Finite Group Databasebeta

← #124306

Subgroups — #124306

Order 512.

494 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 122 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×#1984C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3#12599C211yesmaximal, characteristic
4#7128C214yesmaximal, complemented
5C22×C4×Q8C411yescharacteristic
6C22×C4⋊C8C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C2×C4⋊C16—21no—
9C2×C22⋊C16—21no—
10C2×C22⋊C16—21no—
11C2×#346—21no—
12#1984—21no—
13#1984—21no—
14#1984C412yes—
15#1984C2212yes—
16C2×#336—21no—
17C2×#336—21no—
18#1374C2212yes—
19#1374C2212yes—
20C23×Q8C811yescharacteristic
21C23×C8—21no—
22C23×C8D411yescharacteristic
23C23×C8D411yescharacteristic
24C2×C4×Q8—21no—
25C2×C4×Q8—21no—
26C2×C4×Q8—42no—
27C2×C4×Q8—42no—
28C2×C4×Q8C812yes—
29C2×C4×Q8C2×C412yes—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C811yescharacteristic
32C22×C42—21no—
33C22×C42C2×C411yescharacteristic
34C22×C16—22no—
35C22×C16—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—42no—
38C2×C4⋊C8C2×C412yes—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44C4⋊C16—42no—
45C22⋊C16—24no—
46C22⋊C16—42no—
47#346—42no—
48#336—22no—
49#336—22no—
50#336—22no—
51#336—22no—
52C22×Q8—24no—
53C22×Q8—21no—
54C22×Q8—42no—
55C22×Q8—42no—
56C22×Q8C2×C812yes—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C2×C811yescharacteristic
61C23×C4M1611yescharacteristic
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—42no—
69C22×C8—42no—
70C22×C8—42no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8SD1612yes—
74C22×C8Q1612yes—
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C4×Q8—44no—
78C4×Q8—44no—
79C4×Q8—42no—
80C4×Q8—42no—
81C4×Q8—42no—
82C4×Q8—42no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4C2×C812yes—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—42no—
94C2×C42—42no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C2×C811yescharacteristic
97C2×C42M1611yescharacteristic
98C2×C42C22×C411yescharacteristic
99C2×C16—24no—
100C2×C16—42no—
101C2×C16—42no—
102C2×C16—42no—
103C2×C16—42no—
104C2×C16—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4×C8—22no—
108C4×C8—22no—
109C4×C8—41no—
110C4×C8—41no—
111C24C22⋊C811yescharacteristic
112C2×Q8—24no—
113C2×Q8—22no—
114C2×Q8—416no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8M3214yes—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4C22⋊C811yescharacteristic
139C22×C4C22⋊C811yescharacteristic
140C22×C4C22⋊C811yescharacteristic
141C22×C4#10312yes—
142C22×C4#10412yes—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C22×C811yescharacteristic
145C22×C4C2×M1611yescharacteristic
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—84no—
162C2×C8#10312yes—
163C2×C8C2×SD1611yescharacteristic
164C2×C8C2×Q1611yescharacteristic
165C2×C8#13712yes—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C42—22no—
171C42—22no—
172C42—21no—
173C42—21no—
174C42—44no—
175C42—42no—
176C42—42no—
177C42#9912yes—
178C42#10012yes—
179C16—44no—
180C16—82no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23Q8⋊C812yes—
186C23#34912yes—
187C23#35012yes—
188C23C2×C22⋊C811yescharacteristic
189Q8—44no—
190Q8—48no—
191Q8—44no—
192C2×C4—28no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—44no—
210C2×C4—48no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—84no—
217C2×C4#32312yes—
218C2×C4#32312yes—
219C2×C4Q8⋊C812yes—
220C2×C4C2×#9911yescharacteristic
221C2×C4C2×#10011yescharacteristic
222C2×C4C2×#10311yescharacteristic
223C2×C4#41911yescharacteristic
224C2×C4C2×#10411yescharacteristic
225C2×C4#42111yescharacteristic
226C2×C4C2×M3212yes—
227C8—22no—
228C8—21no—
229C8—22no—
230C8—21no—
231C8—41no—
232C8—82no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22#133814yes—
240C22C2×#32311yescharacteristic
241C22C2×Q8⋊C811yescharacteristic
242C22#147911yescharacteristic
243C22#211211yescharacteristic
244C22C2×#34911yescharacteristic
245C22C2×#35011yescharacteristic
246C22#211711yescharacteristic
247C4—22no—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—44no—
252C4—41no—
253C4—82no—
254C4#150811yescharacteristic
255C4#152011yescharacteristic
256C4#152711yescharacteristic
257C4#154811yescharacteristic
258C2—24no—
259C2#757511yescharacteristic
260C2#1141811yescharacteristic
261C2#1142211yescharacteristic
262C2#1160711yescharacteristic
263C2C2×#133811yescharacteristic
264C2#1163911yescharacteristic
265C2#1164411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7128C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
#127832
#125992
#71282