Finite Group Databasebeta

← #99148

Subgroups — #99148

Order 512.

622 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 102 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#13796C211yesmaximal, characteristic
2#8698C212yesmaximal
3#7032C214yesmaximal
4C2×C4×C22⋊C4C411yescharacteristic
5C2×C4×C22⋊C4C411yescharacteristic
6C22×#95C2211yescharacteristic
7#1779C2214yes—
8#1434—22no—
9#1434—22no—
10C22⋊#95C412yes—
11C22⋊#95C2212yes—
12#1280—22no—
13#1280—22no—
14C24×C4C2×C411yescharacteristic
15C24×C4C2×C411yescharacteristic
16C24×C4C2311yescharacteristic
17C22×C22⋊C4C2×C412yes—
18C22×C22⋊C4C2×C412yes—
19C22×C42—21no—
20C22×C42—21no—
21C2×C22⋊C8—24no—
22C2×C22⋊C8—24no—
23C2×C22⋊C8—24no—
24C4×C22⋊C4—22no—
25C4×C22⋊C4—24no—
26C4×C22⋊C4C814yescomplemented
27C2×#95—21no—
28C2×#95—21no—
29C2×#95—24no—
30C2×#95—22no—
31C2×#95—22no—
32C2×#95—22no—
33C2×#95—22no—
34C2×#95D412yes—
35C2×#95D412yes—
36C25C22×C411yescharacteristic
37C23×C4—22no—
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—22no—
42C23×C4—42no—
43C23×C4—42no—
44C23×C4—81no—
45C23×C4—81no—
46C23×C4—81no—
47C23×C4—81no—
48C23×C4—81no—
49C23×C4—81no—
50C23×C4C22⋊C412yes—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C22⋊C411yescharacteristic
53C23×C4C2×C812yes—
54C23×C4M1612yes—
55C23×C4C2×D412yes—
56C23×C4#4114yes—
57C22×C8—44no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—42no—
61C2×C22⋊C4—42no—
62C2×C22⋊C4—42no—
63C2×C22⋊C4—42no—
64C2×C22⋊C4C2×C814yes—
65C2×C22⋊C4M1614yes—
66C2×C42—24no—
67C2×C42—24no—
68C2×C42—81no—
69C2×C42—81no—
70C2×C42—81no—
71C2×C42—81no—
72C22⋊C8—48no—
73C22⋊C8—84no—
74C22⋊C8—84no—
75#95—24no—
76#95—24no—
77#95—24no—
78#95—24no—
79#95—82no—
80#95—82no—
81#95—82no—
82#95—82no—
83C24—22no—
84C24—22no—
85C24—41no—
86C24—41no—
87C24—41no—
88C24—41no—
89C24C2×C22⋊C411yescharacteristic
90C24#11712yes—
91C24C22×C811yescharacteristic
92C24C2×M1611yescharacteristic
93C24C2×M1612yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—24no—
99C22×C4—24no—
100C22×C4—24no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—44no—
108C22×C4—44no—
109C22×C4—44no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—82no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—84no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4—88no—
137C22×C4—88no—
138C22×C4C22⋊C812yes—
139C22×C4C22⋊C812yes—
140C22×C4#12314yes—
141C2×C8—84no—
142C2×C8—84no—
143C2×C8—84no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C42—84no—
149C42—84no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—41no—
166C23—42no—
167C23—41no—
168C23—41no—
169C23—42no—
170C23—44no—
171C23—44no—
172C23—44no—
173C23—44no—
174C23#38111yescharacteristic
175C23C2×C22⋊C811yescharacteristic
176C23#40711yescharacteristic
177C23#41012yes—
178C23#41312yes—
179C23#43112yes—
180C23#43212yes—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—84no—
204C2×C4—84no—
205C2×C4—84no—
206C2×C4—88no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—88no—
216C2×C4—84no—
217C8—164no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—44no—
233C22—44no—
234C22—44no—
235C22—44no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22#145812yes—
241C22#146412yes—
242C22#181711yescharacteristic
243C22#182011yescharacteristic
244C22#182211yescharacteristic
245C22#182811yescharacteristic
246C22#182811yescharacteristic
247C22#182911yescharacteristic
248C22#183011yescharacteristic
249C4—42no—
250C4—42no—
251C4—44no—
252C4—84no—
253C4—84no—
254C4—82no—
255C2—22no—
256C2—22no—
257C2—42no—
258C2—42no—
259C2#869911yescharacteristic
260C2#873411yescharacteristic
261C2#873511yescharacteristic
262C2#875011yescharacteristic
263C2#875511yescharacteristic
264C2#875811yescharacteristic
265C2#875911yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
26. C4×C22⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
#137962
#86982
#70322