Finite Group Databasebeta

← #120550

Subgroups — #120550

Order 512.

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

normal subgroups only

This group has 466 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#21973C211yesmaximal, characteristic
2#17818C211yesmaximal, characteristic
3#8044C212yesmaximal
4#7984C212yesmaximal
5#7981C211yesmaximal, characteristic
6C2×#432C2212yes—
7C2×#431C2211yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9#2285C2212yes—
10#2283C2211yescharacteristic
11#1939—22no—
12#1850—22no—
13#1843C412yescomplemented
14#1832C412yescomplemented
15#1573—24no—
16#1568—22no—
17#1296—24no—
18#1289—22no—
19#1289—22no—
20#1289—22no—
21C23×C8—21no—
22C2×#117—22no—
23C22×C42C2311yescharacteristic
24#432—22no—
25#432—22no—
26#432—22no—
27#432—22no—
28#431—22no—
29#431—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—24no—
34C2×C4⋊C8C2×C412yes—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8D414yes—
38C2×C22⋊C8—22no—
39C2×C22⋊C8D412yes—
40C2×#97—22no—
41C2×C4×C8D412yes—
42#384—22no—
43#383Q812yes—
44#382—21no—
45#382Q814yes—
46#381—21no—
47#380—22no—
48#380—22no—
49#380D411yescharacteristic
50#380D411yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4—22no—
54C4×C22⋊C4—21no—
55#376C2×C412yes—
56#336—24no—
57#335—24no—
58#334—24no—
59C23×C4C2×D411yescharacteristic
60C23×C4C2×D411yescharacteristic
61C23×C4#4111yescharacteristic
62C22×C8—22no—
63C22×C8—24no—
64C22×C8—42no—
65C22×C8—82no—
66C22×C8C22⋊C412yes—
67C22×C8C22⋊C412yes—
68#117—82no—
69#117—82no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—42no—
78C2×C42—22no—
79C2×C42—42no—
80C2×C42C22×C411yescharacteristic
81C2×C42C2×D412yes—
82C2×C42C2×D411yescharacteristic
83C2×C42C2×Q812yes—
84C2×C42C2×Q811yescharacteristic
85C2×C42#4112yes—
86C2×C42#4112yes—
87C2×C42#4111yescharacteristic
88C4⋊C8—22no—
89C4⋊C8—22no—
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—24no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95C4⋊C8—84no—
96C22⋊C8—24no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99#97—42no—
100#97—42no—
101C4×C8—22no—
102C4×C8—22no—
103#95—22no—
104#95—22no—
105#95—22no—
106#95—24no—
107#95—24no—
108#95C4⋊C412yes—
109#95C4⋊C412yes—
110C24#12111yescharacteristic
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4—41no—
119C22×C4—44no—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C2×C4⋊C411yescharacteristic
122C22×C4#11711yescharacteristic
123C22×C4#12112yes—
124C22×C4#12112yes—
125C22×C4#12111yescharacteristic
126C22×C4C22⋊Q812yes—
127C22×C4C22⋊Q811yescharacteristic
128C22×C4C22⋊Q812yes—
129C22×C4C22⋊Q812yes—
130C22×C4C22⋊Q811yescharacteristic
131C22×C4#12612yes—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—44no—
137C2×C8—42no—
138C2×C8—44no—
139C2×C8—84no—
140C2×C8—84no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—84no—
144C4⋊C4—84no—
145C4⋊C4—82no—
146C4⋊C4—82no—
147C22⋊C4—42no—
148C22⋊C4—41no—
149C22⋊C4—41no—
150C22⋊C4—82no—
151C22⋊C4—82no—
152C42—42no—
153C42—42no—
154C42—84no—
155C42C4×D412yes—
156C42C4×Q812yes—
157C42#12011yescharacteristic
158C42#12311yescharacteristic
159C42#12312yes—
160C42#12411yescharacteristic
161C42#12512yes—
162C42#12512yes—
163C42C4⋊Q812yes—
164C42C4⋊Q811yescharacteristic
165C23—41no—
166C23—41no—
167C23#38011yescharacteristic
168C23#39612yes—
169C23#47112yes—
170C23#47211yescharacteristic
171C23#47311yescharacteristic
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—22no—
181C2×C4—24no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—84no—
197C2×C4—84no—
198C2×C4—84no—
199C2×C4#38611yescharacteristic
200C2×C4#38712yes—
201C2×C4#38912yes—
202C2×C4#39111yescharacteristic
203C2×C4#39411yescharacteristic
204C2×C4#39511yescharacteristic
205C2×C4#39812yes—
206C2×C4#39811yescharacteristic
207C2×C4#40011yescharacteristic
208C8—42no—
209C8—42no—
210C8—84no—
211C8—82no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—22no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22#144111yescharacteristic
220C22#181511yescharacteristic
221C22#181611yescharacteristic
222C22#206312yes—
223C22#206412yes—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—21no—
228C4—22no—
229C4—21no—
230C4—21no—
231C4—82no—
232C4—82no—
233C4—164no—
234C4#190911yescharacteristic
235C4#191011yescharacteristic
236C4#205312yes—
237C2—21no—
238C2—21no—
239C2—22no—
240C2—22no—
241C2#1122511yescharacteristic
242C2#1123611yescharacteristic
243C2#1124211yescharacteristic
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
13. #1843C4semidirect
14. #1832C4semidirect

Maximal subgroups

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

subgroupindex
#219732
#178182
#80442
#79842
#79812