Finite Group Databasebeta

← #120288

Subgroups — #120288

Order 512.

474 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 132 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#21960C211yesmaximal, characteristic
2#20755C211yesmaximal, characteristic
3#8052C212yesmaximal
4#7988C211yesmaximal, characteristic
5#7960C212yesmaximal
6#2920C2212yes—
7C2×#423C2211yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9#2283C2212yes—
10C4×#117C2211yescharacteristic
11#1949—24no—
12#1832C414yescomplemented
13#1831—22no—
14#1808—21no—
15#1807—21no—
16#1729—21no—
17#1729—21no—
18#1297—24no—
19#1290—22no—
20#1286—24no—
21C22×M16—21no—
22C23×C8D411yescharacteristic
23C23×C8D411yescharacteristic
24C2×#117—21no—
25C22×C42C2311yescharacteristic
26#432—22no—
27#432—22no—
28#431—22no—
29#431—22no—
30C2×#107—22no—
31C2×#106—22no—
32#423—22no—
33#423—22no—
34#423—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8C2×C412yes—
39C2×C4⋊C8C2×C412yes—
40C2×C4⋊C8D412yes—
41#407—22no—
42C4×M16Q814yes—
43#383—22no—
44#382D414yes—
45#381—22no—
46#380—22no—
47C4×C4⋊C4—21no—
48C4×C4⋊C4—22no—
49C4×C4⋊C4Q812yes—
50C4×C22⋊C4—21no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4D412yes—
53#376—22no—
54#376C2×C412yes—
55C43—21no—
56#340—22no—
57#340—22no—
58#336—24no—
59#336—24no—
60C23×C4C2×D411yescharacteristic
61C23×C4C2×D411yescharacteristic
62C23×C4#4111yescharacteristic
63C2×M16—24no—
64C2×M16—44no—
65C2×M16—41no—
66C2×M16—41no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—22no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8C22⋊C412yes—
75C22×C8C22⋊C412yes—
76#117—42no—
77#117—42no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—42no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—42no—
86C2×C42C22×C411yescharacteristic
87C2×C42C2×D412yes—
88C2×C42C2×D411yescharacteristic
89C2×C42C2×Q812yes—
90C2×C42C2×Q811yescharacteristic
91C2×C42#4112yes—
92C2×C42#4112yes—
93C2×C42#4111yescharacteristic
94#107—82no—
95#106—82no—
96C4⋊C8—24no—
97C4⋊C8—24no—
98C4⋊C8—24no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104#97—24no—
105C4×C8—24no—
106#95—22no—
107#95—24no—
108#95—24no—
109#95—42no—
110#95—42no—
111#95C4⋊C414yes—
112C24#12111yescharacteristic
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—42no—
117C22×C4—44no—
118C22×C4—81no—
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4C2×C4⋊C411yescharacteristic
121C22×C4#11711yescharacteristic
122C22×C4#12011yescharacteristic
123C22×C4#12112yes—
124C22×C4#12111yescharacteristic
125C22×C4C22⋊Q811yescharacteristic
126C22×C4C22⋊Q811yescharacteristic
127C22×C4C22⋊Q812yes—
128C22×C4#12311yescharacteristic
129C22×C4#12312yes—
130C22×C4#12411yescharacteristic
131C22×C4#12512yes—
132C22×C4C4⋊Q811yescharacteristic
133M16—82no—
134M16—82no—
135M16—84no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—44no—
143C2×C8—42no—
144C2×C8—44no—
145C2×C8—81no—
146C2×C8—81no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C22⋊C4—44no—
151C22⋊C4—84no—
152C42—44no—
153C42—42no—
154C42—42no—
155C42—82no—
156C42—84no—
157C42C4×D412yes—
158C42C4×Q812yes—
159C42#12112yes—
160C42C22⋊Q812yes—
161C42C22⋊Q812yes—
162C42#12512yes—
163C42#12612yes—
164C42C4⋊Q812yes—
165C23—22no—
166C23—22no—
167C23—41no—
168C23#38011yescharacteristic
169C23#39411yescharacteristic
170C23#39811yescharacteristic
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—84no—
190C2×C4—84no—
191C2×C4—84no—
192C2×C4#38611yescharacteristic
193C2×C4#38712yes—
194C2×C4#38912yes—
195C2×C4#39111yescharacteristic
196C2×C4#39511yescharacteristic
197C2×C4#39612yes—
198C2×C4#39812yes—
199C2×C4#40011yescharacteristic
200C2×C4#45412yes—
201C2×C4#45512yes—
202C8—42no—
203C8—42no—
204C8—84no—
205C8—82no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—44no—
214C22#144111yescharacteristic
215C22#181412yes—
216C22#204012yes—
217C4—21no—
218C4—21no—
219C4—22no—
220C4—22no—
221C4—22no—
222C4—22no—
223C4—84no—
224C4—164no—
225C4#188511yescharacteristic
226C4#188811yescharacteristic
227C4#206112yes—
228C2—22no—
229C2—22no—
230C2—41no—
231C2#1121111yescharacteristic
232C2#1123411yescharacteristic
233C2#1124111yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #1832C4semidirect

Maximal subgroups

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

subgroupindex
#219602
#207552
#80522
#79882
#79602