Finite Group Databasebeta

← #140702

Subgroups — #140702

Order 512.

378 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 88 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#21147C211yesmaximal, characteristic
2#12789C211yesmaximal, characteristic
3#12606C211yesmaximal, characteristic
4#11443C212yesmaximal
5#11440C212yesmaximal
6C2×#424C411yescharacteristic
7C2×#424C2211yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9C2×#405C2211yescharacteristic
10#2296C411yescharacteristic
11C2×#364—21no—
12#2112C2212yes—
13#2112C2212yes—
14C2×#347—21no—
15#1940—21no—
16#1940—21no—
17#1939—21no—
18#1939—21no—
19#1729C411yescharacteristic
20#1728C411yescharacteristic
21#1376—21no—
22#1375—21no—
23#1323—24no—
24#1322—24no—
25C22×M16C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×#117C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C2×M32—22no—
33C2×M32—22no—
34#424—22no—
35#424—22no—
36#424—42no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45#405—42no—
46C2×#97D411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#389—21no—
51#389—21no—
52#385—21no—
53#385—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—21no—
56#364—42no—
57C22⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16—22no—
60C22⋊C16D414yes—
61#347—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66C23×C4M1611yescharacteristic
67C23×C4M1611yescharacteristic
68C23×C4C22×C411yescharacteristic
69C2×M16—22no—
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D412yes—
87C22×C8#4112yes—
88#117—22no—
89#117—22no—
90#117—22no—
91#117—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4M1611yescharacteristic
98C2×C4⋊C4M1611yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—22no—
104C2×C42—21no—
105C2×C42—22no—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C22⋊C411yescharacteristic
108M32—42no—
109M32—42no—
110M32—42no—
111M32—42no—
112C2×C16—24no—
113C2×C16—42no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—42no—
122#97—22no—
123#97—41no—
124C4×C8—22no—
125C4×C8—41no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130C24C2×M1611yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—81no—
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C4×D412yes—
147C22×C4C2×M1611yescharacteristic
148C22×C4#13112yes—
149M16—44no—
150M16—82no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—82no—
169C2×C8#10112yes—
170C2×C8#12112yes—
171C2×C8#12411yescharacteristic
172C2×C8#12711yescharacteristic
173C4⋊C4—24no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—24no—
181C22⋊C4—42no—
182C42—21no—
183C42—21no—
184C42—41no—
185C42—41no—
186C42—41no—
187C42—41no—
188C42—42no—
189C42#9912yes—
190C42#10112yes—
191C16—44no—
192C16—82no—
193C23—21no—
194C23—21no—
195C23—41no—
196C23—41no—
197C23#40711yescharacteristic
198C23#43512yes—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
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—82no—
226C2×C4#39011yescharacteristic
227C2×C4#40811yescharacteristic
228C2×C4C2×#9911yescharacteristic
229C2×C4#41011yescharacteristic
230C2×C4C2×#10111yescharacteristic
231C2×C4C2×#10111yescharacteristic
232C2×C4#43512yes—
233C8—22no—
234C8—21no—
235C8—21no—
236C8—22no—
237C8—41no—
238C8—81no—
239C8—81no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22#146311yescharacteristic
248C22#146711yescharacteristic
249C22#195511yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4#147011yescharacteristic
258C4#147211yescharacteristic
259C4#196411yescharacteristic
260C4#196611yescharacteristic
261C2—21no—
262C2—21no—
263C2—42no—
264C2#965411yescharacteristic
265C2#1320011yescharacteristic
266C2#1320111yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC23×C4C22×C4
Frattini, cosocleC23×C8C23
socleC22#1955
prime radicalC24C2×M16
LCSG ▷ C23×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
#211472
#127892
#126062
#114432
#114402