Finite Group Databasebeta

← #124526

Subgroups — #124526

Order 512.

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

normal subgroups only

This group has 412 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
1C2×#1939C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#12844C211yesmaximal, characteristic
4#7118C214yesmaximal, complemented
5C22×C4⋊C8—21no—
6C22×C4⋊C8C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C22⋊C16—21no—
10C2×C22⋊C16—21no—
11#1939—21no—
12#1939—21no—
13#1939C412yes—
14#1939C2212yes—
15C2×#336—21no—
16#1374—21no—
17#1374—21no—
18#1374C2212yes—
19#1374C2212yes—
20C23×C8—21no—
21C23×C8D411yescharacteristic
22C23×C8D411yescharacteristic
23C22×C4⋊C4C811yescharacteristic
24C22×C4⋊C4C811yescharacteristic
25C22×C42—21no—
26C22×C42C2×C411yescharacteristic
27C22×C16—22no—
28C22×C16—22no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—42no—
33C2×C4⋊C8—44no—
34C2×C4⋊C8C2×C412yes—
35C2×C4×C8—42no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8C2311yescharacteristic
40C4×C4⋊C4—21no—
41C4×C4⋊C4—21no—
42C4×C4⋊C4C812yes—
43C4×C4⋊C4C2×C412yes—
44C2×#95—21no—
45C22⋊C16—42no—
46C22⋊C16—42no—
47#336—22no—
48#336—22no—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C2×C811yescharacteristic
53C23×C4M1611yescharacteristic
54C22×C8—22no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—42no—
62C22×C8—42no—
63C22×C8—42no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C22×C8C2×D411yescharacteristic
68C2×C4⋊C4—21no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4C2×C812yes—
73C2×C4⋊C4C2×C812yes—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—42no—
79C2×C42—42no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C2×C811yescharacteristic
82C2×C42M1611yescharacteristic
83C2×C42C22×C411yescharacteristic
84C2×C16—42no—
85C2×C16—42no—
86C2×C16—42no—
87C2×C16—42no—
88C2×C16—42no—
89C2×C16—42no—
90C4⋊C8—42no—
91C4⋊C8—42no—
92C4⋊C8—44no—
93C4⋊C8—44no—
94C4×C8—41no—
95C4×C8—41no—
96C4×C8—41no—
97C4×C8—41no—
98#95—22no—
99#95—22no—
100C24C22⋊C811yescharacteristic
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C22⋊C811yescharacteristic
120C22×C4C22⋊C811yescharacteristic
121C22×C4C22⋊C811yescharacteristic
122C22×C4#9912yes—
123C22×C4#10112yes—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C22×C811yescharacteristic
126C22×C4C2×M1611yescharacteristic
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—44no—
138C2×C8—44no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—84no—
143C2×C8AGL(1,ℤ/8)12yes—
144C2×C8#13712yes—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4M3214yes—
150C42—22no—
151C42—22no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—44no—
157C42—42no—
158C42—42no—
159C16—82no—
160C16—82no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23#32412yes—
166C23#35012yes—
167C23#35012yes—
168C23C2×C22⋊C811yescharacteristic
169C2×C4—28no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—48no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—84no—
195C2×C4#32412yes—
196C2×C4#34912yes—
197C2×C4C2×#9911yescharacteristic
198C2×C4C2×#10111yescharacteristic
199C2×C4#41811yescharacteristic
200C2×C4#41911yescharacteristic
201C2×C4#42111yescharacteristic
202C2×C4#42111yescharacteristic
203C2×C4C2×M3212yes—
204C8—22no—
205C8—22no—
206C8—41no—
207C8—41no—
208C8—82no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22#132314yes—
216C22C2×#32411yescharacteristic
217C22#147811yescharacteristic
218C22#147911yescharacteristic
219C22#211211yescharacteristic
220C22C2×#34911yescharacteristic
221C22C2×#35011yescharacteristic
222C22C2×#35011yescharacteristic
223C4—22no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—41no—
228C4—41no—
229C4—82no—
230C4#151011yescharacteristic
231C4#151111yescharacteristic
232C4#153411yescharacteristic
233C4#155611yescharacteristic
234C2—24no—
235C2#759311yescharacteristic
236C2#1144011yescharacteristic
237C2C2×#132311yescharacteristic
238C2#1163111yescharacteristic
239C2#1163411yescharacteristic
240C2#1163911yescharacteristic
241C2#1164211yescharacteristic
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. #7118C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19392
#128442
#71182