Finite Group Databasebeta

← #124374

Subgroups — #124374

Order 512.

278 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 94 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#21341C211yesmaximal, characteristic
2#12894C211yesmaximal, characteristic
3#12590C211yesmaximal, characteristic
4#7128C212yesmaximal
5#7118C212yesmaximal
6#2295C411yescharacteristic
7C2×C4×C16—21no—
8#1984C411yescharacteristic
9#1984C2211yescharacteristic
10#1939C411yescharacteristic
11#1939C2211yescharacteristic
12#1916—21no—
13#1916—21no—
14#1768—21no—
15#1767C411yescharacteristic
16#1746—21no—
17C42×C8C2211yescharacteristic
18#1374—21no—
19#1374—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C2×C4×Q8C2×C411yescharacteristic
24C22×C16—22no—
25C22×C16—22no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×#97—21no—
31C2×C4×C8C2×C411yescharacteristic
32C2×C4×C8D411yescharacteristic
33C2×C4×C8D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8C2311yescharacteristic
37#391C811yescharacteristic
38#391C811yescharacteristic
39#389C811yescharacteristic
40#389C811yescharacteristic
41#387—21no—
42#387—21no—
43C4×C4⋊C4—21no—
44C4×C4⋊C4C2×C411yescharacteristic
45#376—21no—
46C43C2×C411yescharacteristic
47C4×C16—24no—
48#336—21no—
49#336—21no—
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55C22×Q8C2×C811yescharacteristic
56C22×C8—21no—
57C22×C8—21no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8C2×D411yescharacteristic
61C22×C8C2×D411yescharacteristic
62C4×Q8—41no—
63C4×Q8—41no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2×C811yescharacteristic
68C2×C4⋊C4C2×C811yescharacteristic
69C2×C4⋊C4C2×C811yescharacteristic
70C2×C42—21no—
71C2×C42—21no—
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22⋊C411yescharacteristic
74C2×C42C2×C811yescharacteristic
75C2×C42C2×C811yescharacteristic
76C2×C42M1611yescharacteristic
77C2×C42M1611yescharacteristic
78C2×C42C22×C411yescharacteristic
79C2×C16—24no—
80C2×C16—42no—
81C2×C16—42no—
82C2×C16—42no—
83C2×C16—42no—
84C2×C16—42no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C4⋊C8—42no—
88C4⋊C8—42no—
89#97—42no—
90C4×C8—22no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—41no—
94C4×C8—41no—
95C4×C8—41no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103C2×Q8—22no—
104C2×Q8—41no—
105C2×Q8—41no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4C22⋊C811yescharacteristic
111C22×C4C22⋊C811yescharacteristic
112C22×C4C22⋊C811yescharacteristic
113C22×C4C22⋊C811yescharacteristic
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4C22×C811yescharacteristic
116C22×C4C2×M1611yescharacteristic
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8#10412yes—
133C2×C8#13512yes—
134C2×C8#13712yes—
135C4⋊C4—22no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C42—22no—
142C42—22no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—22no—
150C42—41no—
151C42—41no—
152C42—41no—
153C42—41no—
154C42—41no—
155C16—44no—
156C16—82no—
157C23C2×C22⋊C811yescharacteristic
158Q8—82no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4#35012yes—
187C2×C4#35012yes—
188C2×C4#41011yescharacteristic
189C2×C4#41311yescharacteristic
190C2×C4#41611yescharacteristic
191C2×C4#41911yescharacteristic
192C2×C4C2×#10411yescharacteristic
193C2×C4#42111yescharacteristic
194C2×C4#50412yes—
195C8—22no—
196C8—22no—
197C8—22no—
198C8—41no—
199C8—81no—
200C8—81no—
201C22#145811yescharacteristic
202C22#147611yescharacteristic
203C22#147911yescharacteristic
204C22#211311yescharacteristic
205C22#211511yescharacteristic
206C22C2×#35011yescharacteristic
207C22C2×#35011yescharacteristic
208C4—22no—
209C4—22no—
210C4—22no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—81no—
215C4—81no—
216C4—81no—
217C4—81no—
218C4#150611yescharacteristic
219C4#152111yescharacteristic
220C4#153211yescharacteristic
221C4#155411yescharacteristic
222C2#758111yescharacteristic
223C2#1146111yescharacteristic
224C2#1146211yescharacteristic
225C2#1161611yescharacteristic
226C2#1161811yescharacteristic
227C2#1163811yescharacteristic
228C2#1164111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socle, prime radicalC23C2×C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#213412
#128942
#125902
#71282
#71182