Finite Group Databasebeta

← #120228

Subgroups — #120228

Order 512.

460 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 134 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#21974C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic
3#8129C212yesmaximal
4#8097C212yesmaximal
5#8001C211yesmaximal, characteristic
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C2×#423C2212yes—
9C2×#423C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1939—22no—
12#1851C414yes—
13#1842—21no—
14#1831—21no—
15#1574—24no—
16#1568—22no—
17#1309—24no—
18#1303—22no—
19#1303—22no—
20#1292—21no—
21#1292—21no—
22C22×M16—22no—
23C22×M16C2×C412yes—
24C2×#117—21no—
25C22×C42C2311yescharacteristic
26#432—22no—
27#432—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31#431—22no—
32#423—22no—
33#423—22no—
34#423—22no—
35#423—22no—
36#423—22no—
37#423D414yes—
38#423Q814yes—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—24no—
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46#407—22no—
47C2×C22⋊C8C2×C412yes—
48C4×M16—22no—
49C2×C4×C8D412yes—
50C4×C4⋊C4—21no—
51C4×C4⋊C4D411yescharacteristic
52C4×C4⋊C4D411yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2×C412yes—
55#376Q812yes—
56C43—21no—
57#336—24no—
58#335—24no—
59#334—24no—
60C23×C4C22×C411yescharacteristic
61C23×C4#4111yescharacteristic
62C23×C4#4111yescharacteristic
63C2×M16—24no—
64C2×M16—24no—
65C2×M16—24no—
66C2×M16—42no—
67C2×M16—82no—
68C2×M16—82no—
69C2×M16—82no—
70C22×C8—22no—
71C22×C8—42no—
72C22×C8C22⋊C412yes—
73C22×C8C22⋊C412yes—
74#117—44no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C42—22no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—22no—
82C2×C42C2×D412yes—
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C42C2×Q812yes—
87C2×C42C2×Q811yescharacteristic
88C2×C42#4112yes—
89C2×C42#4112yes—
90C4⋊C8—24no—
91C4⋊C8—22no—
92C4⋊C8—22no—
93C4⋊C8—24no—
94C4⋊C8—24no—
95C4⋊C8—24no—
96C4⋊C8—41no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4⋊C8—42no—
100C4⋊C8—41no—
101C4⋊C8—84no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105#97—42no—
106C4×C8—22no—
107C4×C8—22no—
108C4×C8—42no—
109#95—22no—
110#95C4⋊C412yes—
111#95C4⋊C412yes—
112C24#11711yescharacteristic
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—42no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4C2×C22⋊C411yescharacteristic
122C22×C4C2×C4⋊C411yescharacteristic
123C22×C4C4×D412yes—
124C22×C4C4×Q812yes—
125C22×C4#12111yescharacteristic
126C22×C4#12111yescharacteristic
127C22×C4C22⋊Q812yes—
128C22×C4C22⋊Q811yescharacteristic
129C22×C4C22⋊Q811yescharacteristic
130C22×C4#12312yes—
131C22×C4#12512yes—
132C22×C4#12612yes—
133M16—82no—
134M16—82no—
135M16—84no—
136M16—84no—
137M16—84no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—44no—
143C2×C8—42no—
144C2×C8—44no—
145C2×C8—44no—
146C2×C8—84no—
147C4⋊C4—24no—
148C4⋊C4—42no—
149C22⋊C4—44no—
150C42—42no—
151C42—42no—
152C42—44no—
153C42—42no—
154C42—41no—
155C42—41no—
156C42#12011yescharacteristic
157C42#12112yes—
158C42#12112yes—
159C42C22⋊Q812yes—
160C42C22⋊Q812yes—
161C42#12311yescharacteristic
162C42#12411yescharacteristic
163C42#12512yes—
164C42C4⋊Q812yes—
165C42C4⋊Q811yescharacteristic
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23#38011yescharacteristic
171C23#38712yes—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4#38611yescharacteristic
195C2×C4#38912yes—
196C2×C4#39111yescharacteristic
197C2×C4#39411yescharacteristic
198C2×C4#39511yescharacteristic
199C2×C4#39612yes—
200C2×C4#39812yes—
201C2×C4#39811yescharacteristic
202C2×C4#40011yescharacteristic
203C2×C4#47112yes—
204C2×C4#47211yescharacteristic
205C2×C4#47311yescharacteristic
206C8—42no—
207C8—42no—
208C8—84no—
209C8—84no—
210C8—82no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—81no—
217C22—81no—
218C22#144111yescharacteristic
219C22#181511yescharacteristic
220C22#181611yescharacteristic
221C4—22no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—22no—
227C4—44no—
228C4—82no—
229C4#191411yescharacteristic
230C4#191511yescharacteristic
231C4#205012yes—
232C4#206312yes—
233C4#206412yes—
234C2—21no—
235C2—21no—
236C2—42no—
237C2#1120611yescharacteristic
238C2#1124211yescharacteristic
239C2#1124311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#117
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#219742
#178192
#81292
#80972
#80012