Finite Group Databasebeta

← #124565

Subgroups — #124565

Order 512.

492 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 110 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#21373C211yesmaximal, characteristic
2#12783C212yesmaximal, complemented
3#7128C212yesmaximal, complemented
4#7102C212yesmaximal, complemented
5C2×C4×#41C411yescharacteristic
6C2×#431C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C2×C4⋊C16—22no—
9C2×C22⋊C16—22no—
10#1984C411yescharacteristic
11#1984C2211yescharacteristic
12#1954C411yescharacteristic
13#1954C2211yescharacteristic
14#1869—22no—
15C4×C22⋊C8—22no—
16#1374C2214yes—
17C22×#41C811yescharacteristic
18C23×C8D412yes—
19C4×#41—42no—
20C4×#41—42no—
21C2×C4×Q8C811yescharacteristic
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4—21no—
25C2×C4×D4C811yescharacteristic
26C2×C4×D4C2×C411yescharacteristic
27C2×#117—21no—
28C2×#117C811yescharacteristic
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C16—24no—
32#431—42no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C22⋊C8—21no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C4×C8—21no—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D412yes—
42C2×C4×C8C2311yescharacteristic
43C4⋊C16—44no—
44C22⋊C16—44no—
45#336—22no—
46#336—22no—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—42no—
50C2×#41—42no—
51C22×Q8C2×C811yescharacteristic
52C22×D4—21no—
53C22×D4C2×C811yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C2×C811yescharacteristic
59C23×C4M1611yescharacteristic
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—42no—
65C22×C8—42no—
66C22×C8C22⋊C412yes—
67C22×C8C2×D412yes—
68C4×Q8—42no—
69C4×Q8—41no—
70C4×Q8—41no—
71C4×D4—42no—
72C4×D4—41no—
73C4×D4—41no—
74C4×D4—44no—
75C4×D4—42no—
76C4×D4—42no—
77#117—44no—
78#117—42no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2×C811yescharacteristic
82C2×C4⋊C4C2×C811yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42C22⋊C411yescharacteristic
94C2×C42C2×C811yescharacteristic
95C2×C42M1611yescharacteristic
96C2×C42C22×C411yescharacteristic
97C2×C16—44no—
98C2×C16—44no—
99C2×C16—44no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C22⋊C8—44no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C4×C8—24no—
107C4×C8—41no—
108C4×C8—41no—
109C4×C8—42no—
110C24—21no—
111C24C22⋊C811yescharacteristic
112#41—48no—
113#41—48no—
114C2×Q8—21no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8M3212yes—
118C2×D4—24no—
119C2×D4—21no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4M3212yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—24no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C22⋊C811yescharacteristic
148C22×C4C22⋊C811yescharacteristic
149C22×C4C22⋊C811yescharacteristic
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4C22×C811yescharacteristic
152C22×C4C2×M1611yescharacteristic
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8AGL(1,ℤ/8)12yes—
165C2×C8#13712yes—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C42—22no—
174C42—22no—
175C42—21no—
176C42—21no—
177C42—42no—
178C42—42no—
179C42—41no—
180C42—41no—
181C42#9912yes—
182C42#10012yes—
183C16—84no—
184C23—21no—
185C23—22no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—42no—
191C23—41no—
192C23—42no—
193C23#34912yes—
194C23C2×C22⋊C811yescharacteristic
195Q8—42no—
196Q8—42no—
197D4—42no—
198D4—42no—
199D4—48no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—48no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—84no—
225C2×C4—82no—
226C2×C4#32312yes—
227C2×C4#32312yes—
228C2×C4#35014yes—
229C2×C4C2×#9911yescharacteristic
230C2×C4C2×#10011yescharacteristic
231C2×C4#41712yes—
232C2×C4#42112yes—
233C2×C4C2×M3211yescharacteristic
234C2×C4C2×M3211yescharacteristic
235C8—22no—
236C8—22no—
237C8—42no—
238C8—81no—
239C8—81no—
240C22—24no—
241C22—22no—
242C22—42no—
243C22—41no—
244C22—41no—
245C22—44no—
246C22—42no—
247C22—82no—
248C22—81no—
249C22—81no—
250C22C2×#32311yescharacteristic
251C22#147712yes—
252C22#211211yescharacteristic
253C22C2×#34911yescharacteristic
254C22C2×#35012yes—
255C4—22no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—81no—
262C4—81no—
263C4#131514yes—
264C4#151312yes—
265C4#153611yescharacteristic
266C4#156011yescharacteristic
267C2—42no—
268C2—82no—
269C2#759711yescharacteristic
270C2#1141811yescharacteristic
271C2C2×#131511yescharacteristic
272C2#1162312yes—
273C2#1162612yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC22×#41C8
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
2. #12783C2semidirect
3. #7128C2semidirect
4. #7102C2semidirect

Maximal subgroups

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

subgroupindex
#213732
#127832
#71282
#71022