Finite Group Databasebeta

← #4868347

Subgroups — #4868347

Order 512.

2306 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C212yesmaximal
2#17229C211yesmaximal, characteristic, complemented
3#17168C211yesmaximal, characteristic, complemented
4#17160C212yesmaximal, complemented
5#16697C214yesmaximal, complemented
6#16591C214yesmaximal, complemented
7#16582C214yesmaximal, complemented
8#16391C214yesmaximal
9#16107C214yesmaximal, complemented
10#14892C214yesmaximal, complemented
11#14341C211yesmaximal, characteristic
12C22×#117C2212yes—
13C23×C22⋊C4C2211yescharacteristic
14#2482—216no—
15C2×#399C2212yes—
16C2×#399C2212yes—
17C2×#397C2212yes—
18C2×#397C2212yes—
19C2×#397C2214yes—
20C2×#388C2214yes—
21C2×#388C2214yes—
22C2×#385C2214yes—
23C2×#381C2212yes—
24C2×#379C2214yes—
25C2×#379C2212yes—
26#1445C2214yescomplemented
27#1444C2214yescomplemented
28#1443C2214yescomplemented
29#1443C2214yescomplemented
30#1442C2214yescomplemented
31#1442C2214yescomplemented
32#1442C2214yescomplemented
33#1442C2214yescomplemented
34#1441C2214yescomplemented
35#1441C2214yescomplemented
36#1441C2214yescomplemented
37#1441C2214yescomplemented
38#1440C2211yescharacteristic
39#1440C2211yescharacteristic
40#1440C2214yes—
41#1440C2212yes—
42#1440C2214yes—
43#1440C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1436C2214yescomplemented
54#1436C2214yescomplemented
55#1436C2214yescomplemented
56#1436C2214yescomplemented
57C22⋊#95C2214yes—
58C26C2311yescharacteristic
59C24×C4C2312yes—
60C2×#117—28no—
61C2×#117—44no—
62C2×#117—44no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C22⋊C4—44no—
70C22×C22⋊C4D418yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2311yescharacteristic
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2311yescharacteristic
84C22×C42C2314yes—
85#399—28no—
86#399—28no—
87#397—28no—
88#397—28no—
89#397—48no—
90#388—216no—
91#388—216no—
92#385—216no—
93#381—28no—
94#379—216no—
95#379—44no—
96C2×#95C2312yes—
97C2×#95C2312yes—
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×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C25—24no—
120C25—44no—
121C25—44no—
122C25—44no—
123C25C2×D414yes—
124C25C2411yescharacteristic
125C25C2411yescharacteristic
126C25C2411yescharacteristic
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4C2×D418yes—
132C23×C4C2414yes—
133C23×C4C2414yes—
134C23×C4C2414yes—
135C23×C4C2414yes—
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2412yes—
139C23×C4C2412yes—
140#117—432no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
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×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—416no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4C2×D414yes—
193C2×C22⋊C4C2×D418yes—
194C2×C22⋊C4C2×D414yes—
195C2×C42—28no—
196C2×C42—44no—
197C2×C42—44no—
198#95—28no—
199#95—24no—
200#95—24no—
201#95—28no—
202#95—28no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221C24—24no—
222C24—24no—
223C24—24no—
224C24—22no—
225C24—22no—
226C24—24no—
227C24—24no—
228C24—24no—
229C24—24no—
230C24—24no—
231C24—24no—
232C24—28no—
233C24—48no—
234C24—416no—
235C24—416no—
236C24—84no—
237C24—88no—
238C24—84no—
239C24—88no—
240C24—88no—
241C24—88no—
242C24—88no—
243C24C22×D412yes—
244C24C22×D412yes—
245C24C22×D412yes—
246C24H(5,𝔽2)14yes—
247C24C2511yescharacteristic
248C22×C4—216no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
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—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
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—24no—
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—416no—
300C22×C4—48no—
301C22×C4—416no—
302C22×C4—48no—
303C22×C4C22×D414yes—
304C22×C4C22×D414yes—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—88no—
308C4⋊C4—88no—
309C4⋊C4—88no—
310C4⋊C4—88no—
311C22⋊C4—44no—
312C22⋊C4—48no—
313C22⋊C4—44no—
314C22⋊C4—416no—
315C22⋊C4—416no—
316C22⋊C4—416no—
317C22⋊C4—416no—
318C22⋊C4—88no—
319C22⋊C4—88no—
320C22⋊C4—88no—
321C22⋊C4—84no—
322C22⋊C4—88no—
323C22⋊C4—84no—
324C42—416no—
325C23—24no—
326C23—28no—
327C23—28no—
328C23—24no—
329C23—28no—
330C23—24no—
331C23—22no—
332C23—22no—
333C23—24no—
334C23—24no—
335C23—28no—
336C23—28no—
337C23—416no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23—42no—
343C23—44no—
344C23—42no—
345C23—44no—
346C23—44no—
347C23—42no—
348C23—44no—
349C23—42no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—816no—
359C23—832no—
360C23—832no—
361C23—832no—
362C23#53414yes—
363C23C23×D411yescharacteristic
364C23C2×H(5,𝔽2)12yes—
365C23#58514yes—
366C23#58514yes—
367C23#58514yes—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—28no—
371C2×C4—28no—
372C2×C4—432no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
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—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
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—
421C22—24no—
422C22—28no—
423C22—48no—
424C22—48no—
425C22—48no—
426C22—42no—
427C22—42no—
428C22—44no—
429C22—44no—
430C22—48no—
431C22—48no—
432C22—832no—
433C22—84no—
434C22—84no—
435C22—84no—
436C22—84no—
437C22C2×#53411yescharacteristic
438C22#344912yes—
439C22#348514yes—
440C22#353314yes—
441C22#353312yes—
442C22#353314yes—
443C22#353312yes—
444C22#356614yes—
445C22#356614yes—
446C22#356614yes—
447C22#356814yes—
448C4—44no—
449C4—44no—
450C4—84no—
451C4—84no—
452C4—84no—
453C4—84no—
454C4—84no—
455C4—84no—
456C2—44no—
457C2—82no—
458C2—82no—
459C2#3695512yes—
460C2#3745411yescharacteristic
461C2#4141414yes—
462C2#5665514yes—
463C2#5759214yes—
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
2. #17229C2semidirect
3. #17168C2semidirect
4. #17160C2semidirect
5. #16697C2semidirect
6. #16591C2semidirect
7. #16582C2semidirect
9. #16107C2semidirect
10. #14892C2semidirect
26. #1445C22semidirect
27. #1444C22semidirect
28. #1443C22semidirect
29. #1443C22semidirect
30. #1442C22semidirect
31. #1442C22semidirect
32. #1442C22semidirect
33. #1442C22semidirect
34. #1441C22semidirect
35. #1441C22semidirect
36. #1441C22semidirect
37. #1441C22semidirect
53. #1436C22semidirect
54. #1436C22semidirect
55. #1436C22semidirect
56. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
#172292
#171682
#171602
#166972
#165912
#165822
#163912
#161072
#148922
#143412