Finite Group Databasebeta

← #151838

Subgroups — #151838

Order 512.

658 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C2×#2132C212yesmaximal
2#21195C211yesmaximal, characteristic
3#7499C214yesmaximal, complemented
4C2×#532C2211yescharacteristic
5C2×#416—22no—
6C2×#413—22no—
7C22×C4×C8C411yescharacteristic
8#2285C411yescharacteristic
9#2132—24no—
10#2132C418yescomplemented
11C2×#356C2214yes—
12C2×#355—22no—
13#1958—22no—
14#1957C412yes—
15#1956C2212yes—
16C22×#41D412yes—
17C22×M16—42no—
18C23×C8—21no—
19#532—42no—
20#532C2×C414yes—
21#532D412yes—
22#532Q812yes—
23C2×C4⋊Q8C2311yescharacteristic
24C2×#127C2×C411yescharacteristic
25C2×#124D412yes—
26C2×#121—41no—
27C2×C4×D4—41no—
28C2×#117—21no—
29C22×C42C2×C411yescharacteristic
30#416—44no—
31#416—44no—
32C2×#103—24no—
33C2×#102—24no—
34#413—28no—
35#413—44no—
36C2×#101—24no—
37C2×#100—42no—
38C2×C4×C8—21no—
39C2×C4×C8—42no—
40C2×C4×C8C2×C412yes—
41#384C2×C412yes—
42#382—21no—
43#380—41no—
44C4×C4⋊C4—41no—
45#356—24no—
46#356C2×C418yes—
47#355—28no—
48C2×#41—24no—
49C2×#41—44no—
50C2×#41—42no—
51C2×#41—42no—
52C2×#41C22⋊C414yes—
53C2×#41C4⋊C414yes—
54C22×Q8C2×D412yes—
55C22×D4—22no—
56C22×D4—41no—
57C22×D4C22⋊C412yes—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4C22⋊C411yescharacteristic
61C2×M16—44no—
62C2×M16—48no—
63C2×M16—82no—
64C2×M16—84no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—42no—
69C22×C8—42no—
70C22×C8—82no—
71C4⋊Q8—41no—
72C4⋊Q8C22×C412yes—
73C4⋊Q8C2×D411yescharacteristic
74C4⋊Q8C2×Q811yescharacteristic
75#127—41no—
76#127C4212yes—
77#127C4⋊C412yes—
78#124—42no—
79#124C22⋊C414yes—
80#124C4⋊C414yes—
81#121—44no—
82#121—42no—
83#121—42no—
84C4×D4—44no—
85C4×D4—42no—
86C4×D4—42no—
87#117—42no—
88#117—42no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—41no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—41no—
97C2×C42C4212yes—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C4⋊C412yes—
100C2×C42C22×C411yescharacteristic
101#103—48no—
102#102—48no—
103#101—28no—
104#101—44no—
105#100—48no—
106C4×C8—42no—
107C4×C8—42no—
108#95—42no—
109C24—22no—
110C24#9911yescharacteristic
111#41—44no—
112#41—44no—
113#41—44no—
114#41—44no—
115#41—84no—
116#41—84no—
117C2×Q8—22no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8C2×C22⋊C412yes—
121C2×Q8C2×C4⋊C412yes—
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—84no—
130C2×D4—82no—
131C2×D4—82no—
132C2×D4#9514yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4#9512yes—
149C22×C4#9912yes—
150C22×C4#9911yescharacteristic
151C22×C4C2×C22⋊C411yescharacteristic
152M16—48no—
153M16—84no—
154M16—88no—
155C2×C8—24no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—42no—
160C2×C8—84no—
161C2×C8—88no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—44no—
166C4⋊C4—84no—
167C22⋊C4—44no—
168C22⋊C4—44no—
169C22⋊C4—82no—
170C42—22no—
171C42—22no—
172C42—21no—
173C42—21no—
174C42—81no—
175C42—81no—
176C42#9512yes—
177C42C2×C4211yescharacteristic
178C42C2×C4⋊C411yescharacteristic
179C23—21no—
180C23—21no—
181C23—24no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—84no—
186C23#34212yes—
187C23C2×#9911yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—82no—
191D4—44no—
192D4—82no—
193D4—84no—
194D4—84no—
195D4—84no—
196D4—84no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—84no—
219C2×C4—82no—
220C2×C4#34212yes—
221C2×C4#35311yescharacteristic
222C2×C4#35311yescharacteristic
223C2×C4#35412yes—
224C2×C4C2×#9511yescharacteristic
225C2×C4C2×#9911yescharacteristic
226C8—44no—
227C8—88no—
228C22—22no—
229C22—22no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—44no—
234C22—84no—
235C22—84no—
236C22—162no—
237C22C2×#34211yescharacteristic
238C22C2×#35311yescharacteristic
239C22C2×#35411yescharacteristic
240C4—22no—
241C4—22no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—162no—
248C4#139312yes—
249C4#139312yes—
250C2—21no—
251C2—21no—
252C2—42no—
253C2—84no—
254C2C2×#139311yescharacteristic
255C2#1358912yes—
subgroupG/H
centerC2×C4#353
commutatorC42C2×C42
Frattini, cosocleC2×C4⋊Q8C23
socleC22C2×#353
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G
LECSG ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7499C2semidirect
10. #2132C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21322
#211952
#74992