Finite Group Databasebeta

← #140621

Subgroups — #140621

Order 512.

470 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 142 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×#1939C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C4×C22⋊C16C211yesmaximal, characteristic
4C2×#1322C214yesmaximal
5C22×C4⋊C8C411yescharacteristic
6C22×C4⋊C8C2212yes—
7C22×C4×C8C2211yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4⋊C16—21no—
10C2×C22⋊C16C2212yes—
11C2×C22⋊C16C2212yes—
12C2×C4×C16—21no—
13#1939—21no—
14#1939—21no—
15#1939—21no—
16#1939—21no—
17C2×#336C412yes—
18#1374—21no—
19#1374—21no—
20#1322—28no—
21C23×C8C2×C411yescharacteristic
22C23×C8C2×C412yes—
23C23×C8C2311yescharacteristic
24C22×C4⋊C4C811yescharacteristic
25C22×C4⋊C4C811yescharacteristic
26C22×C42C2×C412yes—
27C22×C42C2×C411yescharacteristic
28C22×C16—22no—
29C22×C16—22no—
30C2×C4⋊C8—24no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—24no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—42no—
37C2×C4×C8—42no—
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C4×C4⋊C4—21no—
43C4×C4⋊C4—21no—
44C4×C4⋊C4—21no—
45C4×C4⋊C4—21no—
46C2×#95C812yes—
47C4⋊C16—42no—
48C22⋊C16—22no—
49C22⋊C16—24no—
50C22⋊C16D414yes—
51C4×C16—42no—
52#336—22no—
53#336—22no—
54C23×C4C2×C812yes—
55C23×C4C2×C811yescharacteristic
56C23×C4C2×C811yescharacteristic
57C23×C4C2×C812yes—
58C23×C4C22×C411yescharacteristic
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—24no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—42no—
67C22×C8—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D412yes—
73C22×C8#4112yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4C1614yescomplemented
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—24no—
85C2×C42—22no—
86C2×C42—24no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89C2×C42M1611yescharacteristic
90C2×C42M1611yescharacteristic
91C2×C16—24no—
92C2×C16—42no—
93C2×C16—42no—
94C2×C16—42no—
95C2×C16—42no—
96C2×C16—42no—
97C4⋊C8—42no—
98C4⋊C8—44no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4×C8—22no—
103C4×C8—22no—
104C4×C8—41no—
105C4×C8—41no—
106#95—22no—
107#95—22no—
108C24C22×C811yescharacteristic
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—41no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C22⋊C811yescharacteristic
127C22×C4C22⋊C811yescharacteristic
128C22×C4C22⋊C811yescharacteristic
129C22×C4C22⋊C811yescharacteristic
130C22×C4C2×C1614yes—
131C22×C4C2×C1612yes—
132C22×C4M3212yes—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C4×D412yes—
135C22×C4C2×M1611yescharacteristic
136C22×C4#13112yes—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—44no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—84no—
153C2×C8#10112yes—
154C2×C8#12112yes—
155C2×C8#12411yescharacteristic
156C2×C8#12711yescharacteristic
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C42—22no—
162C42—22no—
163C42—21no—
164C42—21no—
165C42—42no—
166C42—42no—
167C42—44no—
168C42#9912yes—
169C42#10112yes—
170C16—44no—
171C16—82no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23C2×C22⋊C811yescharacteristic
177C23C8×D412yes—
178C23C22×C1611yescharacteristic
179C23C2×M3211yescharacteristic
180C23#50412yes—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—28no—
197C2×C4—21no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—84no—
206C2×C4#32412yes—
207C2×C4#32412yes—
208C2×C4C22⋊C1612yes—
209C2×C4C22⋊C1612yes—
210C2×C4#34912yes—
211C2×C4#39011yescharacteristic
212C2×C4#40811yescharacteristic
213C2×C4C2×#9911yescharacteristic
214C2×C4#41011yescharacteristic
215C2×C4C2×#10111yescharacteristic
216C2×C4C2×#10111yescharacteristic
217C2×C4#43612yes—
218C8—22no—
219C8—21no—
220C8—21no—
221C8—22no—
222C8—41no—
223C8—82no—
224C22—22no—
225C22—21no—
226C22—21no—
227C22—22no—
228C22—22no—
229C22—21no—
230C22—21no—
231C22—22no—
232C22C2×#32411yescharacteristic
233C22#146111yescharacteristic
234C22#195411yescharacteristic
235C22C2×C22⋊C1611yescharacteristic
236C22#211311yescharacteristic
237C22C2×#34911yescharacteristic
238C22#211511yescharacteristic
239C22D4×C1612yes—
240C22#216912yes—
241C4—22no—
242C4—22no—
243C4—22no—
244C4—44no—
245C4—41no—
246C4—82no—
247C4#132214yescomplemented
248C4#147011yescharacteristic
249C4#147211yescharacteristic
250C4#196411yescharacteristic
251C4#196611yescharacteristic
252C2—22no—
253C2—22no—
254C2#965011yescharacteristic
255C2C2×#132211yescharacteristic
256C2#1144111yescharacteristic
257C2#1149311yescharacteristic
258C2#1149511yescharacteristic
259C2#1318911yescharacteristic
260C2#1319411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C16
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22×C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
80. C2×C4⋊C4C16semidirect
247. C4#1322semidirect

Maximal subgroups

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

subgroupindex
C2×#19392
#127832
C4×C22⋊C162
C2×#13222