Finite Group Databasebeta

← #135467

Subgroups — #135467

Order 512.

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

normal subgroups only

This group has 344 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
1#20546C211yesmaximal, characteristic
2#11440C211yesmaximal, characteristic
3C2×#1322C211yesmaximal, characteristic
4#7202C212yesmaximal
5#7200C212yesmaximal
6C2×#432C411yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8C4×#117C411yescharacteristic
9#2112C2212yes—
10C2×C22⋊C16C2212yes—
11#1771—21no—
12#1771—21no—
13#1767—21no—
14C4×C4⋊C8—21no—
15#1752C411yescharacteristic
16#1752C2211yescharacteristic
17C8×C22⋊C4C411yescharacteristic
18C8×C22⋊C4C2211yescharacteristic
19#1322—21no—
20#1322—21no—
21#1322—21no—
22#1322—21no—
23C23×C8C2×C411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#117—21no—
26C22×C42C2×C411yescharacteristic
27C2×M32—22no—
28C22×C16—22no—
29#432—22no—
30#432—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8D411yescharacteristic
35C2×C4⋊C8D411yescharacteristic
36C2×C4⋊C8D411yescharacteristic
37C2×C4⋊C8D411yescharacteristic
38C2×C22⋊C8C2×C411yescharacteristic
39C2×C22⋊C8C2×C411yescharacteristic
40C2×#97—21no—
41C2×#97—21no—
42C2×#97—21no—
43C2×C4×C8—21no—
44C2×C4×C8—21no—
45C4×C4⋊C4—21no—
46C4×C4⋊C4C811yescharacteristic
47C4×C4⋊C4C811yescharacteristic
48C4×C22⋊C4—21no—
49C4×C22⋊C4C2×C411yescharacteristic
50C4×C22⋊C4C2×C411yescharacteristic
51#376C811yescharacteristic
52#376C811yescharacteristic
53C43—21no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16—22no—
57C22⋊C16—22no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C23×C4M1611yescharacteristic
63C23×C4M1611yescharacteristic
64C23×C4C22×C411yescharacteristic
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—81no—
70C22×C8—81no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8#4112yes—
76C22×C8#4112yes—
77#117—44no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22⋊C411yescharacteristic
94C2×C42C2×C811yescharacteristic
95C2×C42C2×C811yescharacteristic
96M32—82no—
97C2×C16—42no—
98C2×C16—42no—
99C2×C16—82no—
100C4⋊C8—24no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—22no—
104C4⋊C8—22no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C22⋊C8M1612yes—
108C22⋊C8M1612yes—
109#97—24no—
110#97—81no—
111#97—81no—
112C4×C8—81no—
113C4×C8—81no—
114#95—21no—
115#95—21no—
116#95C2×C811yescharacteristic
117#95C2×C811yescharacteristic
118#95C2×C811yescharacteristic
119#95C2×C811yescharacteristic
120C24C2×M1611yescharacteristic
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4C22⋊C811yescharacteristic
135C22×C4C22⋊C811yescharacteristic
136C22×C4C22⋊C811yescharacteristic
137C22×C4C22⋊C811yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4#11711yescharacteristic
140C22×C4#11711yescharacteristic
141C22×C4C22×C811yescharacteristic
142C22×C4C2×M1611yescharacteristic
143C22×C4C2×M1611yescharacteristic
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—81no—
159C2×C8—81no—
160C2×C8#12312yes—
161C2×C8#12312yes—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C22⋊C4—24no—
165C22⋊C4—42no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—41no—
175C42—41no—
176C42—42no—
177C42—41no—
178C42—41no—
179C42—42no—
180C42—42no—
181C42—42no—
182C16—82no—
183C16—82no—
184C23—21no—
185C23—21no—
186C23—22no—
187C23—41no—
188C23C2×C22⋊C811yescharacteristic
189C23#43211yescharacteristic
190C23#43211yescharacteristic
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—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4#35012yes—
226C2×C4#38111yescharacteristic
227C2×C4#40711yescharacteristic
228C2×C4#43111yescharacteristic
229C2×C4#43111yescharacteristic
230C8—22no—
231C8—22no—
232C8—42no—
233C8—42no—
234C8—41no—
235C8—41no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—42no—
244C22—42no—
245C22#181711yescharacteristic
246C22C2×#35011yescharacteristic
247C22#211711yescharacteristic
248C22#216512yes—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—22no—
253C4—42no—
254C4—42no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4#182811yescharacteristic
261C4#183011yescharacteristic
262C2—21no—
263C2—21no—
264C2—22no—
265C2—41no—
266C2#873511yescharacteristic
267C2#1284811yescharacteristic
268C2#1284911yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1817
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×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
#205462
#114402
C2×#13222
#72022
#72002