Finite Group Databasebeta

← #318091

Subgroups — #318091

Order 512.

600 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 158 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#28196C212yesmaximal
2#28187C212yesmaximal
3C2×#1591C211yesmaximal, characteristic
4C2×C8×Q16C211yesmaximal, characteristic
5#10438C214yesmaximal
6#8817C211yesmaximal, characteristic
7#8655C212yesmaximal
8#8649C212yesmaximal
9C2×#446C2212yes—
10C2×C8×Q8C2212yes—
11C2×C4×Q16C2211yescharacteristic
12C2×C4×Q16C2211yescharacteristic
13#2582C2214yes—
14#2087—22no—
15#2086—22no—
16#2056—22no—
17#2056—22no—
18#1935—24no—
19#1933—24no—
20#1834C2212yes—
21#1834C2212yes—
22#1834C2212yes—
23#1803C2214yes—
24#1803C2214yes—
25#1591—24no—
26C8×Q16—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—
39C8×Q8—24no—
40C8×Q8—24no—
41C4×Q16—22no—
42C4×Q16—22no—
43C4×Q16—22no—
44C4×Q16—22no—
45C2×#107D411yescharacteristic
46C2×#107D411yescharacteristic
47C2×C4⋊C8—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#103—24no—
52C2×#103—24no—
53C2×#103D412yes—
54C2×#103D412yes—
55C2×#97—22no—
56C2×C4×C8—22no—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2311yescharacteristic
60#398—44no—
61#391—24no—
62#391—24no—
63#389—24no—
64#389—24no—
65#384C2314yes—
66#340—22no—
67#340—22no—
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—24no—
81C2×Q16—22no—
82C2×Q16—22no—
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—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—44no—
117C4⋊C8—44no—
118#103—24no—
119#103—24no—
120#103—48no—
121#103—48no—
122#97—44no—
123C4×C8—21no—
124C4×C8—44no—
125C4×C8C2×D412yes—
126C4×C8C2×D412yes—
127C4×C8C2×D412yes—
128C4×C8#4112yes—
129C4×C8#4112yes—
130#95—44no—
131#95—44no—
132#95—44no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—24no—
140C2×Q8—22no—
141C2×Q8—24no—
142C2×Q8#13514yes—
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—28no—
152Q16—44no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—44no—
162C2×C8—44no—
163C2×C8#12112yes—
164C2×C8#12112yes—
165C2×C8#12112yes—
166C2×C8#12112yes—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—84no—
180C4⋊C4—164no—
181C42—42no—
182C42—42no—
183C42—42no—
184C42—42no—
185C42C22×D411yescharacteristic
186C42C2×#4111yescharacteristic
187C42C2×#4111yescharacteristic
188C422−1+411yescharacteristic
189C23D4211yescharacteristic
190Q8—24no—
191Q8—24no—
192Q8—42no—
193Q8—42no—
194Q8—44no—
195Q8—42no—
196Q8—42no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4C2×#12111yescharacteristic
212C2×C4C2×#12111yescharacteristic
213C2×C4#53711yescharacteristic
214C2×C4#54711yescharacteristic
215C2×C4#54711yescharacteristic
216C2×C4#55511yescharacteristic
217C2×C4C2×#13512yes—
218C2×C4#57512yes—
219C2×C4#57812yes—
220C2×C4#57812yes—
221C8—22no—
222C8—22no—
223C8—22no—
224C8—22no—
225C8—42no—
226C8—82no—
227C8—82no—
228C22#253711yescharacteristic
229C22#327712yes—
230C22#328512yes—
231C22#329012yes—
232C4—21no—
233C4—21no—
234C4—44no—
235C4—82no—
236C4—82no—
237C4—82no—
238C4—164no—
239C4#305011yescharacteristic
240C4#305411yescharacteristic
241C4#305911yescharacteristic
242C4#305911yescharacteristic
243C4#333612yes—
244C4#335212yes—
245C2#1994114yes—
246C2#2822611yescharacteristic
247C2#2823411yescharacteristic
248C2#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
#281962
#281872
C2×#15912
C2×C8×Q162
#104382
#88172
#86552
#86492