Finite Group Databasebeta

← #6629637

Subgroups — #6629637

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2#33996C211yesmaximal, characteristic
3#33105C218yesmaximal, complemented
4#33104C218yesmaximal, complemented
5#21061C214yesmaximal, complemented
6#21034C214yesmaximal, complemented
7#20911C212yesmaximal
8#20846C212yesmaximal
9#20683C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×#117C2212yes—
12C22×#117C2211yescharacteristic
13C2×#428C2214yes—
14C2×#419C2214yes—
15C2×#407C2212yes—
16#2625—24no—
17#2625C2218yes—
18C2×#400—21no—
19C2×#398—21no—
20C2×#397—22no—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#380C2211yescharacteristic
24#2274—21no—
25#2274—21no—
26#2274C2214yes—
27#2274C2214yes—
28#2273—22no—
29C4×#117—24no—
30C4×#117C2214yes—
31#1928C2218yescomplemented
32#1928C2218yescomplemented
33#1927C2218yescomplemented
34#1927C2218yescomplemented
35#1921C2218yescomplemented
36#1920C2218yescomplemented
37#1919C2218yescomplemented
38#1918C2218yescomplemented
39#1873C2218yes—
40#1872C2214yes—
41#1872C2214yes—
42#1854C2218yes—
43#1853C2214yes—
44#1852C2214yes—
45#1818C2218yes—
46#1736C2214yes—
47#1736C2214yes—
48#1728C2214yes—
49C23×Q8C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16C2314yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8C2314yes—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—28no—
58C2×#117—216no—
59C2×#117D414yes—
60C2×#117C2314yes—
61C2×#117C2314yes—
62C2×#117C2314yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42—22no—
76C22×C42C2312yes—
77#428—216no—
78C2×#107C2318yes—
79C2×#106C2318yes—
80#419—216no—
81C2×#103C2318yes—
82C2×#103C2318yes—
83#407—28no—
84C2×C22⋊C8C2314yes—
85C2×C22⋊C8C2314yes—
86#400—28no—
87#398—28no—
88#397—28no—
89#397—28no—
90#391—24no—
91#391C2318yes—
92#389—24no—
93#389C2318yes—
94#380—22no—
95#380C2314yes—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4C2314yes—
101C4×C4⋊C4C2314yes—
102C4×C4⋊C4C2314yes—
103C4×C4⋊C4C2314yes—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—24no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4—24no—
110#376—24no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C43—24no—
116#340C2318yes—
117#340C2318yes—
118#340C2318yes—
119#340C2318yes—
120#336C2318yes—
121#336C2318yes—
122C25C2×D411yescharacteristic
123C22×Q8—21no—
124C22×Q8—24no—
125C22×Q8—28no—
126C22×Q8C2412yes—
127C23×C4—28no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4#4118yes—
138C23×C4C2412yes—
139C23×C4C2411yescharacteristic
140C2×M16—28no—
141C2×M16—28no—
142C2×M16—28no—
143C22×C8C2418yes—
144C22⋊Q8—28no—
145C22⋊Q8—28no—
146C22⋊Q8—48no—
147#117—216no—
148#117—216no—
149#117—216no—
150#117—28no—
151#117—28no—
152#117—28no—
153#117—28no—
154#117—28no—
155#117#41116yes—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—84no—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2412yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2414yes—
176C2×C22⋊C4—216no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4C2×D414yes—
185C2×C22⋊C4C2×D414yes—
186C2×C42—216no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—28no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42C2×D414yes—
195C2×C42C2414yes—
196#107—216no—
197#106—216no—
198#103—216no—
199#103—216no—
200C22⋊C8—28no—
201C22⋊C8—28no—
202#95—24no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210C24—24no—
211C24—41no—
212C24—42no—
213C24—41no—
214C24#12314yes—
215C24C22×D412yes—
216C24C22×D411yescharacteristic
217C2×Q8—28no—
218C2×Q8—24no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—416no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—232no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—28no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4#12314yes—
254C22×C4#12318yes—
255C22×C4C22×D414yes—
256C22×C4C22×D414yes—
257C22×C4C22×D412yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4114yes—
260C22×C4C2×#4114yes—
261C22×C4C2×#4114yes—
262C22×C4C2×#4118yes—
263C22×C4C2511yescharacteristic
264M16—416no—
265C2×C8—28no—
266C2×C8—28no—
267C2×C8—28no—
268C4⋊C4—28no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—28no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—48no—
278C4⋊C4—44no—
279C4⋊C4—88no—
280C4⋊C4C2×#4118yes—
281C4⋊C4C2×#4118yes—
282C22⋊C4—216no—
283C22⋊C4—28no—
284C22⋊C4—28no—
285C22⋊C4—216no—
286C22⋊C4—48no—
287C22⋊C4—84no—
288C42—216no—
289C42—28no—
290C42—216no—
291C42—28no—
292C42—84no—
293C42—84no—
294C42—84no—
295C42—84no—
296C23—21no—
297C23—22no—
298C23—21no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—28no—
306C23—44no—
307C23—44no—
308C23—84no—
309C23—84no—
310C23C2×#12314yes—
311C23C2×#12312yes—
312C23#52514yes—
313C23C23×D411yescharacteristic
314Q8—48no—
315Q8—84no—
316C2×C4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—216no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—88no—
345C2×C4—88no—
346C2×C4—84no—
347C2×C4—88no—
348C2×C4—84no—
349C2×C4C2×#12314yes—
350C2×C4C2×#12312yes—
351C2×C4#52514yes—
352C2×C4#529116yes—
353C2×C4C22×#4112yes—
354C2×C4C22×#4114yes—
355C8—48no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—24no—
360C22—24no—
361C22—24no—
362C22—24no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—88no—
368C22#285314yes—
369C22C22×#12311yescharacteristic
370C22C2×#52512yes—
371C22#358511yescharacteristic
372C22#358612yes—
373C22#358611yescharacteristic
374C4—24no—
375C4—44no—
376C4—44no—
377C4—48no—
378C4—48no—
379C4—82no—
380C4—82no—
381C4—84no—
382C4#285314yes—
383C4C2×#52914yes—
384C2—24no—
385C2—44no—
386C2C2×#285311yescharacteristic
387C2#6051911yescharacteristic
388C2#6052011yescharacteristic
389C2#6056712yes—
390C2#6056912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33105C2semidirect
4. #33104C2semidirect
5. #21061C2semidirect
6. #21034C2semidirect
31. #1928C22semidirect
32. #1928C22semidirect
33. #1927C22semidirect
34. #1927C22semidirect
35. #1921C22semidirect
36. #1920C22semidirect
37. #1919C22semidirect
38. #1918C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26252
#339962
#331052
#331042
#210612
#210342
#209112
#208462
#206832