Finite Group Databasebeta

← #345533

Subgroups — #345533

Order 512.

536 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 150 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#26527C212yesmaximal
2#26440C212yesmaximal
3#10600C212yesmaximal
4#10594C212yesmaximal
5#10577C211yesmaximal, characteristic
6#10574C211yesmaximal, characteristic
7#9819C211yesmaximal, characteristic
8#9594C212yesmaximal
9#9059C212yesmaximal
10C2×#498C2211yescharacteristic
11#2571C2211yescharacteristic
12#2551C2211yescharacteristic
13#2365C2212yes—
14#2097—22no—
15#2086—22no—
16#2066—22no—
17#2057—22no—
18#1990C2212yes—
19#1988—22no—
20#1988C2212yes—
21#1981C2212yes—
22#1956C2212yes—
23#1948C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1939C2212yes—
27#1918—22no—
28#1863—22no—
29#1844C2211yescharacteristic
30#1805—22no—
31C4×#107C2212yes—
32C2×#335C2212yes—
33C2×#332C2212yes—
34C2×C82C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1279C2212yes—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8C2312yes—
41C2×C4×Q8—22no—
42#498—41no—
43#498Q814yescomplemented
44C2×#107—22no—
45C2×#107D412yes—
46C2×#107D412yes—
47C2×#106—22no—
48C2×#106—41no—
49C2×#106—41no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×#103—22no—
58C2×C4×C8C2312yes—
59C2×C4×C8C2311yescharacteristic
60#400—41no—
61#395—41no—
62#391—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#387—42no—
67#384C2311yescharacteristic
68#384C2311yescharacteristic
69#382—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2312yes—
72C4×C4⋊C4C2312yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340D412yes—
80#340D412yes—
81#336—22no—
82#336—22no—
83#335—42no—
84#332—42no—
85C82—41no—
86C22×Q8—22no—
87C22×Q8—41no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—44no—
97C4⋊Q8—82no—
98C4⋊Q8C2×Q814yes—
99C4×Q8—44no—
100C4×Q8—82no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—42no—
117C2×C42C2411yescharacteristic
118#107—24no—
119#107—24no—
120#107—82no—
121#106—44no—
122#106—82no—
123#106—82no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—82no—
129C4⋊C8—82no—
130#103—44no—
131#103—82no—
132#103—82no—
133#103—82no—
134C4×C8—22no—
135C4×C8—21no—
136C4×C8—22no—
137C4×C8C2×Q812yes—
138#95—22no—
139#95—22no—
140#95—42no—
141#95—41no—
142#95—42no—
143#95—42no—
144#95—41no—
145C2×Q8—24no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—84no—
152C2×Q8—161no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22×D412yes—
161C22×C4H(5,𝔽2)11yescharacteristic
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8#13714yes—
177C4⋊C4—22no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—84no—
193C4⋊C4—84no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—161no—
197C4⋊C4#13514yes—
198C42—42no—
199C42—42no—
200C42—42no—
201C42—42no—
202C42—84no—
203C42C22×Q811yescharacteristic
204C42H(5,𝔽2)11yescharacteristic
205C422−1+412yes—
206C23D4211yescharacteristic
207Q8—44no—
208Q8—44no—
209Q8—82no—
210Q8—82no—
211Q8—164no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—24no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4#53712yes—
234C2×C4#54311yescharacteristic
235C2×C4D4×Q812yes—
236C2×C4#56311yescharacteristic
237C2×C4C2×#13512yes—
238C2×C4C2×#13712yes—
239C2×C4#57612yes—
240C2×C4#57712yes—
241C8—24no—
242C8—24no—
243C8—42no—
244C8—82no—
245C8—82no—
246C22#268311yescharacteristic
247C22#327812yes—
248C22#328012yes—
249C22#329111yescharacteristic
250C22#329311yescharacteristic
251C4—21no—
252C4—21no—
253C4—44no—
254C4—84no—
255C4—82no—
256C4—82no—
257C4—82no—
258C4—161no—
259C4—161no—
260C4#324812yes—
261C4#325712yes—
262C4#338112yes—
263C4#338512yes—
264C2#1990712yes—
265C2#1998711yescharacteristic
266C2#2032311yescharacteristic
267C2#3123512yes—
268C2#3125611yescharacteristic
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
43. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#265272
#264402
#106002
#105942
#105772
#105742
#98192
#95942
#90592