Finite Group Databasebeta

← #345711

Subgroups — #345711

Order 512.

491 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 144 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#26511C212yesmaximal
2#26507C212yesmaximal
3#10601C212yesmaximal
4#10593C212yesmaximal
5#10578C212yesmaximal
6#9835C211yesmaximal, characteristic
7#9598C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic
9#9066C212yesmaximal
10C2×#494C2211yescharacteristic
11#2571C2212yes—
12#2369C2212yes—
13#2101—22no—
14#2101—22no—
15#2088—22no—
16#2086—22no—
17#1990C2212yes—
18#1990C2212yes—
19#1987C2212yes—
20#1987C2212yes—
21#1949—22no—
22#1949—22no—
23#1948C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1939C2212yes—
27#1920—22no—
28#1918—22no—
29#1834C2211yescharacteristic
30C4×#107C2211yescharacteristic
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#333C2212yes—
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C2×C4⋊Q8C2312yes—
41#494—41no—
42#494Q814yes—
43C2×#107—22no—
44C2×#107D412yes—
45C2×#106—22no—
46C2×#106D411yescharacteristic
47C2×#106D411yescharacteristic
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×C4×C8C2311yescharacteristic
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#400—42no—
56#391—42no—
57#389—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#384C2312yes—
62#382—22no—
63#382—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4C2312yes—
67C4×C4⋊C4C2312yes—
68#340—22no—
69#340—22no—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—41no—
79#340D411yescharacteristic
80#340D412yes—
81#340D411yescharacteristic
82#336—22no—
83#336—22no—
84#335—41no—
85#335—41no—
86#333—42no—
87#322—41no—
88C22×Q8—22no—
89C2×Q16—44no—
90C2×Q16—84no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8C2×Q814yes—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4C2×D412yes—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—42no—
116C2×C42C2411yescharacteristic
117#107—42no—
118#107—44no—
119#106—22no—
120#106—41no—
121#106—44no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—44no—
127C4⋊C8—44no—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8C2×Q812yes—
134#95—22no—
135#95—22no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—42no—
141C2×Q8—24no—
142C2×Q8—42no—
143C2×Q8—82no—
144C2×Q8—82no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4C22×D411yescharacteristic
152C22×C4C22×D411yescharacteristic
153C22×C4H(5,𝔽2)11yescharacteristic
154Q16—88no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—22no—
164C2×C8—21no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8#13512yes—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—82no—
190C4⋊C4—162no—
191C4⋊C4—162no—
192C42—42no—
193C42—42no—
194C42—42no—
195C42—42no—
196C42—162no—
197C42C22×Q811yescharacteristic
198C42H(5,𝔽2)11yescharacteristic
199C422−1+411yescharacteristic
200C422−1+411yescharacteristic
201C23D4211yescharacteristic
202Q8—84no—
203Q8—162no—
204C2×C4—22no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4#53711yescharacteristic
224C2×C4#53711yescharacteristic
225C2×C4#54311yescharacteristic
226C2×C4D4×Q811yescharacteristic
227C2×C4D4×Q811yescharacteristic
228C2×C4#56311yescharacteristic
229C2×C4C2×#13511yescharacteristic
230C2×C4#57511yescharacteristic
231C2×C4#57512yes—
232C2×C4#57811yescharacteristic
233C2×C4#57812yes—
234C2×C4#57811yescharacteristic
235C8—22no—
236C8—42no—
237C8—41no—
238C8—41no—
239C8—41no—
240C8—82no—
241C8—82no—
242C22#268311yescharacteristic
243C22#326912yes—
244C22#327711yescharacteristic
245C22#328311yescharacteristic
246C22#328812yes—
247C4—21no—
248C4—21no—
249C4—82no—
250C4—84no—
251C4—82no—
252C4—82no—
253C4—82no—
254C4—162no—
255C4#325011yescharacteristic
256C4#325611yescharacteristic
257C4#326111yescharacteristic
258C4#326111yescharacteristic
259C4#338311yescharacteristic
260C4#339011yescharacteristic
261C4#339012yes—
262C2#1986412yes—
263C2#2023212yes—
264C2#3123611yescharacteristic
265C2#3124911yescharacteristic
266C2#3125811yescharacteristic
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
#265112
#265072
#106012
#105932
#105782
#98352
#95982
#95962
#90662