Finite Group Databasebeta

← #124268

Subgroups — #124268

Order 512.

430 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 122 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#12783C211yesmaximal, characteristic
3#12599C211yesmaximal, characteristic
4#7118C214yesmaximal, complemented
5C22×C4⋊C8—21no—
6C22×C4⋊C8C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4⋊C16—21no—
10C2×C22⋊C16—21no—
11C2×C22⋊C16—21no—
12C2×#346—21no—
13#1939—21no—
14#1939—21no—
15#1939C412yes—
16#1939C2212yes—
17C2×#336—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—42no—
34C2×C4⋊C8—42no—
35C2×C4⋊C8C2×C412yes—
36C2×C4×C8—42no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4—21no—
43C4×C4⋊C4C812yes—
44C4×C4⋊C4C2×C412yes—
45C2×#95—21no—
46C4⋊C16—42no—
47C22⋊C16—24no—
48C22⋊C16—42no—
49#346—42no—
50#336—22no—
51#336—22no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C2×C811yescharacteristic
56C23×C4M1611yescharacteristic
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—42no—
64C22×C8—42no—
65C22×C8—42no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8D812yes—
69C22×C8SD1612yes—
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C2×C4⋊C4—21no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4C2×C812yes—
77C2×C4⋊C4C2×C812yes—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—42no—
83C2×C42—42no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C2×C811yescharacteristic
86C2×C42M1611yescharacteristic
87C2×C42C22×C411yescharacteristic
88C2×C16—24no—
89C2×C16—42no—
90C2×C16—42no—
91C2×C16—42no—
92C2×C16—42no—
93C2×C16—42no—
94C4⋊C8—42no—
95C4⋊C8—44no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—41no—
102C4×C8—41no—
103#95—22no—
104#95—22no—
105C24C22⋊C811yescharacteristic
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C22⋊C811yescharacteristic
125C22×C4C22⋊C811yescharacteristic
126C22×C4C22⋊C811yescharacteristic
127C22×C4#10212yes—
128C22×C4#10412yes—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C22×C811yescharacteristic
131C22×C4C2×M1611yescharacteristic
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—44no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—84no—
148C2×C8#10212yes—
149C2×C8C2×D811yescharacteristic
150C2×C8C2×SD1611yescharacteristic
151C2×C8#13712yes—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4M3214yes—
157C42—22no—
158C42—22no—
159C42—21no—
160C42—21no—
161C42—44no—
162C42—42no—
163C42—42no—
164C42#9912yes—
165C42#10112yes—
166C16—44no—
167C16—82no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23D4⋊C812yes—
173C23#34912yes—
174C23#35012yes—
175C23C2×C22⋊C811yescharacteristic
176C2×C4—28no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—44no—
194C2×C4—48no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—84no—
201C2×C4#32412yes—
202C2×C4#32412yes—
203C2×C4D4⋊C812yes—
204C2×C4C2×#9911yescharacteristic
205C2×C4C2×#10111yescharacteristic
206C2×C4C2×#10211yescharacteristic
207C2×C4#41911yescharacteristic
208C2×C4C2×#10411yescharacteristic
209C2×C4#42111yescharacteristic
210C2×C4C2×M3212yes—
211C8—22no—
212C8—21no—
213C8—22no—
214C8—21no—
215C8—41no—
216C8—82no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22#133014yes—
224C22C2×#32411yescharacteristic
225C22C2×D4⋊C811yescharacteristic
226C22#147911yescharacteristic
227C22#211211yescharacteristic
228C22C2×#34911yescharacteristic
229C22C2×#35011yescharacteristic
230C22#211711yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—22no—
235C4—44no—
236C4—41no—
237C4—82no—
238C4#150811yescharacteristic
239C4#151911yescharacteristic
240C4#152811yescharacteristic
241C4#154611yescharacteristic
242C2—24no—
243C2#757311yescharacteristic
244C2#1144011yescharacteristic
245C2#1144411yescharacteristic
246C2#1158311yescharacteristic
247C2C2×#133011yescharacteristic
248C2#1163911yescharacteristic
249C2#1164411yescharacteristic
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
#127832
#125992
#71182