Finite Group Databasebeta

← #766681

Subgroups — #766681

Order 512.

3298 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
1C22×#399C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic
3#15730C214yesmaximal, complemented
4#15406C214yesmaximal, complemented
5#15292C218yesmaximal, complemented
6#15092C212yesmaximal, complemented
7#14955C212yesmaximal, complemented
8#14127C212yesmaximal
9#14127C212yesmaximal
10#14043C214yesmaximal, complemented
11#13775C211yesmaximal, characteristic
12C23×C4⋊C4C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18C2×#399—28no—
19C2×#399C2214yes—
20C2×#399C2214yes—
21#2315—28no—
22C2×#389C2218yescomplemented
23C2×#385C2212yes—
24C2×#385C2212yes—
25C2×#385C2212yes—
26C2×#385C2212yes—
27C2×#385C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#380C2214yes—
32C2×#379C2212yes—
33C2×C4×C22⋊C4C2212yes—
34C22×#95C2211yescharacteristic
35#1444C2218yescomplemented
36#1444C2218yescomplemented
37#1442C2218yescomplemented
38#1441C2218yescomplemented
39#1441C2218yescomplemented
40#1441C2218yescomplemented
41#1439C2218yes—
42#1439C2218yes—
43#1437C2214yes—
44#1437C2214yes—
45#1436C2218yescomplemented
46C22⋊#95C2214yes—
47C22⋊#95C2214yes—
48C22⋊#95C2214yes—
49C22⋊#95C2214yes—
50C22⋊#95C2212yes—
51C26C2311yescharacteristic
52C24×C4C2312yes—
53C24×C4C2312yes—
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C22×C4⋊C4—28no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—48no—
62C22×C4⋊C4—44no—
63C22×C4⋊C4C2318yes—
64C22×C4⋊C4C2318yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2318yes—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—28no—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—28no—
73C22×C22⋊C4—48no—
74C22×C22⋊C4—44no—
75C22×C22⋊C4—44no—
76C22×C22⋊C4D418yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C42C2314yes—
89#399—216no—
90#399—216no—
91#389—416no—
92#385—28no—
93#385—28no—
94#385—28no—
95#385—28no—
96#385—216no—
97#385—48no—
98#385—48no—
99#385—48no—
100#380—216no—
101#379Q8116yes—
102C4×C22⋊C4—44no—
103C2×#95—24no—
104C2×#95C2314yes—
105C2×#95C2318yes—
106C2×#95C2318yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2318yes—
110C2×#95C2318yes—
111C2×#95C2318yes—
112C2×#95C2318yes—
113C2×#95C2318yes—
114C2×#95C2314yes—
115C25—24no—
116C25—24no—
117C25—44no—
118C25—44no—
119C25C2×D414yes—
120C25C2×Q818yes—
121C25C2411yescharacteristic
122C25C2411yescharacteristic
123C25C2411yescharacteristic
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—24no—
131C23×C4—28no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—28no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—42no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4C2×D414yes—
147C23×C4C2×D414yes—
148C23×C4C2412yes—
149C23×C4C2414yes—
150C23×C4C2412yes—
151C23×C4C2412yes—
152C23×C4C2412yes—
153C23×C4C2418yes—
154C23×C4C2414yes—
155C23×C4C2414yes—
156C2×C4⋊C4—216no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—216no—
159C2×C4⋊C4—216no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—432no—
168C2×C4⋊C4—416no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—48no—
174C2×C22⋊C4—232no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—416no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—432no—
201C2×C22⋊C4—416no—
202C2×C22⋊C4C2×D418yes—
203C2×C22⋊C4C2×D418yes—
204C2×C22⋊C4C2×Q8116yes—
205C2×C22⋊C4C2×Q8116yes—
206C2×C22⋊C4C2×Q8116yes—
207C2×C22⋊C4#4118yes—
208C2×C42—44no—
209C2×C42—48no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—216no—
214#95—216no—
215#95—48no—
216#95—48no—
217#95—44no—
218#95—48no—
219#95—48no—
220#95—48no—
221C24—24no—
222C24—24no—
223C24—22no—
224C24—22no—
225C24—24no—
226C24—24no—
