Finite Group Databasebeta

← #770968

Subgroups — #770968

Order 512.

2646 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16247C211yesmaximal, characteristic, complemented
2#16214C212yesmaximal, complemented
3#15869C214yesmaximal, complemented
4#15821C214yesmaximal, complemented
5#15616C211yesmaximal, characteristic, complemented
6#15553C214yesmaximal, complemented
7#14726C214yesmaximal
8#14028C214yesmaximal, complemented
9#14004C214yesmaximal, complemented
10C2×#1440C211yesmaximal, characteristic
11#13775C212yesmaximal
12C23×C22⋊C4C2212yes—
13C23×C22⋊C4C2211yescharacteristic
14C2×#396C2214yes—
15C2×#396C2214yes—
16C2×#389C2214yescomplemented
17C2×#387C2214yescomplemented
18C2×#385C2214yes—
19C2×#385C2214yes—
20C2×#380C2214yes—
21C2×#379C2214yes—
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2212yes—
24C22×#95C2214yes—
25#1445C2214yescomplemented
26#1444C2214yescomplemented
27#1443C2214yescomplemented
28#1442C2214yescomplemented
29#1442C2214yescomplemented
30#1441C2214yescomplemented
31#1441C2214yescomplemented
32#1441C2214yescomplemented
33#1440C2211yescharacteristic
34#1440C2211yescharacteristic
35#1440C2214yes—
36#1440C2212yes—
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1437C2212yes—
44#1437C2212yes—
45#1437C2212yes—
46#1437C2214yes—
47#1437C2212yes—
48#1437C2214yes—
49#1436C2214yescomplemented
50#1436C2214yescomplemented
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53C4⋊#95C2214yescomplemented
54#1434C2214yescomplemented
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C26C2311yescharacteristic
58C24×C4C2314yes—
59C24×C4C2312yes—
60C22×C4⋊C4C2314yes—
61C22×C4⋊C4C2314yes—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C22⋊C4—48no—
67C22×C22⋊C4—44no—
68C22×C22⋊C4—44no—
69C22×C22⋊C4Q818yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2311yescharacteristic
80C22×C22⋊C4C2312yes—
81C22×C22⋊C4C2311yescharacteristic
82C22×C42C2314yes—
83C22×C42C2312yes—
84C22×C42C2312yes—
85#396—48no—
86#396—48no—
87#389—48no—
88#387—48no—
89#385—216no—
90#385—48no—
91#380—216no—
92#379—216no—
93C4×C22⋊C4—44no—
94C4×C22⋊C4—44no—
95C2×#95—48no—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2312yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C25—24no—
118C25—28no—
119C25—44no—
120C25—44no—
121C25C2×Q814yes—
122C25C2411yescharacteristic
123C25C2411yescharacteristic
124C25C2411yescharacteristic
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—28no—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—28no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4C2×Q818yes—
137C23×C4C2414yes—
138C23×C4C2414yes—
139C23×C4C2414yes—
140C23×C4C2412yes—
141C23×C4C2412yes—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2414yes—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—416no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—432no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4C2×Q814yes—
194C2×C22⋊C4C2×Q818yes—
195C2×C22⋊C4C2×Q814yes—
196C2×C22⋊C4#41116yes—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—44no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—42no—
218#95—42no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225C24—24no—
226C24—24no—
227C24—28no—
228C24—24no—
229C24—24no—
230C24—24no—
231C24—24no—
232C24—22no—
233C24—22no—
234C24—24no—
235C24—24no—
236C24—216no—
237C24—416no—
238C24—48no—
239C24—48no—
240C24—416no—
241C24—416no—
242C24—88no—
243C24—88no—
244C24—88no—
245C24—88no—
246C24C22×Q812yes—
247C24C22×Q812yes—
248C24C22×Q812yes—
249C24C2×#4118yes—
250C24H(5,𝔽2)14yes—
251C24C2511yescharacteristic
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—216no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—416no—
301C22×C4—416no—
302C22×C4—416no—
303C22×C4—416no—
304C22×C4—416no—
305C22×C4—88no—
306C22×C4—88no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4—88no—
310C22×C4—88no—
311C22×C4C22×Q814yes—
312C22×C4C22×Q814yes—
313C22×C4C2×#4118yes—
314C22×C4C2×#4118yes—
315C22×C4H(5,𝔽2)18yes—
316C4⋊C4—416no—
317C4⋊C4—416no—
318C4⋊C4—88no—
319C4⋊C4—88no—
320C4⋊C4—88no—
321C4⋊C4—88no—
322C22⋊C4—232no—
323C22⋊C4—48no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—48no—
327C22⋊C4—416no—
328C22⋊C4—48no—
329C22⋊C4—416no—
330C22⋊C4—88no—
331C22⋊C4—88no—
332C22⋊C4—88no—
333C22⋊C4—88no—
334C42—84no—
335C42—84no—
336C42—88no—
337C23—28no—
338C23—28no—
339C23—28no—
340C23—28no—
341C23—28no—
342C23—24no—
343C23—24no—
344C23—28no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—28no—
350C23—28no—
351C23—432no—
352C23—416no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—832no—
370C23—832no—
371C23—832no—
372C23#53318yes—
373C23#54314yes—
374C23#553116yes—
375C23C23×Q811yescharacteristic
376C23C22×#4114yes—
377C23C2×H(5,𝔽2)12yes—
378C23C2×H(5,𝔽2)14yes—
379C23#58514yes—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—816no—
433C2×C4—832no—
434C22—216no—
435C22—216no—
436C22—216no—
437C22—28no—
438C22—48no—
439C22—48no—
440C22—42no—
441C22—42no—
442C22—44no—
443C22—44no—
444C22—832no—
445C22—84no—
446C22—84no—
447C22—84no—
448C22—84no—
449C22#260818yes—
450C22C2×#53312yes—
451C22C2×#54311yescharacteristic
452C22C2×#55314yes—
453C22#350114yes—
454C22#351314yes—
455C22#351414yes—
456C22#351914yes—
457C22#352114yes—
458C22#353212yes—
459C22#353312yes—
460C22#354214yes—
461C4—48no—
462C4—48no—
463C4—48no—
464C4—44no—
465C4—44no—
466C4—84no—
467C4—84no—
468C4—84no—
469C2—28no—
470C2—82no—
471C2—82no—
472C2C2×#260811yescharacteristic
473C2#3540214yes—
474C2#3674712yes—
475C2#4406314yes—
476C2#4410414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16247C2semidirect
2. #16214C2semidirect
3. #15869C2semidirect
4. #15821C2semidirect
5. #15616C2semidirect
6. #15553C2semidirect
8. #14028C2semidirect
9. #14004C2semidirect
16. C2×#389C22semidirect
17. C2×#387C22semidirect
25. #1445C22semidirect
26. #1444C22semidirect
27. #1443C22semidirect
28. #1442C22semidirect
29. #1442C22semidirect
30. #1441C22semidirect
31. #1441C22semidirect
32. #1441C22semidirect
49. #1436C22semidirect
50. #1436C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect
53. C4⋊#95C22semidirect
54. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#162472
#162142
#158692
#158212
#156162
#155532
#147262
#140282
#140042
C2×#14402
#137752