Finite Group Databasebeta

← #120291

Subgroups — #120291

Order 512.

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

normal subgroups only

This group has 483 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#20814C211yesmaximal, characteristic
2#17818C211yesmaximal, characteristic
3#8022C212yesmaximal
4#7981C211yesmaximal, characteristic
5#7952C212yesmaximal
6C2×#432C2212yes—
7C2×#432C2211yescharacteristic
8C2×#431C2211yescharacteristic
9#2285C2212yes—
10C2×#376C2211yescharacteristic
11#1843C412yescomplemented
12#1832C412yescomplemented
13#1797—22no—
14#1768—22no—
15#1573—24no—
16#1568—22no—
17#1294—22no—
18#1294—22no—
19#1289—22no—
20#1285—24no—
21C2×#117—22no—
22C22×C42—21no—
23C22×C42C2311yescharacteristic
24#432—22no—
25#432—22no—
26#432—22no—
27#432—22no—
28#432—22no—
29#432D414yes—
30#431—22no—
31#431—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—24no—
34C2×C4⋊C8C2×C412yes—
35C2×C4⋊C8C2×C412yes—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8D412yes—
39C2×#97—22no—
40C2×#97—21no—
41C2×C4×C8D412yes—
42#384—22no—
43#382—22no—
44#380—22no—
45#380—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4Q814yes—
48#376—21no—
49#376—21no—
50#376C2×C411yescharacteristic
51#376C2×C411yescharacteristic
52#376Q811yescharacteristic
53#376Q811yescharacteristic
54C2×#95—21no—
55C2×#95D411yescharacteristic
56C2×#95D411yescharacteristic
57#336—24no—
58#335—24no—
59#334—24no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4#4111yescharacteristic
65C22×C8—22no—
66C22×C8—42no—
67C22×C8C22⋊C414yes—
68#117—82no—
69#117—82no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—24no—
79C2×C42—44no—
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—24no—
89C4⋊C8—24no—
90C4⋊C8—24no—
91C4⋊C8—48no—
92C4⋊C8—42no—
93C22⋊C8—24no—
94C22⋊C8—24no—
95C22⋊C8—42no—
96C22⋊C8—42no—
97#97—24no—
98#97—41no—
99#97—41no—
100#97—42no—
101C4×C8—22no—
102C4×C8—22no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—24no—
110#95—42no—
111#95C4⋊C411yescharacteristic
112#95C4⋊C411yescharacteristic
113#95C4⋊C411yescharacteristic
114#95C4⋊C411yescharacteristic
115C24#12111yescharacteristic
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—44no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C2×C4⋊C411yescharacteristic
135C22×C4#11711yescharacteristic
136C22×C4#12112yes—
137C22×C4#12112yes—
138C22×C4#12111yescharacteristic
139C22×C4C22⋊Q812yes—
140C22×C4C22⋊Q811yescharacteristic
141C22×C4C22⋊Q812yes—
142C22×C4C22⋊Q812yes—
143C22×C4C22⋊Q811yescharacteristic
144C22×C4#12612yes—
145C2×C8—24no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—42no—
149C2×C8—44no—
150C2×C8—84no—
151C4⋊C4—84no—
152C4⋊C4—84no—
153C4⋊C4—84no—
154C22⋊C4—82no—
155C22⋊C4—82no—
156C42—24no—
157C42—42no—
158C42—42no—
159C42—42no—
160C42—84no—
161C42C4×D412yes—
162C42C4×Q812yes—
163C42#12011yescharacteristic
164C42#12311yescharacteristic
165C42#12312yes—
166C42#12411yescharacteristic
167C42#12512yes—
168C42#12512yes—
169C42C4⋊Q812yes—
170C42C4⋊Q811yescharacteristic
171C23—41no—
172C23—41no—
173C23#38011yescharacteristic
174C23#39612yes—
175C23#47112yes—
176C23#47211yescharacteristic
177C23#47311yescharacteristic
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—84no—
212C2×C4#38611yescharacteristic
213C2×C4#38712yes—
214C2×C4#38912yes—
215C2×C4#39111yescharacteristic
216C2×C4#39411yescharacteristic
217C2×C4#39511yescharacteristic
218C2×C4#39812yes—
219C2×C4#39811yescharacteristic
220C2×C4#40011yescharacteristic
221C8—44no—
222C8—84no—
223C8—82no—
224C22—21no—
225C22—22no—
226C22—21no—
227C22—22no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22#144111yescharacteristic
232C22#181511yescharacteristic
233C22#181611yescharacteristic
234C22#206312yes—
235C22#206412yes—
236C4—21no—
237C4—22no—
238C4—21no—
239C4—22no—
240C4—21no—
241C4—22no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—81no—
247C4—81no—
248C4—164no—
249C4#188412yes—
250C2—21no—
251C2—21no—
252C2—22no—
253C2—22no—
254C2#1121211yescharacteristic
255C2#1123411yescharacteristic
256C2#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
11. #1843C4semidirect
12. #1832C4semidirect

Maximal subgroups

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

subgroupindex
#208142
#178182
#80222
#79812
#79522