Finite Group Databasebeta

← #124525

Subgroups — #124525

Order 512.

360 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 98 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#21143C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#12844C211yesmaximal, characteristic
4#7118C214yesmaximal, complemented
5C2×#424—21no—
6C2×#423C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2285C411yescharacteristic
9C2×C22⋊C16—21no—
10C2×C22⋊C16—21no—
11#1939—21no—
12#1939—21no—
13#1939C412yes—
14#1939C2212yes—
15#1729—21no—
16#1374—21no—
17#1374—21no—
18#1374C2212yes—
19#1374C2212yes—
20C22×M16—21no—
21C23×C8D411yescharacteristic
22C23×C8D411yescharacteristic
23C2×#117—21no—
24C22×C42C2×C411yescharacteristic
25C22×C16—22no—
26C22×C16—22no—
27#424—44no—
28#423—42no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8C2×C412yes—
33C2×C4×C8—42no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#384C811yescharacteristic
39#384C811yescharacteristic
40#382—21no—
41#380—21no—
42#380C812yes—
43C4×C4⋊C4—21no—
44C4×C4⋊C4C2×C412yes—
45C22⋊C16—42no—
46C22⋊C16—42no—
47#336—22no—
48#336—22no—
49C23×C4C22⋊C411yescharacteristic
50C23×C4C2×C811yescharacteristic
51C23×C4M1611yescharacteristic
52C2×M16—42no—
53C2×M16—42no—
54C2×M16—42no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8C22⋊C411yescharacteristic
63C22×C8C22⋊C411yescharacteristic
64C22×C8C2×D411yescharacteristic
65C22×C8C2×D411yescharacteristic
66#117—42no—
67#117—42no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2×C812yes—
73C2×C4⋊C4C2×C812yes—
74C2×C22⋊C4—22no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42C22⋊C411yescharacteristic
79C2×C42C2×C811yescharacteristic
80C2×C42M1611yescharacteristic
81C2×C42C22×C411yescharacteristic
82C2×C16—42no—
83C2×C16—42no—
84C2×C16—42no—
85C2×C16—42no—
86C2×C16—42no—
87C2×C16—42no—
88C4⋊C8—42no—
89C4⋊C8—42no—
90C4⋊C8—44no—
91C4⋊C8—44no—
92C4×C8—41no—
93C4×C8—41no—
94C4×C8—41no—
95C4×C8—41no—
96#95—22no—
97#95—22no—
98C24C22⋊C811yescharacteristic
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4C22⋊C811yescharacteristic
112C22×C4C22⋊C811yescharacteristic
113C22×C4C22⋊C811yescharacteristic
114C22×C4#9912yes—
115C22×C4#10112yes—
116C22×C4C2×C22⋊C411yescharacteristic
117C22×C4C22×C811yescharacteristic
118C22×C4C2×M1611yescharacteristic
119M16—84no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—44no—
131C2×C8—44no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8AGL(1,ℤ/8)12yes—
136C2×C8#13712yes—
137C4⋊C4—24no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C22⋊C4—44no—
144C42—22no—
145C42—22no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—42no—
151C42—42no—
152C16—82no—
153C16—82no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23#32412yes—
161C23C2×C22⋊C811yescharacteristic
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4#32412yes—
187C2×C4#35012yes—
188C2×C4C2×#9911yescharacteristic
189C2×C4C2×#10111yescharacteristic
190C2×C4#41811yescharacteristic
191C2×C4#41911yescharacteristic
192C2×C4#42111yescharacteristic
193C2×C4#42111yescharacteristic
194C2×C4#50412yes—
195C8—22no—
196C8—22no—
197C8—41no—
198C8—41no—
199C8—82no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22C2×#32411yescharacteristic
208C22#147811yescharacteristic
209C22#147911yescharacteristic
210C22#211311yescharacteristic
211C22#211511yescharacteristic
212C22C2×#35011yescharacteristic
213C22#211711yescharacteristic
214C4—22no—
215C4—22no—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—82no—
220C4—82no—
221C4#151011yescharacteristic
222C4#151111yescharacteristic
223C4#153411yescharacteristic
224C4#155611yescharacteristic
225C2—42no—
226C2#759311yescharacteristic
227C2#1144111yescharacteristic
228C2#1144411yescharacteristic
229C2#1163211yescharacteristic
230C2#1163311yescharacteristic
231C2#1163811yescharacteristic
232C2#1164311yescharacteristic
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. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#211432
#128442
#71182