4—28no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—28no—
183C23×C4—48no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—48no—
187C23×C4—84no—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
193C22×C8C2414yes—
194C22×C8C2414yes—
195#125—48no—
196C4×D4—216no—
197C4×D4—28no—
198C4×D4—28no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—48no—
202C4×D4—416no—
203C4×D4C2×Q818yes—
204C4×D4C2×Q818yes—
205#117—44no—
206#117—44no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—416no—
231C2×C4⋊C4—416no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C4⋊C4C2414yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—48no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—44no—
256C2×C42C2×D414yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×Q814yes—
260C2×C42#4114yes—
261C2×C42C2414yes—
262C2×C42C2411yescharacteristic
263C2×C42C2411yescharacteristic
264#107—48no—
265#106—48no—
266C4⋊C8—24no—
267C4⋊C8—24no—
268C4⋊C8—48no—
269#102—28no—
270#102—28no—
271#102—28no—
272#102—28no—
273C22⋊C8—24no—
274C22⋊C8—24no—
275C4×C8—24no—
276C4×C8—24no—
277#95—28no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95C2×D414yes—
287#95C2×D414yes—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—22no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24C22×D411yescharacteristic
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—216no—
300C2×D4—216no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4C22×Q814yes—
306C2×D4H(5,𝔽2)14yes—
307C2×D42−1+414yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—416no—
328C22×C4—416no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—432no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—88no—
340C22×C4—832no—
341C22×C4C22⋊Q814yes—
342C22×C4C22⋊Q814yes—
343C22×C4#13514yes—
344C22×C4AGL(1,ℤ/8)14yes—
345C22×C4C22×D414yes—
346C22×C4C22×D414yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q812yes—
353C22×C4C2×#4112yes—
354C22×C4C2511yescharacteristic
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—28no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—416no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—81no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4C22×Q814yes—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—416no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C42—28no—
397C42—44no—
398C42—44no—
399C42—44no—
400C42C22⋊Q814yes—
401C42C22⋊Q814yes—
402C42C22×Q812yes—
403C42C22×Q812yes—
404C42C2×#4112yes—
405C42C2×#4112yes—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—216no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—48no—
415C23—48no—
416C23—48no—
417C23—48no—
418C23—48no—
419C23—48no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23C2×C22⋊Q812yes—
425C23C2×#13512yes—
426C23C2×AGL(1,ℤ/8)12yes—
427C23C23×D411yescharacteristic
428D4—216no—
429D4—24no—
430D4—24no—
431D4—24no—
432D4#54418yes—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—416no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—84no—
471C2×C4—832no—
472C2×C4—88no—
473C2×C4#478116yescomplemented
474C2×C4C2×C22⋊Q812yes—
475C2×C4C2×C22⋊Q812yes—
476C2×C4C2×C22⋊Q812yes—
477C2×C4C2×C22⋊Q812yes—
478C2×C4C2×C22⋊Q812yes—
479C2×C4C2×#13512yes—
480C2×C4C2×#13512yes—
481C2×C4C2×AGL(1,ℤ/8)12yes—
482C2×C4C2×AGL(1,ℤ/8)12yes—
483C2×C4C23×Q811yescharacteristic
484C2×C4C22×#4111yescharacteristic
485C2×C4C2×H(5,𝔽2)12yes—
486C2×C4C2×2−1+412yes—
487C8—44no—
488C8—44no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—28no—
496C22—432no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22C2×#47814yes—
504C22C22×C22⋊Q811yescharacteristic
505C22#348411yescharacteristic
506C22#348611yescharacteristic
507C22C22×#13511yescharacteristic
508C22C22×AGL(1,ℤ/8)11yescharacteristic
509C22#358511yescharacteristic
510C22#358611yescharacteristic
511C4—22no—
512C4—22no—
513C4—22no—
514C4—44no—
515C4—44no—
516C4—44no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4—84no—
522C4C2×#47814yes—
523C4C2×#47814yes—
524C4C2×#54412yes—
525C4#349212yes—
526C2—24no—
527C2—48no—
528C2#3522411yescharacteristic
529C2C22×#47811yescharacteristic
530C2#6049211yescharacteristic
531C2#6156511yescharacteristic
532C2#6158211yescharacteristic
533C2#6181911yescharacteristic
534C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #35181C2semidirect
4. #29972C2semidirect
5. #26111C2semidirect
6. #26107C2semidirect
7. #26045C2semidirect
8. #26041C2semidirect
9. #21982C2semidirect
10. #21981C2semidirect
11. #21951C2semidirect
22. #2571C22semidirect
23. #2571C22semidirect
26. #2552C22semidirect
27. #2552C22semidirect
41. #2101C22semidirect
42. #2088C22semidirect
43. #2087C22semidirect
44. #2059C22semidirect
45. #1948C22semidirect
46. #1947C22semidirect
47. #1939C22semidirect
48. #1939C22semidirect
53. #1845C22semidirect
54. #1844C22semidirect
55. #1834C22semidirect
56. #1834C22semidirect
473. C2×C4#478semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×D4×C4⋊C42
#329592
#299722
#261112
#261072
#260452
#260412
#219822
#219812
#219512
#207322