Finite Group Databasebeta

← #129259

Subgroups — #129259

Order 512.

867 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 867 conjugacy classes of subgroups — too many to draw. Only its 134 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#34249C211yesmaximal, characteristic
2#11448C214yesmaximal
3#8303C212yesmaximal
4C2×#519C411yescharacteristic
5C2×#519C411yescharacteristic
6C22×C4×Q8C2211yescharacteristic
7#2351—24no—
8Q8×C4⋊C4—24no—
9#2317—22no—
10#2317C2212yes—
11#2296—22no—
12#2296C412yes—
13#1817—22no—
14C2×#324—22no—
15C2×#324C2214yes—
16#1325—24no—
17#1324—24no—
18C23×Q8C2×C411yescharacteristic
19#519—22no—
20#519—24no—
21#519—44no—
22#519C814yescomplemented
23C2×C4×Q8—22no—
24C2×C4×Q8—22no—
25C2×C4×Q8—24no—
26C2×C4×Q8—24no—
27C2×C4×Q8—44no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117C2×C412yes—
31C2×#117C2×C412yes—
32C22×C4⋊C4D412yes—
33C22×C4⋊C4C2311yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42D412yes—
36C2×C22⋊C8—24no—
37#389—28no—
38#389—24no—
39#389—24no—
40#389—22no—
41#389—22no—
42#389—24no—
43#385—22no—
44#385—22no—
45#385—24no—
46#384—24no—
47#384D414yes—
48#382—24no—
49#382D414yes—
50C4×C4⋊C4—28no—
51C4×C4⋊C4—24no—
52#336—44no—
53#324—28no—
54#324—44no—
55C22×Q8—24no—
56C22×Q8—24no—
57C22×Q8—22no—
58C22×Q8—22no—
59C22×Q8C2×C812yes—
60C22×Q8M1612yes—
61C23×C4C22⋊C412yes—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D412yes—
66C22×C8—44no—
67C4×Q8—216no—
68C4×Q8—216no—
69C4×Q8—416no—
70C4×Q8—48no—
71C4×Q8—48no—
72#117—28no—
73#117—28no—
74#117—22no—
75#117—22no—
76#117—44no—
77#117—44no—
78#117—44no—
79#117C2×C814yes—
80#117M1614yes—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4C22⋊C412yes—
93C2×C4⋊C4C22⋊C412yes—
94C2×C4⋊C4C2×D414yes—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C42—24no—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—24no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42C22⋊C414yes—
106C22⋊C8—44no—
107C22⋊C8—44no—
108#95—24no—
109#95—24no—
110#95—28no—
111C24C2×C22⋊C411yescharacteristic
112C2×Q8—28no—
113C2×Q8—216no—
114C2×Q8—28no—
115C2×Q8—28no—
116C2×Q8—28no—
117C2×Q8—24no—
118C2×Q8—24no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—82no—
136C22×C4—81no—
137C22×C4C22⋊C814yes—
138C22×C4#9914yes—
139C22×C4#10114yes—
140C22×C4C2×C22⋊C412yes—
141C22×C4#12014yes—
142C22×C4C22×C811yescharacteristic
143C22×C4C2×M1612yes—
144C22×C4C2×M1611yescharacteristic
145C2×C8—84no—
146C2×C8—88no—
147C4⋊C4—216no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—44no—
151C4⋊C4—48no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—48no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—88no—
158C4⋊C4—84no—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—44no—
162C42—216no—
163C42—24no—
164C42—48no—
165C42—44no—
166C42—48no—
167C42—44no—
168C42C22⋊C818yes—
169C23—41no—
170C23—41no—
171C23#37911yescharacteristic
172C23C2×C22⋊C811yescharacteristic
173C23#40711yescharacteristic
174C23C2×#9912yes—
175C23C2×#10112yes—
176Q8—216no—
177Q8—216no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—28no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4#32418yes—
209C2×C4C2×C22⋊C812yes—
210C2×C4#40712yes—
211C8—164no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—22no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22C2×#32412yes—
220C22#146312yes—
221C22#177911yescharacteristic
222C22#178111yescharacteristic
223C22#178411yescharacteristic
224C4—28no—
225C4—22no—
226C4—22no—
227C4—48no—
228C4—44no—
229C4—42no—
230C4—82no—
231C4—81no—
232C4—82no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2—21no—
237C2#829911yescharacteristic
238C2#1270811yescharacteristic
239C2#1271011yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #519C8semidirect

Maximal subgroups

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

subgroupindex
#342492
#114482
#83032