Finite Group Databasebeta

← #129683

Subgroups — #129683

Order 512.

745 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 116 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×#2975C211yesmaximal, characteristic
2#11636C214yesmaximal, complemented
3#11460C212yesmaximal, complemented
4C22×C4×D4C411yescharacteristic
5#2975—28no—
6C2×#436—22no—
7C2×#436C412yes—
8C2×#436C2212yes—
9C2×#431C411yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#407—22no—
12#2163C2214yescomplemented
13#1330—28no—
14#1322—22no—
15#1314—22no—
16C23×D4C811yescharacteristic
17C24×C4—21no—
18C22×M16—22no—
19C22×M16—22no—
20C2×C4×D4—24no—
21C2×C4×D4—24no—
22C2×C4×D4—21no—
23C2×C4×D4—21no—
24C2×C4×D4C812yes—
25C2×C4×D4C2×C412yes—
26C22×C4⋊C4C811yescharacteristic
27C22×C22⋊C4—21no—
28C22×C42C2×C411yescharacteristic
29#436—216no—
30#436—24no—
31#436—24no—
32#436—24no—
33#436D418yes—
34#431—24no—
35#431—24no—
36C2×C4⋊C8C2×C412yes—
37C2×C4⋊C8D412yes—
38#407—216no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8D412yes—
43C2×C4×C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46C4⋊C16—24no—
47C22⋊C16—24no—
48#346—24no—
49C25—21no—
50C22×D4—24no—
51C22×D4—21no—
52C22×D4—42no—
53C22×D4—42no—
54C22×D4C2×C812yes—
55C23×C4—24no—
56C23×C4—24no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—84no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C2×C811yescharacteristic
63C23×C4M1611yescharacteristic
64C2×M16—28no—
65C2×M16—28no—
66C2×M16—84no—
67C2×M16—84no—
68C22×C8—22no—
69C22×C8C22⋊C412yes—
70C22×C8C2×D412yes—
71C4×D4—24no—
72C4×D4—28no—
73C4×D4—24no—
74C4×D4—24no—
75C4×D4—24no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4C22⋊C414yes—
79C4×D4M1614yes—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4C2×C812yes—
83C2×C22⋊C4—28no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C2×C811yescharacteristic
90C2×C42M1611yescharacteristic
91C2×C42C22×C411yescharacteristic
92C2×C16—44no—
93C4⋊C8—24no—
94C4⋊C8—22no—
95C4⋊C8C22⋊C414yes—
96C22⋊C8—24no—
97C22⋊C8—28no—
98C22⋊C8—28no—
99C22⋊C8—24no—
100C4×C8—24no—
101C4×C8—22no—
102C4×C8C2×D414yes—
103C24—24no—
104C24—22no—
105C24—42no—
106C24—42no—
107C24—82no—
108C24—82no—
109C24C22⋊C811yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—416no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4C22⋊C814yes—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—28no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—21no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—42no—
132C22×C4—88no—
133C22×C4—88no—
134C22×C4—82no—
135C22×C4—84no—
136C22×C4—81no—
137C22×C4C22⋊C811yescharacteristic
138C22×C4C22⋊C811yescharacteristic
139C22×C4C22⋊C811yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C22×C811yescharacteristic
142C22×C4C2×M1611yescharacteristic
143M16—88no—
144M16—88no—
145C2×C8—22no—
146C2×C8—24no—
147C2×C8—42no—
148C2×C8—44no—
149C2×C8—42no—
150C2×C8#12014yes—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4C22⋊C814yes—
154C22⋊C4—28no—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C42—21no—
158C42—22no—
159C42—21no—
160C42—22no—
161C42C2×C22⋊C412yes—
162C42C2×M1612yes—
163C16—84no—
164C23—21no—
165C23—21no—
166C23—24no—
167C23—41no—
168C23—41no—
169C23—44no—
170C23—48no—
171C23—44no—
172C23—42no—
173C23—42no—
174C23—84no—
175C23—84no—
176C23—88no—
177C23—88no—
178C23#34912yes—
179C23C2×C22⋊C811yescharacteristic
180D4—44no—
181D4—48no—
182D4—44no—
183C2×C4—28no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—41no—
193C2×C4—48no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—82no—
202C2×C4—84no—
203C2×C4—88no—
204C2×C4#34912yes—
205C2×C4#37911yescharacteristic
206C2×C4C2×C22⋊C812yes—
207C2×C4#40712yes—
208C2×C4#40711yescharacteristic
209C8—24no—
210C8—44no—
211C8—42no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—42no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—44no—
220C22—44no—
221C22—44no—
222C22—84no—
223C22—84no—
224C22—816no—
225C22#177911yescharacteristic
226C22C2×#34911yescharacteristic
227C22C2×#34911yescharacteristic
228C4—21no—
229C4—22no—
230C4—22no—
231C4—22no—
232C4—44no—
233C4—41no—
234C4—41no—
235C4#179512yes—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—84no—
240C2#832211yescharacteristic
241C2#1268111yescharacteristic
242C2#1268611yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC23×D4C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11636C2semidirect
3. #11460C2semidirect
12. #2163C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29752
#116362
#114602