Finite Group Databasebeta

← #124576

Subgroups — #124576

Order 512.

386 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 86 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#21376C211yesmaximal, characteristic
2#12785C212yesmaximal, complemented
3#7133C212yesmaximal, complemented
4#7105C212yesmaximal, complemented
5C2×#529C411yescharacteristic
6C2×#432C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8#2150—22no—
9C2×#349—22no—
10#1985C411yescharacteristic
11#1985C2211yescharacteristic
12#1955C411yescharacteristic
13#1955C2211yescharacteristic
14#1870—22no—
15#1750—22no—
16#1377C2214yes—
17C22×M16D412yes—
18#529—42no—
19#529—42no—
20C2×#126—21no—
21C2×#125C811yescharacteristic
22C2×#124C811yescharacteristic
23C2×#123—21no—
24C2×C22⋊Q8C811yescharacteristic
25C2×#121C811yescharacteristic
26C2×C4×Q8C2×C411yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2×C411yescharacteristic
29C2×#117—21no—
30C22×C42C2×C411yescharacteristic
31C2×M32—24no—
32#432—42no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C22⋊C8—21no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C4×M16—42no—
39C2×#97—21no—
40C2×#97C2×C411yescharacteristic
41C2×#97C2311yescharacteristic
42C2×C4×C8D412yes—
43C4⋊C16—44no—
44#349—44no—
45#336—22no—
46#336—22no—
47C22×Q8C2×C811yescharacteristic
48C22×D4—21no—
49C22×D4C2×C811yescharacteristic
50C23×C4—21no—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C2×C811yescharacteristic
53C23×C4M1611yescharacteristic
54C2×M16—22no—
55C2×M16—22no—
56C2×M16—42no—
57C2×M16—42no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8C22⋊C412yes—
61C22×C8C2×D412yes—
62#126—44no—
63#125—42no—
64#124—42no—
65#123—44no—
66C22⋊Q8—42no—
67#121—42no—
68C4×Q8—41no—
69C4×Q8—41no—
70C4×D4—41no—
71C4×D4—41no—
72C4×D4—42no—
73C4×D4—42no—
74#117—42no—
75#117—42no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4C2×C811yescharacteristic
81C2×C4⋊C4C2×C811yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C2×C811yescharacteristic
92C2×C42M1611yescharacteristic
93C2×C42C22×C411yescharacteristic
94M32—44no—
95M32—44no—
96C2×C16—44no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C22⋊C8—44no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103#97—41no—
104#97—41no—
105#97—42no—
106C4×C8—24no—
107C24—21no—
108C24C22⋊C811yescharacteristic
109C2×Q8—41no—
110C2×Q8—81no—
111C2×D4—41no—
112C2×D4—42no—
113C2×D4—82no—
114C2×D4—81no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4C22⋊C811yescharacteristic
131C22×C4C22⋊C811yescharacteristic
132C22×C4C22⋊C811yescharacteristic
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C22×C811yescharacteristic
135C22×C4C2×M1611yescharacteristic
136M16—44no—
137M16—44no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8AGL(1,ℤ/8)12yes—
148C2×C8#13712yes—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C42—22no—
161C42—22no—
162C42—21no—
163C42—21no—
164C42—41no—
165C42—41no—
166C42#9912yes—
167C42#10012yes—
168C16—84no—
169C23—21no—
170C23—22no—
171C23—41no—
172C23—41no—
173C23—42no—
174C23—42no—
175C23—41no—
176C23—42no—
177C23C2×C22⋊C811yescharacteristic
178Q8—82no—
179D4—82no—
180D4—84no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4#32312yes—
204C2×C4#32312yes—
205C2×C4C2×#9911yescharacteristic
206C2×C4C2×#10011yescharacteristic
207C2×C4#41712yes—
208C2×C4#42112yes—
209C8—22no—
210C8—22no—
211C8—42no—
212C8—81no—
213C8—81no—
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—82no—
220C22—84no—
221C22—81no—
222C22—81no—
223C22—81no—
224C22C2×#32311yescharacteristic
225C22#147712yes—
226C4—22no—
227C4—22no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—81no—
233C4—81no—
234C4#151312yes—
235C4#153611yescharacteristic
236C4#156011yescharacteristic
237C2—22no—
238C2—81no—
239C2—161no—
240C2#759711yescharacteristic
241C2#1143611yescharacteristic
242C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#323
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12785C2semidirect
3. #7133C2semidirect
4. #7105C2semidirect

Maximal subgroups

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

subgroupindex
#213762
#127852
#71332
#71052