Finite Group Databasebeta

← #124146

Subgroups — #124146

Order 512.

404 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 90 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#21376C211yesmaximal, characteristic
2C4×#349C212yesmaximal, complemented
3#7133C212yesmaximal, complemented
4#7105C212yesmaximal, complemented
5C2×#529C411yescharacteristic
6C2×#432C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8C2×#349—22no—
9C4×M32—22no—
10#1985C411yescharacteristic
11#1985C2211yescharacteristic
12#1955C411yescharacteristic
13#1955C2211yescharacteristic
14#1870—22no—
15#1750—22no—
16#1377C2214yes—
17C22×M16D412yes—
18#529—42no—
19#529—42no—
20C2×#126—21no—
21C2×#125C811yescharacteristic
22C2×#124C811yescharacteristic
23C2×#123—21no—
24C2×C22⋊Q8C811yescharacteristic
25C2×#121C811yescharacteristic
26C2×C4×Q8C2×C411yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2×C411yescharacteristic
29C2×#117—21no—
30C22×C42C2×C411yescharacteristic
31C2×M32—24no—
32#432—42no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C22⋊C8—21no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C4×M16—42no—
39C2×#97—21no—
40C2×#97C2×C411yescharacteristic
41C2×#97C2311yescharacteristic
42C2×C4×C8D412yes—
43#349—28no—
44#346—42no—
45C4×C16—42no—
46#336—22no—
47#336—22no—
48C22×Q8C2×C811yescharacteristic
49C22×D4—21no—
50C22×D4C2×C811yescharacteristic
51C23×C4—21no—
52C23×C4C22⋊C411yescharacteristic
53C23×C4C2×C811yescharacteristic
54C23×C4M1611yescharacteristic
55C2×M16—24no—
56C2×M16—24no—
57C2×M16—42no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8C22⋊C412yes—
61C22×C8C2×D412yes—
62#126—44no—
63#125—42no—
64#124—42no—
65#123—44no—
66C22⋊Q8—42no—
67#121—42no—
68C4×Q8—41no—
69C4×Q8—41no—
70C4×D4—41no—
71C4×D4—41no—
72C4×D4—42no—
73C4×D4—42no—
74#117—42no—
75#117—42no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4C2×C811yescharacteristic
81C2×C4⋊C4C2×C811yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C2×C811yescharacteristic
92C2×C42M1611yescharacteristic
93C2×C42C22×C411yescharacteristic
94M32—28no—
95M32—44no—
96C2×C16—44no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C22⋊C8—42no—
100C22⋊C8—44no—
101C22⋊C8—44no—
102#97—41no—
103#97—41no—
104#97—42no—
105C4×C8—22no—
106C4×C8—22no—
107C24—21no—
108C24C22⋊C811yescharacteristic
109C2×Q8—41no—
110C2×Q8—81no—
111C2×D4—41no—
112C2×D4—42no—
113C2×D4—82no—
114C2×D4—81no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4C22⋊C811yescharacteristic
131C22×C4C22⋊C811yescharacteristic
132C22×C4C22⋊C811yescharacteristic
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C22×C811yescharacteristic
135C22×C4C2×M1611yescharacteristic
136M16—48no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8#10414yes—
146C2×C8#13514yes—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C42—22no—
159C42—22no—
160C42—21no—
161C42—21no—
162C42—41no—
163C42—41no—
164C42#9912yes—
165C42#10012yes—
166C16—48no—
167C23—21no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23—42no—
175C23C2×C22⋊C811yescharacteristic
176Q8—82no—
177D4—82no—
178D4—84no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4#32312yes—
202C2×C4#32312yes—
203C2×C4C2×#9911yescharacteristic
204C2×C4C2×#10011yescharacteristic
205C2×C4#41612yes—
206C2×C4C2×#10412yes—
207C8—24no—
208C8—24no—
209C8—81no—
210C8—81no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—82no—
217C22—84no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22C2×#32311yescharacteristic
222C22#147612yes—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—81no—
229C4—81no—
230C4—81no—
231C4#150112yes—
232C4#152511yescharacteristic
233C4#154311yescharacteristic
234C2—22no—
235C2—81no—
236C2—161no—
237C2#756311yescharacteristic
238C2#1143611yescharacteristic
239C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#323
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#349C2semidirect
3. #7133C2semidirect
4. #7105C2semidirect

Maximal subgroups

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

subgroupindex
#213762
C4×#3492
#71332
#71052