Finite Group Databasebeta

← #5228721

Subgroups — #5228721

Order 512.

2191 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17168C211yesmaximal, characteristic, complemented
2#16587C214yesmaximal, complemented
3#16575C214yesmaximal, complemented
4#16497C214yesmaximal, complemented
5#15757C214yesmaximal, complemented
6#15616C211yesmaximal, characteristic, complemented
7#15348C214yesmaximal, complemented
8#15297C214yesmaximal, complemented
9#15283C212yesmaximal
10#14657C211yesmaximal, characteristic
11#14618C212yesmaximal, complemented
12C22×#120C2211yescharacteristic
13C2×#399C2212yes—
14C2×#397C2212yes—
15C2×#397C2214yes—
16C2×#396C2214yes—
17C2×#394C2214yes—
18C2×#390C2212yes—
19C2×#389C2214yescomplemented
20C2×#388C2214yes—
21C2×#388C2214yescomplemented
22C2×#379C2212yes—
23C2×#379C2214yes—
24#1445C2214yescomplemented
25#1445C2214yescomplemented
26#1443C2214yescomplemented
27#1442C2214yescomplemented
28#1442C2214yescomplemented
29#1441C2214yescomplemented
30#1441C2214yescomplemented
31#1440C2211yescharacteristic
32#1440C2214yes—
33#1440C2211yescharacteristic
34#1440C2214yes—
35#1440C2214yes—
36#1440C2214yes—
37#1440C2214yescomplemented
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1438C2212yes—
46#1438C2214yes—
47#1438C2214yes—
48#1438C2212yes—
49#1437C2212yes—
50#1437C2214yes—
51#1437C2214yes—
52#1437C2212yes—
53#1436C2214yescomplemented
54#1436C2214yescomplemented
55#1436C2214yescomplemented
56#1434C2214yescomplemented
57C26C2311yescharacteristic
58C23×D4C2312yes—
59C23×D4C2311yescharacteristic
60C2×#120—24no—
61C2×#120—44no—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2314yes—
67C22×C22⋊C4C2312yes—
68C22×C22⋊C4C2314yes—
69C22×C22⋊C4C2314yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2311yescharacteristic
82C22×C42C2314yes—
83C22×C42C2312yes—
84#399—28no—
85#397—48no—
86#397—44no—
87#396—48no—
88#394—48no—
89#390—44no—
90#389—48no—
91#388—48no—
92#388—48no—
93#379—48no—
94#379—44no—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2312yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2312yes—
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—
115C25—22no—
116C25—24no—
117C25—44no—
118C25—44no—
119C25—44no—
120C25C2411yescharacteristic
121C25C2411yescharacteristic
122C25C2412yes—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—44no—
131C23×C4C2414yes—
132C23×C4C2414yes—
133C23×C4C2414yes—
134C23×C4C2414yes—
135C23×C4C2414yes—
136C23×C4C2411yescharacteristic
137C23×C4C2412yes—
138C23×C4C2414yes—
139#120—416no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—28no—
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—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
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—42no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
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⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4#4114yes—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42—42no—
201C2×C42—44no—
202#95—24no—
203#95—28no—
204#95—28no—
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—42no—
220#95—44no—
221#95—44no—
222C24—28no—
223C24—22no—
224C24—24no—
225C24—24no—
226C24—24no—
227C24—22no—
228C24—22no—
229C24—24no—
230C24—24no—
231C24—24no—
232C24—24no—
233C24—24no—
234C24—24no—
235C24—21no—
236C24—24no—
237C24—22no—
238C24—24no—
239C24—24no—
240C24—416no—
241C24—84no—
242C24—88no—
243C24—84no—
244C24—88no—
245C24—84no—
246C24—88no—
247C24—88no—
248C24—88no—
249C24—88no—
250C24C2×#4112yes—
251C24H(5,𝔽2)14yes—
252C24C2511yescharacteristic
253C2×D4—48no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—88no—
257C2×D4—84no—
258C2×D4—84no—
259C2×D4—88no—
260C2×D4—88no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
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—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4C2×#4114yes—
310C22×C4H(5,𝔽2)18yes—
311C4⋊C4—416no—
312C4⋊C4—84no—
313C4⋊C4—88no—
314C4⋊C4—88no—
315C4⋊C4—88no—
316C22⋊C4—44no—
317C22⋊C4—416no—
318C22⋊C4—48no—
319C22⋊C4—416no—
320C22⋊C4—88no—
321C22⋊C4—88no—
322C22⋊C4—88no—
323C22⋊C4—88no—
324C22⋊C4—88no—
325C22⋊C4—88no—
326C22⋊C4—84no—
327C22⋊C4—88no—
328C22⋊C4—88no—
329C22⋊C4—84no—
330C22⋊C4—88no—
331C42—84no—
332C42—88no—
333C23—22no—
334C23—24no—
335C23—24no—
336C23—28no—
337C23—24no—
338C23—28no—
339C23—24no—
340C23—28no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—42no—
359C23—42no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—41no—
366C23—42no—
367C23—44no—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—816no—
377C23—88no—
378C23—832no—
379C23—832no—
380C23—832no—
381C23C22×#4111yescharacteristic
382C23C2×H(5,𝔽2)12yes—
383C23C2×H(5,𝔽2)14yes—
384C23#58514yes—
385C23#58514yes—
386D4—816no—
387D4—88no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
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—
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—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C22—48no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22—42no—
443C22—44no—
444C22—48no—
445C22—48no—
446C22—832no—
447C22—84no—
448C22—84no—
449C22—84no—
450C22—84no—
451C22—84no—
452C22—84no—
453C22—84no—
454C22—82no—
455C22—81no—
456C22—84no—
457C22—84no—
458C22#349011yescharacteristic
459C22#351012yes—
460C22#351414yes—
461C22#353312yes—
462C22#353414yes—
463C22#353414yes—
464C22#353914yes—
465C22#354014yes—
466C22#354312yes—
467C22#356014yes—
468C22#356614yes—
469C4—44no—
470C4—48no—
471C4—84no—
472C4—84no—
473C4—81no—
474C4—84no—
475C4—84no—
476C4—84no—
477C2—84no—
478C2—82no—
479C2—161no—
480C2#4645711yescharacteristic
481C2#4897112yes—
482C2#5114814yes—
483C2#5448114yes—
484C2#5812114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17168C2semidirect
2. #16587C2semidirect
3. #16575C2semidirect
4. #16497C2semidirect
5. #15757C2semidirect
6. #15616C2semidirect
7. #15348C2semidirect
8. #15297C2semidirect
11. #14618C2semidirect
19. C2×#389C22semidirect
21. C2×#388C22semidirect
24. #1445C22semidirect
25. #1445C22semidirect
26. #1443C22semidirect
27. #1442C22semidirect
28. #1442C22semidirect
29. #1441C22semidirect
30. #1441C22semidirect
37. #1440C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
53. #1436C22semidirect
54. #1436C22semidirect
55. #1436C22semidirect
56. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#171682
#165872
#165752
#164972
#157572
#156162
#153482
#152972
#152832
#146572
#146182