Finite Group Databasebeta

← #124149

Subgroups — #124149

Order 512.

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

normal subgroups only

This group has 506 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#13854C211yesmaximal, characteristic
2C4×#324C212yesmaximal
3#7122C214yesmaximal
4C2×C4×C4⋊C4C411yescharacteristic
5C2×C4×C4⋊C4C2211yescharacteristic
6C2×#376C411yescharacteristic
7C4×C22⋊C8—22no—
8C2×#324—22no—
9#1442—22no—
10#1442C412yes—
11#1442C2212yes—
12C4×#95—22no—
13#1280C2214yes—
14C22×C4⋊C4C2×C412yes—
15C22×C4⋊C4D412yes—
16C22×C42—21no—
17C22×C42—21no—
18C22×C42C2×C411yescharacteristic
19C22×C42D412yes—
20C2×C22⋊C8—24no—
21C2×C4×C8—42no—
22C4×C4⋊C4—41no—
23C4×C4⋊C4—41no—
24C4×C4⋊C4—41no—
25C4×C4⋊C4—41no—
26#376—22no—
27#376C814yescomplemented
28C2×#95—21no—
29C2×#95—22no—
30C2×#95—22no—
31C2×#95—21no—
32C2×#95—22no—
33C2×#95—22no—
34C2×#95C2×C412yes—
35C2×#95C2×C411yescharacteristic
36C2×#95C2311yescharacteristic
37#336—42no—
38#324—28no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4C22⋊C411yescharacteristic
42C23×C4C22⋊C412yes—
43C23×C4C22⋊C411yescharacteristic
44C23×C4C22×C411yescharacteristic
45C23×C4C2×D412yes—
46C22×C8—44no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4—42no—
51C2×C4⋊C4—44no—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42—22no—
55C2×C42—24no—
56C2×C42—22no—
57C2×C42—22no—
58C2×C42—42no—
59C2×C42—42no—
60C2×C42—44no—
61C2×C42—44no—
62C2×C42—41no—
63C2×C42—41no—
64C2×C42C2×C812yes—
65C2×C42M1612yes—
66C22⋊C8—28no—
67C22⋊C8—44no—
68C4×C8—84no—
69#95—24no—
70#95—22no—
71#95—41no—
72#95—41no—
73#95—44no—
74#95—44no—
75#95—42no—
76#95—44no—
77#95—44no—
78#95C2×C814yes—
79C24C2×C22⋊C411yescharacteristic
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—24no—
93C22×C4—24no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4C22⋊C812yes—
112C22×C4C22⋊C812yes—
113C22×C4#9911yescharacteristic
114C22×C4#9911yescharacteristic
115C22×C4#10112yes—
116C22×C4#10414yes—
117C22×C4C22×C811yescharacteristic
118C22×C4C2×M1611yescharacteristic
119C22×C4#13514yes—
120C2×C8—48no—
121C2×C8—84no—
122C2×C8—84no—
123C4⋊C4—48no—
124C4⋊C4—44no—
125C4⋊C4—44no—
126C42—22no—
127C42—22no—
128C42—22no—
129C42—22no—
130C42—48no—
131C42—44no—
132C42—82no—
133C42—82no—
134C42—82no—
135C23—22no—
136C23—21no—
137C23—21no—
138C23C2×C22⋊C811yescharacteristic
139C23C2×#9911yescharacteristic
140C23C2×#10111yescharacteristic
141C23#41612yes—
142C23C2×#10412yes—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4#32412yes—
183C2×C4#32412yes—
184C2×C4#35412yes—
185C2×C4#35512yes—
186C8—88no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22C2×#32411yescharacteristic
196C22#147612yes—
197C22#150112yes—
198C22#152611yescharacteristic
199C22#154411yescharacteristic
200C22#212012yes—
201C22C2×#35411yescharacteristic
202C22C2×#35511yescharacteristic
203C4—22no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—44no—
208C4—44no—
209C4—84no—
210C4—84no—
211C4#132414yes—
212C2—22no—
213C2—21no—
214C2—21no—
215C2#756411yescharacteristic
216C2C2×#132411yescharacteristic
217C2#1145211yescharacteristic
218C2#1172812yes—
219C2#1173412yes—
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
27. #376C8semidirect

Maximal subgroups

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

subgroupindex
#138542
C4×#3242
#71222