Finite Group Databasebeta

← #324465

Subgroups — #324465

Order 512.

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

normal subgroups only

This group has 562 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#30230C211yesmaximal, characteristic
2#29150C211yesmaximal, characteristic
3#28511C211yesmaximal, characteristic
4#22238C211yesmaximal, characteristic
5#10483C214yesmaximal
6#9897C211yesmaximal, characteristic
7#9702C211yesmaximal, characteristic
8#9216C214yesmaximal
9#8278C211yesmaximal, characteristic
10C2×#477C2214yes—
11C2×C8×Q8C2212yes—
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2100—24no—
16#2095—22no—
17#2026—22no—
18#1990C2212yes—
19#1990C2212yes—
20#1984C2212yes—
21#1960—21no—
22#1956—21no—
23#1916—24no—
24#1845—21no—
25#1836—21no—
26#1831C2211yescharacteristic
27#1803C2214yes—
28#1767C2211yescharacteristic
29C4×C4⋊C8C2211yescharacteristic
30#1744C2211yescharacteristic
31C2×#335C2212yes—
32C2×#333C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8D411yescharacteristic
37C2×C4⋊Q8C2312yes—
38C2×#125—21no—
39C2×C4×Q8C2312yes—
40#477—28no—
41#477—44no—
42C8×Q8—44no—
43C2×#107—24no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8C2312yes—
46C2×C4⋊C8C2312yes—
47C2×C4⋊C8C2312yes—
48C2×#103—24no—
49C2×#103—24no—
50C2×#97—21no—
51C2×C4×C8—22no—
52C2×C4×C8—21no—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#391D412yes—
56#389—22no—
57#389—22no—
58#389—22no—
59#384—21no—
60#384—22no—
61#384—21no—
62#384D411yescharacteristic
63#384C2312yes—
64#382—21no—
65C4×C4⋊C4—22no—
66C43C2311yescharacteristic
67#340—24no—
68#340—24no—
69#340—24no—
70#336—24no—
71#336—22no—
72#336—21no—
73#336—21no—
74#336D414yes—
75#335—24no—
76#333—44no—
77C22×Q8—22no—
78C22×Q8C2×D412yes—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8C2×D412yes—
83C22×C8C2×D412yes—
84C4⋊Q8—22no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—84no—
88C4⋊Q8Q1614yes—
89#125—82no—
90C4×Q8—24no—
91C4×Q8—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C2×D412yes—
104C2×C42—22no—
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109C2×C42C2411yescharacteristic
110#107—48no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—24no—
115C4⋊C8—44no—
116C4⋊C8#4114yes—
117#103—48no—
118#103—84no—
119#97—42no—
120C4×C8—22no—
121C4×C8—22no—
122C4×C8—44no—
123C4×C8—42no—
124#95—24no—
125#95—42no—
126C2×Q8—22no—
127C2×Q8—24no—
128C2×Q8—24no—
129C2×Q8—44no—
130C2×Q8—84no—
131C2×Q8C2×Q1614yes—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4#12112yes—
137C22×C4#12112yes—
138C22×C4C22×D411yescharacteristic
139C22×C4C22×D411yescharacteristic
140C22×C4C2×#4111yescharacteristic
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—24no—
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—44no—
153C4⋊C4—28no—
154C4⋊C4—24no—
155C4⋊C4—44no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C42—22no—
167C42—24no—
168C42—42no—
169C42—42no—
170C42—42no—
171C42C2×Q1612yes—
172C42#13512yes—
173C42C2×#4112yes—
174C42H(5,𝔽2)12yes—
175C23C2×#12111yescharacteristic
176Q8—28no—
177Q8—44no—
178Q8—164no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4#46714yes—
196C2×C4#53211yescharacteristic
197C2×C4#53511yescharacteristic
198C2×C4#54612yes—
199C2×C4#55912yes—
200C2×C4C22×Q1611yescharacteristic
201C2×C4C2×#13511yescharacteristic
202C2×C4#57712yes—
203C2×C4#57814yes—
204C8—42no—
205C8—42no—
206C8—44no—
207C8—82no—
208C8—82no—
209C22#254411yescharacteristic
210C22#304412yes—
211C22C2×#46711yescharacteristic
212C22#305711yescharacteristic
213C22#306812yes—
214C4—24no—
215C4—22no—
216C4—42no—
217C4—44no—
218C4—82no—
219C4—162no—
220C4—162no—
221C4#317712yes—
222C4#322312yes—
223C4#329912yes—
224C4#330312yes—
225C2#1940512yes—
226C2#1952912yes—
227C2#2859811yescharacteristic
228C2#2862511yescharacteristic
229C2#2863511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#302302
#291502
#285112
#222382
#104832
#98972
#97022
#92162
#82782