Finite Group Databasebeta

← #326993

Subgroups — #326993

Order 512.

1567 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1567 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2#32956C212yesmaximal, complemented
3#27905C211yesmaximal, characteristic, complemented
4#10432C214yesmaximal, complemented
5#10235C214yesmaximal, complemented
6#8723C211yesmaximal, characteristic, complemented
7#8310C212yesmaximal, complemented
8C22×#121—22no—
9C22×C4×D4C2212yes—
10C2×#483—22no—
11C2×#476C2214yescomplemented
12#2857C2211yescharacteristic
13#2537—22no—
14#2537—24no—
15#2537C2214yes—
16C2×#394—22no—
17C2×#384C2211yescharacteristic
18C2×#380—22no—
19#2096—22no—
20#1956C2212yescomplemented
21#1864—24no—
22#1853—22no—
23#1844C2212yescomplemented
24#1802C2214yescomplemented
25#1797C2212yes—
26#1786—22no—
27C4×C22⋊C8C2211yescharacteristic
28C2×D4⋊C8C2218yescomplemented
29#1296C2214yescomplemented
30C23×D4—22no—
31C23×D4D412yes—
32C24×C4—22no—
33C2×C4⋊Q8C2311yescharacteristic
34C2×C22⋊Q8—21no—
35C2×#121—28no—
36C2×#121—28no—
37C2×#121—48no—
38C2×#121—48no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—48no—
42C2×C4×D4D414yes—
43C2×C4×D4C2314yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—22no—
49C22×C42C2311yescharacteristic
50#483—48no—
51#476—48no—
52C2×#106—24no—
53C2×#106—24no—
54C2×C4⋊C8C2314yes—
55C2×#103—24no—
56C2×#102—24no—
57C2×#102—24no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—22no—
60C2×C4×C8C2312yes—
61#394—28no—
62#394—28no—
63#391—21no—
64#389—22no—
65#384—22no—
66#384D412yes—
67#384C2311yescharacteristic
68#384C2312yes—
69#380—28no—
70#380—28no—
71#380—22no—
72C2×#95—22no—
73#340—24no—
74#336—22no—
75#336—21no—
76D4⋊C8—48no—
77C25—22no—
78C22×Q8—21no—
79C22×D4—28no—
80C22×D4—22no—
81C22×D4—24no—
82C22×D4—44no—
83C22×D4—44no—
84C22×D4—416no—
85C22×D4—84no—
86C22×D4SD1618yes—
87C22×D4C2×D414yes—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—48no—
93C23×C4—48no—
94C23×C4—44no—
95C23×C4—44no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D412yes—
98C22×C8—22no—
99C22×C8—24no—
100C4⋊Q8—24no—
101C22⋊Q8—44no—
102#121—416no—
103#121—432no—
104#121—416no—
105C4×D4—28no—
106C4×D4—28no—
107C4×D4—44no—
108C4×D4—44no—
109C4×D4—48no—
110C4×D4—48no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D414yes—
125C2×C22⋊C4—216no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—48no—
129C2×C22⋊C4—416no—
130C2×C22⋊C4—41no—
131C2×C42—24no—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D412yes—
134C2×C42C2411yescharacteristic
135#106—48no—
136#106—48no—
137C4⋊C8—28no—
138#103—48no—
139#102—48no—
140#102—48no—
141C22⋊C8—24no—
142C22⋊C8—44no—
143C4×C8—24no—
144#95—28no—
145#95—42no—
146C24—24no—
147C24—28no—
148C24—44no—
149C24—44no—
150C24—44no—
151C24—44no—
152C24—44no—
153C24—44no—
154C24#12011yescharacteristic
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—28no—
164C2×D4—416no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—416no—
168C2×D4—432no—
169C2×D4—88no—
170C2×D4—832no—
171C2×D4C2×SD1618yes—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—41no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—48no—
185C22×C4—44no—
186C22×C4—48no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—816no—
194C22×C4—816no—
195C22×C4#12011yescharacteristic
196C22×C4#12012yes—
197C22×C4C2×SD1614yes—
198C22×C4AGL(1,ℤ/8)14yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D412yes—
201C2×C8—24no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—44no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—48no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C22⋊C4—48no—
220C22⋊C4—432no—
221C22⋊C4—48no—
222C22⋊C4—82no—
223C42—24no—
224C42—22no—
225C42H(5,𝔽2)14yes—
226C23—21no—
227C23—22no—
228C23—21no—
229C23—28no—
230C23—416no—
231C23—416no—
232C23—44no—
233C23—416no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—44no—
239C23—88no—
240C23—88no—
241C23—88no—
242C23—88no—
243C23#45014yes—
244C23#45312yes—
245C23C2×#12011yescharacteristic
246Q8—84no—
247D4—28no—
248D4—28no—
249D4—416no—
250D4—44no—
251D4—44no—
252D4—832no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—28no—
258C2×C4—28no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—48no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—816no—
269C2×C4—88no—
270C2×C4—88no—
271C2×C4—88no—
272C2×C4—88no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4#45014yes—
279C2×C4#53414yes—
280C2×C4#53512yes—
281C2×C4C22×SD1612yes—
282C2×C4C2×AGL(1,ℤ/8)12yes—
283C2×C4#57514yes—
284C8—44no—
285C8—84no—
286C22—24no—
287C22—22no—
288C22—24no—
289C22—22no—
290C22—48no—
291C22—48no—
292C22—48no—
293C22—832no—
294C22—84no—
295C22—84no—
296C22—84no—
297C22—84no—
298C22#162914yes—
299C22#261311yescharacteristic
300C22C2×#45012yes—
301C22#300512yes—
302C22C2×#45311yescharacteristic
303C22#302011yescharacteristic
304C4—24no—
305C4—22no—
306C4—41no—
307C4—84no—
308C4—84no—
309C4—161no—
310C4—161no—
311C4#318714yes—
312C4#319414yes—
313C2—24no—
314C2—88no—
315C2C2×#162911yescharacteristic
316C2#1854711yescharacteristic
317C2#1875712yes—
318C2#2959612yes—
319C2#2961611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#2537C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #32956C2semidirect
3. #27905C2semidirect
4. #10432C2semidirect
5. #10235C2semidirect
6. #8723C2semidirect
7. #8310C2semidirect
11. C2×#476C22semidirect
20. #1956C22semidirect
23. #1844C22semidirect
24. #1802C22semidirect
28. C2×D4⋊C8C22semidirect
29. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
#329562
#279052
#104322
#102352
#87232
#83102