Finite Group Databasebeta

← #124305

Subgroups — #124305

Order 512.

424 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 118 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#21300C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C4×C22⋊C16C211yesmaximal, characteristic
4#7128C214yesmaximal, complemented
5C2×#533C411yescharacteristic
6C2×#423C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C2×C4⋊C16—21no—
9C2×C22⋊C16—21no—
10C2×C22⋊C16—21no—
11C2×C4×C16—21no—
12#1984—21no—
13#1984—21no—
14#1984C412yes—
15#1984C2212yes—
16#1729—21no—
17#1729—21no—
18#1374C2212yes—
19#1374C2212yes—
20C22×M16—21no—
21C23×C8D411yescharacteristic
22C23×C8D411yescharacteristic
23#533—42no—
24#533—42no—
25C2×C4⋊Q8C811yescharacteristic
26C2×C4⋊Q8C811yescharacteristic
27C2×#125—21no—
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8C812yes—
30C2×C4×Q8—21no—
31C2×C4×Q8C2×C412yes—
32C2×#117—21no—
33C22×C42C2×C411yescharacteristic
34C22×C16—22no—
35C22×C16—22no—
36#423—42no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8C2×C412yes—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44C4⋊C16—42no—
45C22⋊C16—24no—
46C22⋊C16—42no—
47C4×C16—42no—
48#336—22no—
49#336—22no—
50#336—22no—
51#336—22no—
52C22×Q8—21no—
53C22×Q8C2×C812yes—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C2×C811yescharacteristic
56C23×C4M1611yescharacteristic
57C2×M16—42no—
58C2×M16—42no—
59C2×M16—42no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8SD1612yes—
69C22×C8Q1612yes—
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C4⋊Q8—42no—
73C4⋊Q8—42no—
74#125—44no—
75C22⋊Q8—44no—
76C22⋊Q8—44no—
77C4×Q8—42no—
78C4×Q8—42no—
79C4×Q8—42no—
80C4×Q8—42no—
81#117—42no—
82#117—42no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4C2×C812yes—
89C2×C22⋊C4—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42C22⋊C411yescharacteristic
94C2×C42C2×C811yescharacteristic
95C2×C42M1611yescharacteristic
96C2×C42C22×C411yescharacteristic
97C2×C16—24no—
98C2×C16—42no—
99C2×C16—42no—
100C2×C16—42no—
101C2×C16—42no—
102C2×C16—42no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4×C8—22no—
106C4×C8—22no—
107C4×C8—41no—
108C4×C8—41no—
109C24C22⋊C811yescharacteristic
110C2×Q8—24no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—41no—
128C22×C4C22⋊C811yescharacteristic
129C22×C4C22⋊C811yescharacteristic
130C22×C4C22⋊C811yescharacteristic
131C22×C4#10312yes—
132C22×C4#10412yes—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C22×C811yescharacteristic
135C22×C4C2×M1611yescharacteristic
136M16—84no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—44no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8#10312yes—
153C2×C8C2×SD1611yescharacteristic
154C2×C8C2×Q1611yescharacteristic
155C2×C8#13712yes—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—44no—
162C22⋊C4—44no—
163C42—22no—
164C42—22no—
165C42—21no—
166C42—21no—
167C42—42no—
168C42—42no—
169C42#9912yes—
170C42#10012yes—
171C16—44no—
172C16—82no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23Q8⋊C812yes—
180C23C2×C22⋊C811yescharacteristic
181Q8—84no—
182Q8—84no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4#32312yes—
205C2×C4#32312yes—
206C2×C4Q8⋊C812yes—
207C2×C4#35012yes—
208C2×C4#35012yes—
209C2×C4C2×#9911yescharacteristic
210C2×C4C2×#10011yescharacteristic
211C2×C4C2×#10311yescharacteristic
212C2×C4#41911yescharacteristic
213C2×C4C2×#10411yescharacteristic
214C2×C4#42111yescharacteristic
215C2×C4#50412yes—
216C8—22no—
217C8—21no—
218C8—22no—
219C8—21no—
220C8—41no—
221C8—82no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22C2×#32311yescharacteristic
230C22C2×Q8⋊C811yescharacteristic
231C22#147911yescharacteristic
232C22#211311yescharacteristic
233C22#211511yescharacteristic
234C22C2×#35011yescharacteristic
235C22C2×#35011yescharacteristic
236C4—22no—
237C4—22no—
238C4—22no—
239C4—41no—
240C4—82no—
241C4—82no—
242C4#131514yes—
243C4#150811yescharacteristic
244C4#152011yescharacteristic
245C4#152711yescharacteristic
246C4#154811yescharacteristic
247C2—42no—
248C2#757511yescharacteristic
249C2#1141911yescharacteristic
250C2C2×#131511yescharacteristic
251C2#1160811yescharacteristic
252C2#1161011yescharacteristic
253C2#1163811yescharacteristic
254C2#1164111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7128C2semidirect

Maximal subgroups

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

subgroupindex
#213002
#127832
C4×C22⋊C162
#71282