Finite Group Databasebeta

← #305441

Subgroups — #305441

Order 512.

608 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 158 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
1#26022C212yesmaximal
2#22525C212yesmaximal
3#10676C211yesmaximal, characteristic
4#10601C212yesmaximal
5#10349C211yesmaximal, characteristic
6#8883C212yesmaximal
7#8848C212yesmaximal
8#8655C212yesmaximal
9#8451C211yesmaximal, characteristic
10C2×Q82C2211yescharacteristic
11C2×#446C2212yes—
12#2551C2211yescharacteristic
13#2355C2212yes—
14#2066—22no—
15#2024—22no—
16#1993—22no—
17#1990C2212yes—
18#1988—22no—
19#1988—22no—
20#1987C2212yes—
21#1956C2211yescharacteristic
22#1948C2212yes—
23#1939C2212yes—
24#1919—22no—
25#1845C2211yescharacteristic
26#1834C2212yes—
27#1805—22no—
28#1803C2212yes—
29#1803C2212yes—
30#1763—22no—
31C4×#103C2212yes—
32C2×#335C2212yes—
33C2×#333C2211yescharacteristic
34C2×Q8⋊C8C2212yes—
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1273C2211yescharacteristic
39Q82—24no—
40Q82—41no—
41Q82Q814yescomplemented
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4×Q8—22no—
47C2×C4×Q8C2312yes—
48#446—24no—
49#446—42no—
50C2×#107—41no—
51C2×#106—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#103—22no—
57C2×#103—22no—
58C2×#103—41no—
59C2×#103D412yes—
60C2×#103D412yes—
61C2×#97—22no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2312yes—
64#395—41no—
65#391—22no—
66#389—22no—
67#387—42no—
68#384C2311yescharacteristic
69#384C2312yes—
70#382—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4C2312yes—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340D412yes—
84#340D412yes—
85#336—22no—
86#335—42no—
87#333—41no—
88Q8⋊C8—24no—
89C22×Q8—21no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8C2×D412yes—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8C2×D412yes—
97C22×C8C2×D412yes—
98C4⋊Q8—28no—
99C4⋊Q8—22no—
100C4⋊Q8—21no—
101C4⋊Q8—44no—
102C4⋊Q8—44no—
103C4⋊Q8—44no—
104C4⋊Q8—44no—
105C4⋊Q8—82no—
106C4⋊Q8C2×Q812yes—
107C4×Q8—22no—
108C4×Q8—22no—
109C4×Q8—28no—
110C4×Q8—44no—
111C4×Q8C2×Q814yes—
112C4×Q8#4114yes—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D412yes—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—41no—
128C2×C42—42no—
129C2×C42C2411yescharacteristic
130#107—82no—
131#106—44no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—44no—
137C4⋊C8—82no—
138#103—24no—
139#103—42no—
140#103—44no—
141#103—82no—
142#103—82no—
143#97—44no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147#95—22no—
148#95—42no—
149#95—42no—
150#95—42no—
151#95—41no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8—41no—
160C2×Q8—84no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8—161no—
164C2×Q8—161no—
165C2×Q8—161no—
166C2×Q8C22⋊Q814yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4C22×D412yes—
175C22×C4H(5,𝔽2)11yescharacteristic
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—24no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—48no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—84no—
207C4⋊C4—81no—
208C4⋊C4—82no—
209C4⋊C4C22⋊Q814yes—
210C4⋊C4#13714yes—
211C42—24no—
212C42—24no—
213C42—44no—
214C42—42no—
215C42—42no—
216C42—162no—
217C42C22×Q811yescharacteristic
218C42C2×#4112yes—
219C42H(5,𝔽2)11yescharacteristic
220C23D4211yescharacteristic
221Q8—24no—
222Q8—24no—
223Q8—42no—
224Q8—42no—
225Q8—82no—
226Q8—161no—
227Q8—164no—
228Q8—164no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4C2×C22⋊Q812yes—
252C2×C4#54311yescharacteristic
253C2×C4#54612yes—
254C2×C4#55111yescharacteristic
255C2×C4C2×#13712yes—
256C2×C4#57512yes—
257C2×C4#57712yes—
258C2×C4#57812yes—
259C8—42no—
260C8—42no—
261C8—42no—
262C8—82no—
263C8—82no—
264C22#243711yescharacteristic
265C22#326612yes—
266C22#327712yes—
267C22#328911yescharacteristic
268C22#329211yescharacteristic
269C4—21no—
270C4—21no—
271C4—44no—
272C4—44no—
273C4—44no—
274C4—82no—
275C4—82no—
276C4—161no—
277C4—162no—
278C4#307812yes—
279C4#308212yes—
280C4#332112yes—
281C4#332812yes—
282C2#2004011yescharacteristic
283C2#2018412yes—
284C2#2031911yescharacteristic
285C2#2597112yes—
286C2#2598211yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
41. Q82Q8semidirect

Maximal subgroups

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

subgroupindex
#260222
#225252
#106762
#106012
#103492
#88832
#88482
#86552
#84512