Finite Group Databasebeta

← #124114

Subgroups — #124114

Order 512.

544 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 154 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×#1984C211yesmaximal, characteristic
2#12544C212yesmaximal
3#7128C214yesmaximal, complemented
4C22×C4×Q8C411yescharacteristic
5C22×M32—22no—
6C22×C4⋊C8C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#1984—22no—
9#1984C412yes—
10#1984C2212yes—
11C2×#336—22no—
12#1374—22no—
13#1374C2214yes—
14C23×Q8C811yescharacteristic
15C23×C8—21no—
16C23×C8D412yes—
17C2×C4×Q8—21no—
18C2×C4×Q8—21no—
19C2×C4×Q8—42no—
20C2×C4×Q8—42no—
21C2×C4×Q8C812yes—
22C2×C4×Q8C2×C412yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C811yescharacteristic
25C22×C42—21no—
26C22×C42C2×C411yescharacteristic
27C2×M32—28no—
28C2×M32—44no—
29C2×M32—44no—
30C22×C16—24no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—42no—
33C2×C4⋊C8C2×C412yes—
34C2×C4×C8—42no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8D412yes—
37C2×C4×C8C2311yescharacteristic
38#336—24no—
39#336—24no—
40C22×Q8—24no—
41C22×Q8—21no—
42C22×Q8—42no—
43C22×Q8—42no—
44C22×Q8C2×C812yes—
45C23×C4—21no—
46C23×C4—21no—
47C23×C4C22⋊C411yescharacteristic
48C23×C4C2×C811yescharacteristic
49C23×C4M1611yescharacteristic
50C22×C8—24no—
51C22×C8—24no—
52C22×C8—22no—
53C22×C8—42no—
54C22×C8—44no—
55C22×C8C22⋊C412yes—
56C22×C8SD1614yes—
57C22×C8Q1614yes—
58C22×C8C2×D412yes—
59C4×Q8—44no—
60C4×Q8—42no—
61C4×Q8—42no—
62C4×Q8—44no—
63C4×Q8—42no—
64C4×Q8—42no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—44no—
70C2×C4⋊C4C2×C812yes—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42—42no—
76C2×C42—42no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C2×C811yescharacteristic
79C2×C42M1611yescharacteristic
80C2×C42C22×C411yescharacteristic
81M32—416no—
82C2×C16—28no—
83C2×C16—44no—
84C2×C16—44no—
85C4⋊C8—42no—
86C4⋊C8—42no—
87C4×C8—41no—
88C4×C8—42no—
89C4×C8—41no—
90C24C22⋊C811yescharacteristic
91C2×Q8—24no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—416no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×Q8M3214yescomplemented
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—44no—
110C22×C4—42no—
111C22×C4C22⋊C811yescharacteristic
112C22×C4C22⋊C811yescharacteristic
113C22×C4C22⋊C811yescharacteristic
114C22×C4#9912yes—
115C22×C4#10012yes—
116C22×C4#10314yes—
117C22×C4#10414yes—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C22×C811yescharacteristic
120C22×C4C2×M1611yescharacteristic
121C2×C8—28no—
122C2×C8—28no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—42no—
128C2×C8—44no—
129C2×C8—84no—
130C2×C8#10314yes—
131C2×C8C2×SD1612yes—
132C2×C8C2×Q1612yes—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—44no—
136C4⋊C4—44no—
137C42—22no—
138C42—22no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—44no—
144C42—42no—
145C42—42no—
146C16—48no—
147C23—21no—
148C23—21no—
149C23—22no—
150C23#32312yes—
151C23Q8⋊C814yes—
152C23C2×C22⋊C811yescharacteristic
153Q8—44no—
154Q8—48no—
155Q8—44no—
156C2×C4—28no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—44no—
167C2×C4—48no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—84no—
173C2×C4#32312yes—
174C2×C4Q8⋊C814yes—
175C2×C4#32818yes—
176C2×C4#32914yes—
177C2×C4#33314yes—
178C2×C4#34912yes—
179C2×C4C2×#9911yescharacteristic
180C2×C4C2×#10011yescharacteristic
181C2×C4C2×#10312yes—
182C2×C4C2×#10412yes—
183C2×C4C2×M3212yes—
184C8—22no—
185C8—24no—
186C8—22no—
187C8—82no—
188C22—24no—
189C22—24no—
190C22—22no—
191C22—22no—
192C22#127214yes—
193C22C2×#32311yescharacteristic
194C22C2×Q8⋊C812yes—
195C22#211211yescharacteristic
196C22C2×#34911yescharacteristic
197C22#211712yes—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—41no—
202C4—41no—
203C4—82no—
204C4#127214yes—
205C4C2×#32812yes—
206C4C2×#32911yescharacteristic
207C4C2×#33311yescharacteristic
208C2—24no—
209C2C2×#127211yescharacteristic
210C2#1141811yescharacteristic
211C2#1142111yescharacteristic
212C2#1160712yes—
213C2#1161112yes—
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
3. #7128C2semidirect
98. C2×Q8M32semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
#125442
#71282