Finite Group Databasebeta

← #122399

Subgroups — #122399

Order 512.

311 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 76 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#21566C211yesmaximal, characteristic
2#11487C211yesmaximal, characteristic
3#11465C211yesmaximal, characteristic
4#7067C212yesmaximal, complemented
5#7059C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C2×#405C411yescharacteristic
8#2395C411yescharacteristic
9#2162—21no—
10C2×C22⋊C16—21no—
11#1924—21no—
12#1917—21no—
13#1870C411yescharacteristic
14#1851C2211yescharacteristic
15#1818C2211yescharacteristic
16C8×C4⋊C4—21no—
17C8×C22⋊C4—21no—
18C8×C22⋊C4C411yescharacteristic
19#1322—21no—
20#1314—21no—
21#1314C2212yes—
22#1314C2212yes—
23C22×M16C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C4×D4C2×C411yescharacteristic
27C2×#117C2×C411yescharacteristic
28C22×C16—41no—
29C8×D4—41no—
30C8×D4—41no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8D411yescharacteristic
33C2×C22⋊C8—21no—
34C2×C22⋊C8C2×C411yescharacteristic
35C2×C22⋊C8D411yescharacteristic
36C2×C22⋊C8C2311yescharacteristic
37#405—41no—
38#405—41no—
39C2×#97—21no—
40C2×C4×C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8D411yescharacteristic
43#400C811yescharacteristic
44#399C811yescharacteristic
45#398C811yescharacteristic
46#397—21no—
47#397C811yescharacteristic
48#394—21no—
49C4×C4⋊C4—21no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4C2×C411yescharacteristic
52C4⋊C16—41no—
53C22⋊C16—22no—
54C22⋊C16—22no—
55C22⋊C16—41no—
56C4×C16—41no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C22×D4M1611yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C2×M16—42no—
66C2×M16—41no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—42no—
74C22×C8—42no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C4×D4—41no—
80C4×D4—41no—
81#117—41no—
82#117—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2×C811yescharacteristic
86C2×C4⋊C4C2×C811yescharacteristic
87C2×C4⋊C4C2×C811yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4C2×C811yescharacteristic
93C2×C22⋊C4M1611yescharacteristic
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C22⋊C411yescharacteristic
97C2×C16—42no—
98C2×C16—42no—
99C2×C16—82no—
100C4⋊C8—41no—
101C4⋊C8—42no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—42no—
105C22⋊C8—41no—
106#97—42no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—42no—
110C4×C8—41no—
111C4×C8—41no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95C2×C811yescharacteristic
117#95C2×C811yescharacteristic
118C24C22⋊C811yescharacteristic
119C24C2×M1611yescharacteristic
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—41no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4C22⋊C811yescharacteristic
136C22×C4C22⋊C811yescharacteristic
137C22×C4C22⋊C811yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C22×C811yescharacteristic
140M16—42no—
141M16—42no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—42no—
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#9911yescharacteristic
160C2×C8#9911yescharacteristic
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C22⋊C4—41no—
167C22⋊C4—41no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—21no—
173C42—21no—
174C42—41no—
175C42—41no—
176C42—41no—
177C42—41no—
178C16—82no—
179C16—82no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23#35012yes—
190C23C2×C22⋊C811yescharacteristic
191D4—44no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—81no—
220C2×C4#35012yes—
221C2×C4C2×#9911yescharacteristic
222C2×C4#41311yescharacteristic
223C8—21no—
224C8—41no—
225C8—41no—
226C8—41no—
227C8—42no—
228C8—81no—
229C8#35212yes—
230C22—21no—
231C22—21no—
232C22—22no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#145911yescharacteristic
243C22C2×#35011yescharacteristic
244C22C2×#35011yescharacteristic
245C4—22no—
246C4—21no—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—81no—
252C4—81no—
253C4—81no—
254C4C2×#35211yescharacteristic
255C4#212011yescharacteristic
256C2—21no—
257C2—21no—
258C2—42no—
259C2—81no—
260C2#1146811yescharacteristic
261C2#1147211yescharacteristic
262C2#1148511yescharacteristic
subgroupG/H
centerC2×C8#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C22⋊C8C23
socleC22#1459
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7067C2semidirect
5. #7059C2semidirect

Maximal subgroups

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

subgroupindex
#215662
#114872
#114652
#70672
#70592