Finite Group Databasebeta

← #318137

Subgroups — #318137

Order 512.

592 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 162 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#28274C212yesmaximal
2#28196C212yesmaximal
3C2×#1591C211yesmaximal, characteristic
4C2×#1591C211yesmaximal, characteristic
5#10438C214yesmaximal
6#8815C211yesmaximal, characteristic
7#8655C212yesmaximal
8#8651C212yesmaximal
9C2×#446C2212yes—
10C2×#446C2212yes—
11C2×C4×Q16C2211yescharacteristic
12C2×C4×Q16C2211yescharacteristic
13#2582C2214yes—
14#2086—22no—
15#2066—24no—
16#2056—22no—
17#1971—22no—
18#1944—22no—
19#1935—24no—
20#1844C2212yes—
21#1834C2212yes—
22#1834C2212yes—
23#1803C2214yes—
24#1803C2214yes—
25#1591—24no—
26#1591—24no—
27C2×#335C2211yescharacteristic
28C2×#335C2211yescharacteristic
29C2×Q8⋊C8C2212yes—
30C2×Q8⋊C8C2212yes—
31C2×C82C2211yescharacteristic
32#1296C2214yes—
33C22×Q16D411yescharacteristic
34C22×Q16D411yescharacteristic
35C2×C4×Q8C2312yes—
36C2×C4×Q8C2312yes—
37#446—24no—
38#446—24no—
39#446—24no—
40#446—24no—
41C4×Q16—22no—
42C4×Q16—22no—
43C4×Q16—22no—
44C4×Q16—22no—
45C2×#107D411yescharacteristic
46C2×#107D411yescharacteristic
47C2×#106—22no—
48C2×#106—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—24no—
54C2×#103—24no—
55C2×#103D412yes—
56C2×#103D412yes—
57C2×#97—22no—
58C2×#97—22no—
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#398—44no—
63#391—24no—
64#391—24no—
65#389—24no—
66#389—24no—
67#384C2314yes—
68#340—22no—
69#340—22no—
70#340—24no—
71#340—24no—
72#340—42no—
73#335—22no—
74#335—22no—
75Q8⋊C8—24no—
76Q8⋊C8—24no—
77C82D414yes—
78C22×Q8C2×D412yes—
79C22×Q8C2×D412yes—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—24no—
83C2×Q16—42no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4C2×D412yes—
106C2×C4⋊C4C2×D412yes—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42C2411yescharacteristic
110#107—22no—
111#107—22no—
112#106—44no—
113#106—82no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—44no—
119C4⋊C8—44no—
120#103—24no—
121#103—24no—
122#103—48no—
123#103—84no—
124#97—44no—
125#97—44no—
126C4×C8—21no—
127C4×C8C2×D412yes—
128C4×C8C2×D412yes—
129C4×C8C2×D412yes—
130C4×C8#4112yes—
131C4×C8#4112yes—
132#95—44no—
133#95—44no—
134#95—44no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—24no—
142C2×Q8—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—44no—
148C22×C4C22×D411yescharacteristic
149C22×C4C22×D411yescharacteristic
150C22×C4H(5,𝔽2)11yescharacteristic
151Q16—44no—
152Q16—44no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—44no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8#12112yes—
164C2×C8#12112yes—
165C2×C8#12112yes—
166C2×C8#12112yes—
167C2×C8#13714yes—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—84no—
181C4⋊C4—164no—
182C42—42no—
183C42—42no—
184C42—42no—
185C42—42no—
186C42C22×D411yescharacteristic
187C42C2×#4111yescharacteristic
188C42C2×#4111yescharacteristic
189C422−1+411yescharacteristic
190C23D4211yescharacteristic
191Q8—44no—
192Q8—44no—
193Q8—42no—
194Q8—42no—
195Q8—42no—
196Q8—42no—
197C2×C4—22no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—84no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4C2×#12111yescharacteristic
211C2×C4C2×#12111yescharacteristic
212C2×C4#53711yescharacteristic
213C2×C4#54711yescharacteristic
214C2×C4#54711yescharacteristic
215C2×C4#55511yescharacteristic
216C2×C4C2×#13712yes—
217C2×C4#57512yes—
218C2×C4#57712yes—
219C2×C4#57812yes—
220C8—22no—
221C8—22no—
222C8—22no—
223C8—42no—
224C8—82no—
225C8—82no—
226C8#47014yes—
227C22#253711yescharacteristic
228C22#327712yes—
229C22#327812yes—
230C22#329012yes—
231C4—21no—
232C4—21no—
233C4—82no—
234C4—82no—
235C4—82no—
236C4—82no—
237C4—164no—
238C4C2×#47011yescharacteristic
239C4#305411yescharacteristic
240C4#305911yescharacteristic
241C4#306011yescharacteristic
242C4#335212yes—
243C4#335612yes—
244C2#2032014yes—
245C2#2823211yescharacteristic
246C2#2823411yescharacteristic
247C2#2824511yescharacteristic
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
#282742
#281962
C2×#15912
#104382
#88152
#86552
#86512