Finite Group Databasebeta

← #128398

Subgroups — #128398

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#20435C211yesmaximal, characteristic
2#11452C212yesmaximal
3#8157C212yesmaximal
4#7188C212yesmaximal
5C2×#424C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2296C2211yescharacteristic
10#1771—22no—
11C8×C4⋊C4—22no—
12#1729C2211yescharacteristic
13#1728C411yescharacteristic
14#1458C2214yes—
15#1452—22no—
16#1326C418yes—
17#1281—22no—
18C22×M16C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C2×#117C2311yescharacteristic
24C22×C42C2×C411yescharacteristic
25#431—24no—
26#424—22no—
27#424—22no—
28C2×C4⋊C8—22no—
29#405—22no—
30#405—22no—
31#405—42no—
32C2×#97—21no—
33C2×#97D412yes—
34C2×C4×C8—21no—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—42no—
38C2×C4×C8D412yes—
39#389—22no—
40#385C2×C414yes—
41C4×C4⋊C4—22no—
42#336—22no—
43#336—22no—
44#324—24no—
45#323C2×C418yes—
46#322—44no—
47C23×C4C4212yes—
48C23×C4C22×C411yescharacteristic
49C2×M16—21no—
50C2×M16—21no—
51C2×M16—21no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16M1612yes—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8C22⋊C412yes—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8M1612yes—
70#117—21no—
71#117—21no—
72#117—41no—
73#117—41no—
74#117M1612yes—
75#117M1612yes—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4#4112yes—
79C2×C22⋊C4C4212yes—
80C2×C22⋊C4C22×C412yes—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42C2×D412yes—
87C4⋊C8—28no—
88C4⋊C8—44no—
89C22⋊C8—28no—
90#97—24no—
91#97—42no—
92C4×C8—24no—
93C4×C8—24no—
94C4×C8—44no—
95C4×C8—42no—
96C4×C8—42no—
97#95—24no—
98C24C2×C4211yescharacteristic
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—44no—
106C22×C4—42no—
107C22×C4C2×C22⋊C411yescharacteristic
108C22×C4#11711yescharacteristic
109C22×C4C4×D414yes—
110C22×C4C2×M1611yescharacteristic
111C22×C4C2×M1611yescharacteristic
112C22×C4C2×M1611yescharacteristic
113C22×C4C2×M1611yescharacteristic
114M16—42no—
115M16—42no—
116M16—42no—
117M16—42no—
118C2×C8—24no—
119C2×C8—24no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—42no—
135C2×C8—48no—
136C2×C8#9912yes—
137C2×C8#10112yes—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4—48no—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4#9714yes—
145C42—22no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—42no—
150C42—41no—
151C42—41no—
152C23—21no—
153C23—21no—
154C23—42no—
155C23C4×C22⋊C411yescharacteristic
156C23C2×#9711yescharacteristic
157C23C4×M1611yescharacteristic
158C2×C4—21no—
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—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4#40711yescharacteristic
178C2×C4#40711yescharacteristic
179C2×C4C2×#9911yescharacteristic
180C2×C4#41011yescharacteristic
181C2×C4C2×#10111yescharacteristic
182C2×C4#41311yescharacteristic
183C2×C4#43211yescharacteristic
184C2×C4#43211yescharacteristic
185C8—22no—
186C8—22no—
187C8—42no—
188C8—42no—
189C8—42no—
190C8—88no—
191C22—22no—
192C22—21no—
193C22—21no—
194C22—41no—
195C22—41no—
196C22—42no—
197C22—42no—
198C22#174911yescharacteristic
199C22C4×#9911yescharacteristic
200C22C4×#10111yescharacteristic
201C4—21no—
202C4—21no—
203C4—44no—
204C4—44no—
205C4—44no—
206C4#146311yescharacteristic
207C4#146411yescharacteristic
208C4#146511yescharacteristic
209C4#146711yescharacteristic
210C2—21no—
211C2—21no—
212C2—41no—
213C2—41no—
214C2#815311yescharacteristic
215C2#1262312yes—
subgroupG/H
centerC2×C4C2×#99
commutator, prime radicalC24C2×C42
Frattini, cosocleC2×#117C23
socleC22C4×#99
LCSG ▷ C24 ▷ 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
#204352
#114522
#81572
#71882