Finite Group Databasebeta

← #128394

Subgroups — #128394

Order 512.

420 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 114 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#20437C211yesmaximal, characteristic
2#11447C212yesmaximal
3#8157C212yesmaximal
4#7190C212yesmaximal
5C2×#423C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2285C2211yescharacteristic
10#1770—22no—
11C8×C4⋊C4—22no—
12#1729C411yescharacteristic
13#1729C2211yescharacteristic
14#1458C2214yes—
15#1452—22no—
16#1325C414yes—
17#1324C414yes—
18#1281—22no—
19C22×M16C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C23×C8C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C2×#117C2311yescharacteristic
25C22×C42C2×C411yescharacteristic
26#431—24no—
27#423—22no—
28#423—22no—
29C2×C4⋊C8—22no—
30#405—42no—
31#405—42no—
32C2×#97—21no—
33C2×#97D412yes—
34C2×C4×C8—21no—
35C2×C4×C8—21no—
36C2×C4×C8—22no—
37C2×C4×C8—22no—
38C2×C4×C8—21no—
39C2×C4×C8D412yes—
40#384C2×C412yes—
41#382C2×C412yes—
42#380—22no—
43C4×C4⋊C4—22no—
44#336—22no—
45#336—22no—
46#324C2×C418yes—
47#323—24no—
48#322—44no—
49C23×C4—21no—
50C23×C4C22×C411yescharacteristic
51C2×M16—21no—
52C2×M16—21no—
53C2×M16—21no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16M1612yes—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C22⋊C412yes—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8M1612yes—
72#117—41no—
73#117—41no—
74#117—41no—
75#117—41no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4C4212yes—
78C2×C4⋊C4C22×C412yes—
79C2×C22⋊C4—21no—
80C2×C22⋊C4#4112yes—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C4211yescharacteristic
86C2×C42C4211yescharacteristic
87C2×C42M1612yes—
88C2×C42M1612yes—
89C2×C42C2×D412yes—
90C4⋊C8—28no—
91C4⋊C8—44no—
92C22⋊C8—28no—
93#97—42no—
94#97—42no—
95C4×C8—24no—
96C4×C8—24no—
97C4×C8—24no—
98C4×C8—44no—
99C4×C8—42no—
100#95—24no—
101C24#11711yescharacteristic
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—44no—
112C22×C4C2×C4211yescharacteristic
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4C4×D414yes—
115C22×C4C2×M1611yescharacteristic
116C22×C4C2×M1611yescharacteristic
117C22×C4C2×M1611yescharacteristic
118C22×C4C2×M1611yescharacteristic
119M16—42no—
120M16—42no—
121M16—42no—
122M16—42no—
123C2×C8—24no—
124C2×C8—24no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—42no—
140C2×C8—48no—
141C2×C8#9912yes—
142C2×C8#10112yes—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—48no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C42—21no—
149C42—21no—
150C42—22no—
151C42—22no—
152C42—22no—
153C42—22no—
154C42—41no—
155C42—41no—
156C42#9712yes—
157C42#9712yes—
158C23—21no—
159C23—21no—
160C23—42no—
161C23C4×C22⋊C411yescharacteristic
162C23#43211yescharacteristic
163C23#43211yescharacteristic
164C2×C4—24no—
165C2×C4—22no—
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—22no—
177C2×C4—24no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4C2×#9711yescharacteristic
185C2×C4C4×M1611yescharacteristic
186C2×C4#40711yescharacteristic
187C2×C4#40711yescharacteristic
188C2×C4C2×#9911yescharacteristic
189C2×C4#41011yescharacteristic
190C2×C4C2×#10111yescharacteristic
191C2×C4#41311yescharacteristic
192C8—22no—
193C8—22no—
194C8—42no—
195C8—42no—
196C8—42no—
197C8—88no—
198C22—22no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22—42no—
205C22#174911yescharacteristic
206C22C4×#9911yescharacteristic
207C22C4×#10111yescharacteristic
208C4—24no—
209C4—21no—
210C4—21no—
211C4—42no—
212C4—44no—
213C4—82no—
214C4#146311yescharacteristic
215C4#146411yescharacteristic
216C4#146511yescharacteristic
217C4#146711yescharacteristic
218C2—21no—
219C2—21no—
220C2—41no—
221C2—41no—
222C2#815311yescharacteristic
223C2#1263312yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C4×#99
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#204372
#114472
#81572
#71902