Finite Group Databasebeta

← #137246

Subgroups — #137246

Order 512.

429 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 92 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×#2086C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic
3#12871C212yesmaximal
4#11440C211yesmaximal, characteristic
5#7264C212yesmaximal
6#2920—21no—
7C22×#107C2211yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9C2×#389—21no—
10#2112C2212yes—
11#2086—21no—
12#2086—21no—
13#2086C412yes—
14#2086C412yes—
15#1950—24no—
16#1810—21no—
17#1810C2212yes—
18C2×#340—21no—
19C2×#340C2211yescharacteristic
20#1387—24no—
21#1322—21no—
22#1322—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—41no—
28C22×C42D411yescharacteristic
29C2×M32—22no—
30#432—41no—
31#432—41no—
32#431—41no—
33#431—41no—
34C2×#108—22no—
35C2×#108D414yes—
36C2×#107—22no—
37C2×#107—22no—
38C2×#107—21no—
39C2×#107C2×C412yes—
40#423—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47#407—22no—
48C4×M16—22no—
49#389—22no—
50#389—22no—
51#389—41no—
52#389—41no—
53C2×#95—41no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56#341—22no—
57#341—22no—
58#340—22no—
59#340—22no—
60#340—21no—
61#340C2×C412yes—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4SD1611yescharacteristic
65C23×C4SD1611yescharacteristic
66C23×C4C2×D411yescharacteristic
67C2×M16—24no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D412yes—
76C22×C8C2×D411yescharacteristic
77C22×C8#4112yes—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—84no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97C2×C42D811yescharacteristic
98C2×C42D811yescharacteristic
99M32—44no—
100C2×C16—42no—
101#108—44no—
102#108—82no—
103#107—24no—
104#107—42no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#97—42no—
114C4×C8—42no—
115#95—42no—
116#95—42no—
117C24C2×SD1611yescharacteristic
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—82no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4#12011yescharacteristic
142C22×C4#12311yescharacteristic
143C22×C4C2×D811yescharacteristic
144C22×C4AGL(1,ℤ/8)11yescharacteristic
145C22×C4#13711yescharacteristic
146M16—44no—
147M16—44no—
148C2×C8—24no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—44no—
158C2×C8—81no—
159C2×C8C4×D412yes—
160C2×C8#12112yes—
161C4⋊C4—24no—
162C4⋊C4—44no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C16—82no—
179C23—22no—
180C23—21no—
181C23—21no—
182C23—22no—
183C23C2×#10211yescharacteristic
184C23#44811yescharacteristic
185C23#48111yescharacteristic
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—44no—
206C2×C4—81no—
207C2×C4—88no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—162no—
214C2×C4—162no—
215C2×C4#36112yes—
216C2×C4#38611yescharacteristic
217C2×C4#41711yescharacteristic
218C2×C4#44711yescharacteristic
219C2×C4#45612yes—
220C2×C4#45812yes—
221C2×C4#48011yescharacteristic
222C8—22no—
223C8—22no—
224C8—41no—
225C8—41no—
226C8—84no—
227C22—24no—
228C22—24no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22#161112yes—
236C22#187411yescharacteristic
237C22C2×#36111yescharacteristic
238C22#214811yescharacteristic
239C22#223912yes—
240C4—21no—
241C4—21no—
242C4—22no—
243C4—41no—
244C4—41no—
245C4—41no—
246C4—84no—
247C4—162no—
248C4#161012yes—
249C4#188611yescharacteristic
250C4#189011yescharacteristic
251C2—21no—
252C2—22no—
253C2—21no—
254C2—22no—
255C2#902711yescharacteristic
256C2#1300411yescharacteristic
257C2#1300911yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20862
#219692
#128712
#114402
#72642