Finite Group Databasebeta

← #128332

Subgroups — #128332

Order 512.

444 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C4×#424C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic
3C4×C22⋊C16C211yesmaximal, characteristic
4C2×#1323C212yesmaximal
5#7186C212yesmaximal
6C2×#424C411yescharacteristic
7C2×#424C2211yescharacteristic
8C2×C4×M16C411yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C4×#117C411yescharacteristic
11C2×C22⋊C16C2212yes—
12C2×C22⋊C16C2212yes—
13C2×#346—21no—
14C2×C4×C16—21no—
15#1768—21no—
16#1768—21no—
17C4×C4⋊C8—21no—
18C4×C4⋊C8—21no—
19#1729C411yescharacteristic
20#1729C2211yescharacteristic
21#1374—21no—
22#1374—21no—
23#1323C418yescomplemented
24C22×M16C2×C411yescharacteristic
25C22×M16C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×#117C2×C411yescharacteristic
29C2×#117C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C16—22no—
32C22×C16—22no—
33#424—22no—
34#424—22no—
35#424C814yes—
36#424C2×C414yes—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C4×M16—42no—
42C2×#97—21no—
43C2×C4×C8—21no—
44C2×C4×C8—42no—
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C4×C4⋊C4C812yes—
50C4×C4⋊C4C812yes—
51C4×C22⋊C4—22no—
52#376—21no—
53C43—21no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16C2×C414yes—
57C22⋊C16C2×C414yes—
58#346—42no—
59C4×C16—42no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64C23×C4M1612yes—
65C23×C4C22×C411yescharacteristic
66C2×M16—21no—
67C2×M16—21no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16C4212yes—
73C2×M16C2×C812yes—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C2×C812yes—
83C22×C8M1612yes—
84C22×C8C22×C412yes—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8#4112yes—
88#117—21no—
89#117—21no—
90#117—41no—
91#117—41no—
92#117C4212yes—
93#117C2×C812yes—
94C2×C4⋊C4C2×C811yescharacteristic
95C2×C4⋊C4C2×C811yescharacteristic
96C2×C4⋊C4C2×C812yes—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—21no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42C22⋊C411yescharacteristic
107C2×C42C2×C812yes—
108C2×C16—24no—
109C2×C16—24no—
110C2×C16—42no—
111C2×C16—42no—
112C2×C16—42no—
113C2×C16—42no—
114C4⋊C8—24no—
115C4⋊C8—24no—
116C4⋊C8—24no—
117C4⋊C8—24no—
118#97—42no—
119C4×C8—22no—
120C4×C8—22no—
121C4×C8—41no—
122C4×C8—41no—
123C4×C8—42no—
124#95—22no—
125#95—22no—
126C24C2×M1611yescharacteristic
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—42no—
137C22×C4C22⋊C812yes—
138C22×C4C22⋊C812yes—
139C22×C4C2×C4211yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4#11711yescharacteristic
142C22×C4C22×C811yescharacteristic
143C22×C4C22×C811yescharacteristic
144C22×C4C2×M1611yescharacteristic
145M16—42no—
146M16—42no—
147M16—42no—
148M16—42no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8C22⋊C812yes—
168C2×C8C22⋊C812yes—
169C2×C8C4×D412yes—
170C2×C8C4×D412yes—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—42no—
174C4⋊C4C4×C812yes—
175C4⋊C4C4×C812yes—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—42no—
179C42—24no—
180C42—22no—
181C42—22no—
182C42—21no—
183C42—21no—
184C42—44no—
185C42—44no—
186C42—41no—
187C42—41no—
188C42#9912yes—
189C42#10112yes—
190C16—44no—
191C16—44no—
192C23—22no—
193C23—22no—
194C23—21no—
195C23—21no—
196C23C4×M1611yescharacteristic
197C23C2×C22⋊C811yescharacteristic
198C23#43111yescharacteristic
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—48no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4#32414yes—
222C2×C4#35014yes—
223C2×C4C4×C22⋊C411yescharacteristic
224C2×C4C2×C4×C811yescharacteristic
225C2×C4C2×C22⋊C811yescharacteristic
226C2×C4C2×#9911yescharacteristic
227C2×C4#41011yescharacteristic
228C2×C4C2×#10111yescharacteristic
229C2×C4#41311yescharacteristic
230C2×C4#43111yescharacteristic
231C8—22no—
232C8—22no—
233C8—22no—
234C8—22no—
235C8—42no—
236C8—42no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—42no—
244C22—42no—
245C22C2×#32411yescharacteristic
246C22#145811yescharacteristic
247C22C4×C22⋊C811yescharacteristic
248C22C2×#35012yes—
249C22#211712yes—
250C4—24no—
251C4—24no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—82no—
257C4#145911yescharacteristic
258C4#146111yescharacteristic
259C4C4×#9911yescharacteristic
260C4C4×#10111yescharacteristic
261C2—41no—
262C2—41no—
263C2C4×#32411yescharacteristic
264C2#1144412yes—
265C2#1146412yes—
266C2C4×#35011yescharacteristic
267C2#1260411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C2×C4×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC24C2×M16
LCSG ▷ C2×C4 ▷ C22 ▷ 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
23. #1323C4semidirect

Maximal subgroups

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

subgroupindex
C4×#4242
#125992
C4×C22⋊C162
C2×#13232
#71862