Finite Group Databasebeta

← #134314

Subgroups — #134314

Order 512.

408 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 110 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#20437C211yesmaximal, characteristic
2#11447C212yesmaximal
3#8602C212yesmaximal
4#7188C212yesmaximal
5C2×#423C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2285C2211yescharacteristic
10#1770—22no—
11C8×C4⋊C4—22no—
12#1729C411yescharacteristic
13#1729C2211yescharacteristic
14#1564—22no—
15#1458C2214yes—
16#1325—22no—
17#1324—22no—
18#1275—22no—
19C22×M16C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C23×C8C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C2×#117C2311yescharacteristic
25C22×C42C2×C411yescharacteristic
26#431—24no—
27#423—22no—
28#423C814yescomplemented
29C2×C4⋊C8—22no—
30#405—41no—
31#405—41no—
32#405—41no—
33#405—41no—
34C2×#97—21no—
35C2×#97D412yes—
36C2×C4×C8—21no—
37C2×C4×C8—21no—
38C2×C4×C8—22no—
39C2×C4×C8—22no—
40C2×C4×C8—21no—
41C2×C4×C8D412yes—
42#384D411yescharacteristic
43#384Q811yescharacteristic
44#382D411yescharacteristic
45#382Q811yescharacteristic
46#380—22no—
47C4×C4⋊C4—22no—
48#336—22no—
49#336—22no—
50#335—42no—
51#334—42no—
52#324—24no—
53#323—24no—
54C23×C4—21no—
55C23×C4C22×C411yescharacteristic
56C2×M16—21no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16C2×C812yes—
62C2×M16C2×C812yes—
63C2×M16M1612yes—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—22no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8C22⋊C412yes—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8M1612yes—
78#117—41no—
79#117—41no—
80#117—41no—
81#117—41no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4C4⋊C411yescharacteristic
85C2×C4⋊C4C4⋊C411yescharacteristic
86C2×C4⋊C4C2×D411yescharacteristic
87C2×C4⋊C4C2×Q811yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4#4112yes—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C4⋊C411yescharacteristic
95C2×C42C4⋊C411yescharacteristic
96C2×C42C2×C812yes—
97C2×C42M1612yes—
98C2×C42C2×D412yes—
99C4⋊C8—28no—
100C4⋊C8—44no—
101C22⋊C8—44no—
102#97—42no—
103#97—42no—
104C4×C8—24no—
105C4×C8—24no—
106C4×C8—22no—
107C4×C8—22no—
108C4×C8—44no—
109C4×C8—41no—
110C4×C8—41no—
111#95—22no—
112#95—22no—
113C24#11711yescharacteristic
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4C2×C4⋊C411yescharacteristic
128C22×C4#12112yes—
129C22×C4C22⋊Q812yes—
130C22×C4C22×C811yescharacteristic
131C22×C4C2×M1611yescharacteristic
132C22×C4C2×M1611yescharacteristic
133C22×C4C2×M1611yescharacteristic
134M16—42no—
135M16—42no—
136M16—42no—
137M16—42no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—22no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—42no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—42no—
156C2×C8—84no—
157C2×C8C22⋊C814yes—
158C2×C8#9911yescharacteristic
159C2×C8#9911yescharacteristic
160C2×C8#10112yes—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—21no—
174C42—21no—
175C42—41no—
176C42—41no—
177C42C4⋊C812yes—
178C42C4⋊C812yes—
179C23—21no—
180C23—21no—
181C23—41no—
182C23—41no—
183C23#38011yescharacteristic
184C23#43111yescharacteristic
185C23#43211yescharacteristic
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4C2×C22⋊C811yescharacteristic
213C2×C4#40711yescharacteristic
214C2×C4C2×#9911yescharacteristic
215C2×C4#41011yescharacteristic
216C2×C4C2×#10111yescharacteristic
217C2×C4#41311yescharacteristic
218C2×C4C2×C4⋊C811yescharacteristic
219C2×C4#42311yescharacteristic
220C8—24no—
221C8—21no—
222C8—21no—
223C8—22no—
224C8—42no—
225C8—42no—
226C8—164no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—41no—
232C22—41no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22#179711yescharacteristic
237C22#179911yescharacteristic
238C22#180111yescharacteristic
239C4—24no—
240C4—21no—
241C4—21no—
242C4—42no—
243C4—42no—
244C4—42no—
245C4—81no—
246C4—81no—
247C4#145911yescharacteristic
248C4#146111yescharacteristic
249C4#146311yescharacteristic
250C4#146411yescharacteristic
251C2—21no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2#856011yescharacteristic
256C2#1280711yescharacteristic
257C2#1280911yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #423C8semidirect

Maximal subgroups

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

subgroupindex
#204372
#114472
#86022
#71882