Finite Group Databasebeta

← #305304

Subgroups — #305304

Order 512.

670 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 186 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#26042C212yesmaximal
2Q8×#103C212yesmaximal
3#10685C211yesmaximal, characteristic
4#10597C212yesmaximal
5#10349C211yesmaximal, characteristic
6#8888C212yesmaximal
7#8846C212yesmaximal
8#8643C212yesmaximal
9#8451C211yesmaximal, characteristic
10C2×Q82C2211yescharacteristic
11C2×#475—22no—
12C2×C8×Q8C2212yes—
13#2551C2211yescharacteristic
14Q8×C4⋊C4C2212yes—
15#2088—22no—
16#2058—22no—
17#1990—22no—
18#1990C2212yes—
19#1987—22no—
20#1987C2212yes—
21#1956C2211yescharacteristic
22#1948C2212yes—
23#1939C2212yes—
24#1865—22no—
25#1844C2212yes—
26#1844C2211yescharacteristic
27#1803—22no—
28#1803C2212yes—
29#1803C2212yes—
30C4×#103—22no—
31C4×#103C2212yes—
32C2×#334C2212yes—
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×Q8—22no—
48C2×C4×Q8C2312yes—
49#475—48no—
50C8×Q8—24no—
51C8×Q8—42no—
52C2×#106—22no—
53C2×#106—22no—
54C2×#106—22no—
55C2×#106—41no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—41no—
63C2×#103D412yes—
64C2×#103D412yes—
65C2×C4×C8—22no—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2312yes—
68#395—41no—
69#389—22no—
70#389—22no—
71#389—42no—
72#384—22no—
73#384C2311yescharacteristic
74#384C2312yes—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2312yes—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340D412yes—
85#340D412yes—
86#336—22no—
87#334—42no—
88#333—41no—
89Q8⋊C8—24no—
90C22×Q8—21no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8C2×D412yes—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8C2×D412yes—
98C22×C8C2×D412yes—
99C4⋊Q8—28no—
100C4⋊Q8—22no—
101C4⋊Q8—21no—
102C4⋊Q8—44no—
103C4⋊Q8—44no—
104C4⋊Q8—44no—
105C4⋊Q8—44no—
106C4⋊Q8—82no—
107C4⋊Q8C2×Q812yes—
108C4×Q8—22no—
109C4×Q8—22no—
110C4×Q8—28no—
111C4×Q8—44no—
112C4×Q8—44no—
113C4×Q8—82no—
114C4×Q8C2×Q814yes—
115C4×Q8#4114yes—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—41no—
133C2×C42—42no—
134C2×C42C2411yescharacteristic
135#106—44no—
136#106—44no—
137#106—82no—
138#106—82no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—44no—
144C4⋊C8—44no—
145#103—24no—
146#103—44no—
147#103—44no—
148#103—82no—
149#103SD1618yes—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—44no—
154#95—22no—
155#95—42no—
156#95—41no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—44no—
162C2×Q8—41no—
163C2×Q8—84no—
164C2×Q8—81no—
165C2×Q8—81no—
166C2×Q8—161no—
167C2×Q8—161no—
168C2×Q8—161no—
169C2×Q8C22⋊Q814yes—
170C2×Q8C2×SD1614yes—
171C2×Q8#13714yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4C22×D412yes—
180C22×C4H(5,𝔽2)11yescharacteristic
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8C2×SD1614yes—
196C4⋊C4—24no—
197C4⋊C4—22no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—48no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—84no—
212C4⋊C4—84no—
213C4⋊C4—81no—
214C4⋊C4—82no—
215C4⋊C4—162no—
216C4⋊C4C22⋊Q814yes—
217C4⋊C4C2×SD1614yes—
218C42—24no—
219C42—24no—
220C42—44no—
221C42—42no—
222C42—42no—
223C42—84no—
224C42C22×Q811yescharacteristic
225C42C2×#4112yes—
226C42H(5,𝔽2)11yescharacteristic
227C23D4211yescharacteristic
228Q8—24no—
229Q8—24no—
230Q8—24no—
231Q8—82no—
232Q8—161no—
233Q8—164no—
234Q8—164no—
235Q8#47518yescomplemented
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4C2×C22⋊Q812yes—
259C2×C4#54311yescharacteristic
260C2×C4#54612yes—
261C2×C4#55111yescharacteristic
262C2×C4C22×SD1612yes—
263C2×C4C2×#13712yes—
264C2×C4#57712yes—
265C2×C4#57812yes—
266C8—24no—
267C8—42no—
268C8—42no—
269C8—82no—
270C8—82no—
271C22#243711yescharacteristic
272C22#327012yes—
273C22D4×SD1612yes—
274C22#328811yescharacteristic
275C22#329111yescharacteristic
276C4—28no—
277C4—21no—
278C4—21no—
279C4—44no—
280C4—44no—
281C4—82no—
282C4—82no—
283C4—161no—
284C4—162no—
285C4C2×#47512yes—
286C4#308212yes—
287C4#329812yes—
288C4#331612yes—
289C2#1980912yes—
290C2SD16211yescharacteristic
291C2#2009711yescharacteristic
292C2#2595512yes—
293C2#2598311yescharacteristic
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
235. Q8#475semidirect

Maximal subgroups

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

subgroupindex
#260422
Q8×#1032
#106852
#105972
#103492
#88882
#88462
#86432
#84512