Finite Group Databasebeta

← #124131

Subgroups — #124131

Order 512.

296 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 106 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#12653C211yesmaximal, characteristic
3#12590C211yesmaximal, characteristic
4#7128C212yesmaximal
5#7118C212yesmaximal
6#2295C411yescharacteristic
7C2×C4⋊C16—21no—
8C2×#346—21no—
9C2×C4×C16—21no—
10#1984C411yescharacteristic
11#1984C2211yescharacteristic
12#1939C411yescharacteristic
13#1939C2211yescharacteristic
14#1916—21no—
15#1916—21no—
16#1768—21no—
17#1767C411yescharacteristic
18#1746—21no—
19C42×C8C2211yescharacteristic
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#346—42no—
49C4×C16—24no—
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57C22×Q8C2×C811yescharacteristic
58C22×C8—21no—
59C22×C8—21no—
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22⋊C411yescharacteristic
62C22×C8C2×D411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C4×Q8—41no—
65C4×Q8—41no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2×C811yescharacteristic
70C2×C4⋊C4C2×C811yescharacteristic
71C2×C4⋊C4C2×C811yescharacteristic
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42C2×C811yescharacteristic
77C2×C42C2×C811yescharacteristic
78C2×C42M1611yescharacteristic
79C2×C42M1611yescharacteristic
80C2×C42C22×C411yescharacteristic
81C2×C16—24no—
82C2×C16—24no—
83C2×C16—42no—
84C2×C16—42no—
85C2×C16—42no—
86C2×C16—42no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—42no—
90C4⋊C8—42no—
91#97—42no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8—22no—
95C4×C8—41no—
96C4×C8—41no—
97C4×C8SD1612yes—
98C4×C8Q1612yes—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106C2×Q8—22no—
107C2×Q8—41no—
108C2×Q8—41no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4C22⋊C811yescharacteristic
114C22×C4C22⋊C811yescharacteristic
115C22×C4C22⋊C811yescharacteristic
116C22×C4C22⋊C811yescharacteristic
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4C22×C811yescharacteristic
119C22×C4C2×M1611yescharacteristic
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8#10312yes—
135C2×C8#10412yes—
136C2×C8C2×SD1611yescharacteristic
137C2×C8C2×Q1611yescharacteristic
138C2×C8#13512yes—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C42—22no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—22no—
151C42—41no—
152C42—41no—
153C42—41no—
154C42—41no—
155C42—41no—
156C42—41no—
157C42#10312yes—
158C42#10412yes—
159C16—44no—
160C16—44no—
161C23C2×C22⋊C811yescharacteristic
162Q8—82no—
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—21no—
175C2×C4—21no—
176C2×C4—21no—
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—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4Q8⋊C812yes—
190C2×C4Q8⋊C812yes—
191C2×C4#35012yes—
192C2×C4#41011yescharacteristic
193C2×C4#41311yescharacteristic
194C2×C4C2×#10311yescharacteristic
195C2×C4#41611yescharacteristic
196C2×C4C2×#10411yescharacteristic
197C2×C4C2×#10411yescharacteristic
198C2×C4#50412yes—
199C8—22no—
200C8—22no—
201C8—21no—
202C8—21no—
203C8—22no—
204C8—81no—
205C8—81no—
206C22#145811yescharacteristic
207C22C2×Q8⋊C811yescharacteristic
208C22#147611yescharacteristic
209C22#211311yescharacteristic
210C22#211511yescharacteristic
211C22C2×#35011yescharacteristic
212C22#211711yescharacteristic
213C4—22no—
214C4—22no—
215C4—22no—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—81no—
220C4—81no—
221C4—81no—
222C4—81no—
223C4#150011yescharacteristic
224C4#150311yescharacteristic
225C4#152411yescharacteristic
226C4#154211yescharacteristic
227C2#756211yescharacteristic
228C2#1146111yescharacteristic
229C2#1146411yescharacteristic
230C2#1160811yescharacteristic
231C2#1161211yescharacteristic
232C2#1161411yescharacteristic
233C2#1161811yescharacteristic
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
#126532
#125902
#71282
#71182