227C24—22no—
228C24—22no—
229C24—24no—
230C24—24no—
231C24—28no—
232C24—28no—
233C24—232no—
234C24—48no—
235C24—48no—
236C24—416no—
237C24—416no—
238C24—816no—
239C24—88no—
240C24—88no—
241C24C4⋊Q8116yes—
242C24C22×D412yes—
243C24C22×D412yes—
244C24C22×D412yes—
245C24C22×Q814yes—
246C24C22×Q814yes—
247C24C22×Q814yes—
248C24C2×#4114yes—
249C24H(5,𝔽2)14yes—
250C24C2511yescharacteristic
251C22×C4—216no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—22no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—22no—
264C22×C4—28no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—216no—
274C22×C4—28no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—432no—
289C22×C4—48no—
290C22×C4—416no—
291C22×C4—48no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—816no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—88no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—816no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4C22×D414yes—
307C22×C4C22×D414yes—
308C22×C4C22×Q8116yes—
309C22×C4C2×#4118yes—
310C22×C4H(5,𝔽2)18yes—
311C4⋊C4—432no—
312C4⋊C4—432no—
313C4⋊C4—416no—
314C4⋊C4—88no—
315C4⋊C4—88no—
316C4⋊C4—816no—
317C22⋊C4—216no—
318C22⋊C4—216no—
319C22⋊C4—232no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C22⋊C4—416no—
323C22⋊C4—416no—
324C22⋊C4—416no—
325C22⋊C4—416no—
326C22⋊C4—48no—
327C22⋊C4—48no—
328C42—88no—
329C23—24no—
330C23—24no—
331C23—28no—
332C23—28no—
333C23—216no—
334C23—28no—
335C23—28no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—28no—
341C23—28no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—24no—
346C23—28no—
347C23—28no—
348C23—28no—
349C23—28no—
350C23—232no—
351C23—432no—
352C23—48no—
353C23—48no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—816no—
359C23—88no—
360C23—88no—
361C23—864no—
362C23#52514yes—
363C23C2×C4⋊Q814yes—
364C23C2×C4⋊Q814yes—
365C23C2×C4⋊Q814yes—
366C23#53414yes—
367C23#554116yes—
368C23C23×D411yescharacteristic
369C23C23×Q812yes—
370C23C22×#4112yes—
371C23C2×H(5,𝔽2)12yes—
372C23C2×H(5,𝔽2)14yes—
373C23#58514yes—
374C2×C4—28no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—216no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—216no—
382C2×C4—216no—
383C2×C4—216no—
384C2×C4—216no—
385C2×C4—432no—
386C2×C4—416no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—48no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—88no—
413C2×C4—816no—
414C2×C4—816no—
415C2×C4—88no—
416C2×C4—88no—
417C22—28no—
418C22—28no—
419C22—24no—
420C22—24no—
421C22—232no—
422C22—24no—
423C22—24no—
424C22—28no—
425C22—28no—
426C22—44no—
427C22—44no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—416no—
433C22—416no—
434C22—832no—
435C22#258318yes—
436C22C2×#52511yescharacteristic
437C22C22×C4⋊Q811yescharacteristic
438C22C2×#53411yescharacteristic
439C22C2×#55414yes—
440C22#348412yes—
441C22#348512yes—
442C22#350114yes—
443C22#350114yes—
444C22#351318yes—
445C22#353314yes—
446C22#354314yes—
447C4—48no—
448C4—48no—
449C4—44no—
450C4—44no—
451C4—416no—
452C4—82no—
453C4—82no—
454C4—84no—
455C2—28no—
456C2—44no—
457C2—44no—
458C2C2×#258311yescharacteristic
459C2#3535412yes—
460C2#3539112yes—
461C2#3696511yescharacteristic
462C2#3723211yescharacteristic
463C2#4312118yes—
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
3. #15730C2semidirect
4. #15406C2semidirect
5. #15292C2semidirect
6. #15092C2semidirect
7. #14955C2semidirect
10. #14043C2semidirect
22. C2×#389C22semidirect
35. #1444C22semidirect
36. #1444C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
45. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
C2×#23152
#157302
#154062
#152922
#150922
#149552
#141272
#140432
#137752