Finite Group Databasebeta

← #135490

Subgroups — #135490

Order 512.

551 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 66 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#14711C211yesmaximal, characteristic
2#8556C211yesmaximal, characteristic
3#8556C211yesmaximal, characteristic
4#7215C214yesmaximal, complemented
5C22×#126C2211yescharacteristic
6C2×#383C411yescharacteristic
7C2×C43C411yescharacteristic
8#1821—21no—
9#1821C2212yes—
10#1463—21no—
11#1463C2212yes—
12#1462—21no—
13#1462—21no—
14#1437—21no—
15#1437—21no—
16#1437C412yes—
17#1436—21no—
18#1436—21no—
19#1436C2212yes—
20C2×#126—24no—
21C2×#126—42no—
22C2×#126—42no—
23C22×C4⋊C4D411yescharacteristic
24C22×C4⋊C4D411yescharacteristic
25C22×C4⋊C4C2311yescharacteristic
26C22×C22⋊C4—22no—
27C22×C22⋊C4C2×C411yescharacteristic
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C412yes—
34#407—22no—
35#407—22no—
36#383—42no—
37C2×#95—22no—
38C2×#95—21no—
39C2×#95—21no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95C2×C412yes—
43C43—42no—
44#343—24no—
45#343—42no—
46#324—24no—
47#324—42no—
48#323—42no—
49#323—42no—
50C25C22⋊C411yescharacteristic
51C23×C4—22no—
52C23×C4—22no—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22×C411yescharacteristic
57C23×C4C2×D411yescharacteristic
58C23×C4C2×D411yescharacteristic
59C2×M16—42no—
60C2×M16—42no—
61#126—416no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4#4114yes—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—44no—
81C2×C22⋊C4—44no—
82C2×C22⋊C4—44no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—42no—
87C2×C42—42no—
88C2×C42—44no—
89C2×C42—44no—
90C2×C42—42no—
91C2×C42—44no—
92C22⋊C8—42no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96#95—42no—
97#95—44no—
98#95—44no—
99#95—44no—
100#95—42no—
101#95—42no—
102C24—21no—
103C24—21no—
104C24—21no—
105C24—42no—
106C24—42no—
107C24—42no—
108C24C2×C22⋊C411yescharacteristic
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4#9912yes—
143C22×C4#9912yes—
144C22×C4#11712yes—
145C22×C4#12312yes—
146C22×C4#12312yes—
147M16—84no—
148M16—84no—
149C2×C8—82no—
150C2×C8—82no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C22⋊C4—44no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C22⋊C4—88no—
158C42—28no—
159C42—44no—
160C42—48no—
161C42—48no—
162C42—44no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—41no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—82no—
173C23—82no—
174C23—82no—
175C23—82no—
176C23—81no—
177C23—81no—
178C23—81no—
179C23—81no—
180C23#38111yescharacteristic
181C23C2×#9911yescharacteristic
182C23C2×#9911yescharacteristic
183C2×C4—28no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C8—162no—
222C8—162no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—84no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—162no—
238C22—162no—
239C22#181911yescharacteristic
240C22#182712yes—
241C22#212012yes—
242C22#213212yes—
243C4—48no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—162no—
251C2—22no—
252C2—22no—
253C2—22no—
254C2—161no—
255C2#873911yescharacteristic
256C2#1285311yescharacteristic
257C2#1286111yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#126 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7215C2semidirect

Maximal subgroups

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

subgroupindex
#147112
#85562
#72152