Finite Group Databasebeta

← #345702

Subgroups — #345702

Order 512.

523 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 148 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#26528C212yesmaximal
2#26447C212yesmaximal
3#10602C212yesmaximal
4#10600C212yesmaximal
5#10582C212yesmaximal
6#9831C211yesmaximal, characteristic
7#9598C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic
9#9058C212yesmaximal
10C2×#500C2211yescharacteristic
11#2552C2212yes—
12#2365C2212yes—
13#2096—22no—
14#2088—22no—
15#2066—22no—
16#2059—22no—
17#1990C2212yes—
18#1989—22no—
19#1988C2212yes—
20#1981C2212yes—
21#1956C2212yes—
22#1948C2212yes—
23#1948C2212yes—
24#1939C2212yes—
25#1939C2212yes—
26#1921—22no—
27#1865—22no—
28#1834C2211yescharacteristic
29#1803—22no—
30C4×#107C2211yescharacteristic
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#332C2212yes—
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C2×C4⋊Q8C2312yes—
40C2×#125—22no—
41C2×C4×Q8—22no—
42#500—41no—
43#500Q814yes—
44C2×#107—41no—
45C2×#107D412yes—
46C2×#106—22no—
47C2×#106—22no—
48C2×#106D411yescharacteristic
49C2×#106D411yescharacteristic
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×C8C2311yescharacteristic
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#398—42no—
62#391—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#387—42no—
67#384—22no—
68#384C2312yes—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2312yes—
71C4×C4⋊C4C2312yes—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—41no—
79#340D412yes—
80#340D411yescharacteristic
81#340D411yescharacteristic
82#336—22no—
83#336—22no—
84#335—41no—
85#335—41no—
86#332—42no—
87#322—41no—
88C22×Q8—22no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8C2×Q814yes—
100#125—44no—
101#125—82no—
102C4×Q8—44no—
103C4×Q8—82no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—42no—
120C2×C42C2411yescharacteristic
121#107—24no—
122#107—44no—
123#106—22no—
124#106—41no—
125#106—82no—
126#106—82no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—82no—
132C4⋊C8—82no—
133#103—44no—
134#103—82no—
135#103—82no—
136#103—82no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8C2×Q812yes—
143#95—22no—
144#95—22no—
145#95—42no—
146#95—42no—
147#95—42no—
148C2×Q8—24no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4C22×D411yescharacteristic
161C22×C4C22×D411yescharacteristic
162C22×C4H(5,𝔽2)11yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—22no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8#13512yes—
181C4⋊C4—22no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—84no—
197C4⋊C4—84no—
198C4⋊C4—82no—
199C4⋊C4—84no—
200C4⋊C4—162no—
201C4⋊C4#13714yes—
202C42—42no—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—84no—
207C42C22×Q811yescharacteristic
208C42H(5,𝔽2)11yescharacteristic
209C422−1+411yescharacteristic
210C422−1+411yescharacteristic
211C23D4211yescharacteristic
212Q8—44no—
213Q8—44no—
214Q8—82no—
215Q8—82no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4#53711yescharacteristic
236C2×C4#53711yescharacteristic
237C2×C4#54311yescharacteristic
238C2×C4D4×Q811yescharacteristic
239C2×C4D4×Q811yescharacteristic
240C2×C4#56311yescharacteristic
241C2×C4C2×#13511yescharacteristic
242C2×C4C2×#13712yes—
243C2×C4#57511yescharacteristic
244C2×C4#57611yescharacteristic
245C2×C4#57611yescharacteristic
246C2×C4#57712yes—
247C8—22no—
248C8—42no—
249C8—41no—
250C8—41no—
251C8—41no—
252C8—82no—
253C8—82no—
254C22#268311yescharacteristic
255C22#326612yes—
256C22#327311yescharacteristic
257C22#328711yescharacteristic
258C22#328912yes—
259C4—21no—
260C4—21no—
261C4—44no—
262C4—84no—
263C4—82no—
264C4—82no—
265C4—82no—
266C4—162no—
267C4#325011yescharacteristic
268C4#325511yescharacteristic
269C4#325911yescharacteristic
270C4#326311yescharacteristic
271C4#338011yescharacteristic
272C4#338712yes—
273C4#338911yescharacteristic
274C2#1986912yes—
275C2#2017812yes—
276C2#3123811yescharacteristic
277C2#3124611yescharacteristic
278C2#3125911yescharacteristic
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

Maximal subgroups

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

subgroupindex
#265282
#264472
#106022
#106002
#105822
#98312
#95982
#95962
#90582