Finite Group Databasebeta

← #345650

Subgroups — #345650

Order 512.

512 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 146 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#26508C212yesmaximal
2#26445C212yesmaximal
3#10601C212yesmaximal
4#10593C212yesmaximal
5#10580C212yesmaximal
6#9839C211yesmaximal, characteristic
7#9600C211yesmaximal, characteristic
8#9600C211yesmaximal, characteristic
9#9066C212yesmaximal
10C2×#494C2211yescharacteristic
11#2552C2212yes—
12#2369C2212yes—
13#2099—22no—
14#2088—22no—
15#2086—22no—
16#2059—22no—
17#1990C2212yes—
18#1990C2212yes—
19#1989—22no—
20#1987C2212yes—
21#1987C2212yes—
22#1948—22no—
23#1948C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1939C2212yes—
27#1920—22no—
28#1919—22no—
29#1844C2211yescharacteristic
30C4×#106C2211yescharacteristic
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#333C2212yes—
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C2×C4⋊Q8C2312yes—
41C2×#125—22no—
42#494—41no—
43#494Q814yes—
44C2×#107—22no—
45C2×#106—22no—
46C2×#106—41no—
47C2×#106D412yes—
48C2×#106D412yes—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#398—42no—
57#391—42no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389—22no—
62#384—22no—
63#384C2312yes—
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#340D411yescharacteristic
79#340D411yescharacteristic
80#340D411yescharacteristic
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—82no—
91C2×Q16—82no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8C2×Q814yes—
103#125—44no—
104#125—82no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D412yes—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—42no—
121C2×C42C2411yescharacteristic
122#107—82no—
123#106—24no—
124#106—42no—
125#106—44no—
126#106—82no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—44no—
132C4⋊C8—82no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8C2×Q812yes—
139#95—22no—
140#95—22no—
141#95—42no—
142#95—42no—
143#95—42no—
144C2×Q8—24no—
145C2×Q8—42no—
146C2×Q8—82no—
147C2×Q8—82no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C22×D411yescharacteristic
155C22×C4C22×D411yescharacteristic
156C22×C4H(5,𝔽2)11yescharacteristic
157Q16—88no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—82no—
193C4⋊C4—162no—
194C4⋊C4—162no—
195C4⋊C4#13714yes—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—42no—
200C42—162no—
201C42C22×Q811yescharacteristic
202C42H(5,𝔽2)11yescharacteristic
203C422−1+411yescharacteristic
204C422−1+411yescharacteristic
205C23D4211yescharacteristic
206Q8—84no—
207Q8—162no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—24no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#53711yescharacteristic
229C2×C4#53711yescharacteristic
230C2×C4#54311yescharacteristic
231C2×C4D4×Q811yescharacteristic
232C2×C4D4×Q811yescharacteristic
233C2×C4#56311yescharacteristic
234C2×C4C2×#13712yes—
235C2×C4#57512yes—
236C2×C4#57712yes—
237C2×C4#57812yes—
238C8—24no—
239C8—41no—
240C8—41no—
241C8—41no—
242C8—41no—
243C8—82no—
244C8—82no—
245C22#268311yescharacteristic
246C22#326612yes—
247C22#326912yes—
248C22#329111yescharacteristic
249C22#329211yescharacteristic
250C4—21no—
251C4—21no—
252C4—44no—
253C4—84no—
254C4—82no—
255C4—82no—
256C4—82no—
257C4—162no—
258C4#325511yescharacteristic
259C4#325611yescharacteristic
260C4#326111yescharacteristic
261C4#326311yescharacteristic
262C4#338712yes—
263C4#339012yes—
264C2#2005012yes—
265C2#2010612yes—
266C2#3124611yescharacteristic
267C2#3124911yescharacteristic
268C2#3125411yescharacteristic
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
#265082
#264452
#106012
#105932
#105802
#98392
#96002
#90662