Finite Group Databasebeta

← #124934

Subgroups — #124934

Order 512.

698 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 94 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×#1957C211yesmaximal, characteristic
2#12788C212yesmaximal, complemented
3#7107C214yesmaximal, complemented
4C22×#127C411yescharacteristic
5C22×#102—22no—
6C22×C4×C8C2211yescharacteristic
7C2×#384C411yescharacteristic
8C2×#364—22no—
9#2112—22no—
10#1957—22no—
11#1957C412yes—
12#1957C2212yes—
13#1376C2214yescomplemented
14C23×D4—21no—
15C23×D4—41no—
16C23×C8D412yes—
17C2×#127—24no—
18C2×#127—21no—
19C2×#127—41no—
20C2×#127—42no—
21C2×#127C2×C412yes—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—41no—
24C22×C42C2×C411yescharacteristic
25C2×M32—24no—
26C2×#102—24no—
27C2×#102—24no—
28C2×#102—42no—
29C2×#102—42no—
30C2×#102—42no—
31C2×#102—42no—
32C2×C4×C8—42no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D412yes—
35C2×C4×C8C2311yescharacteristic
36#384—21no—
37#384—41no—
38#384C2×C412yes—
39#364—44no—
40C22⋊C16—42no—
41C22⋊C16—42no—
42C25—41no—
43C22×D4—24no—
44C22×D4—22no—
45C22×D4—44no—
46C22×D4—42no—
47C22×D4—48no—
48C22×D4—42no—
49C22×D4—42no—
50C22×D4—41no—
51C22×D4—42no—
52C22×D4—42no—
53C22×D4—84no—
54C22×D4—84no—
55C23×C4—21no—
56C23×C4—41no—
57C23×C4C22⋊C411yescharacteristic
58C22×C8—42no—
59C22×C8—42no—
60C22×C8C22⋊C412yes—
61C22×C8D812yes—
62C22×C8D812yes—
63C22×C8SD1612yes—
64C22×C8SD1612yes—
65C22×C8C2×D412yes—
66#127—24no—
67#127—22no—
68#127—42no—
69#127—48no—
70#127—42no—
71#127M1614yes—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—84no—
79C2×C42—21no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22×C411yescharacteristic
84M32—44no—
85M32—44no—
86C2×C16—44no—
87#102—44no—
88#102—44no—
89#102—44no—
90#102—44no—
91C4×C8—24no—
92C4×C8—41no—
93C4×C8—41no—
94C24—44no—
95C24—42no—
96C24—82no—
97C24—84no—
98C24—84no—
99C24—81no—
100C24—81no—
101C24#10011yescharacteristic
102C2×D4—24no—
103C2×D4—48no—
104C2×D4—48no—
105C2×D4—44no—
106C2×D4—48no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4—42no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—82no—
113C2×D4—84no—
114C2×D4—84no—
115C2×D4—84no—
116C2×D4—84no—
117C2×D4—84no—
118C2×D4—832no—
119C2×D4—84no—
120C2×D4—82no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—42no—
130C22×C4—84no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4#10011yescharacteristic
134C22×C4#10212yes—
135C22×C4#10212yes—
136C22×C4C2×C22⋊C411yescharacteristic
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8#10212yes—
146C2×C8#10212yes—
147C2×C8C2×D812yes—
148C2×C8C2×SD1612yes—
149C4⋊C4—44no—
150C4⋊C4—84no—
151C4⋊C4—84no—
152C4⋊C4—82no—
153C42—24no—
154C42—21no—
155C42—21no—
156C42#9912yes—
157C42C2×M1612yes—
158C16—84no—
159C23—22no—
160C23—22no—
161C23—21no—
162C23—21no—
163C23—44no—
164C23—84no—
165C23—84no—
166C23—816no—
167C23—84no—
168C23—84no—
169C23—82no—
170C23—82no—
171C23—168no—
172C23—162no—
173C23—162no—
174C23—162no—
175C23—162no—
176C23#33111yescharacteristic
177C23#33111yescharacteristic
178C23C2×#10011yescharacteristic
179D4—48no—
180D4—88no—
181D4—88no—
182D4—816no—
183D4—88no—
184D4—84no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—216no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—84no—
199C2×C4—82no—
200C2×C4—162no—
201C2×C4—162no—
202C2×C4#33112yes—
203C2×C4#40711yescharacteristic
204C2×C4C2×#9911yescharacteristic
205C2×C4C2×#10212yes—
206C2×C4#42112yes—
207C8—22no—
208C8—22no—
209C8—22no—
210C8—22no—
211C22—24no—
212C22—24no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—88no—
220C22—84no—
221C22—168no—
222C22—168no—
223C22—164no—
224C22—162no—
225C22—162no—
226C22#135712yes—
227C22#135912yes—
228C22#146211yescharacteristic
229C22#153311yescharacteristic
230C22C2×#33111yescharacteristic
231C4—24no—
232C4—24no—
233C4—22no—
234C4—162no—
235C4#148612yes—
236C4#151912yes—
237C2—21no—
238C2—22no—
239C2—22no—
240C2—21no—
241C2—164no—
242C2#764011yescharacteristic
243C2C2×#135711yescharacteristic
244C2C2×#135911yescharacteristic
subgroupG/H
center, socleC22C2×#331
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×#127C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×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. #12788C2semidirect
3. #7107C2semidirect
13. #1376C22semidirect

Maximal subgroups

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

subgroupindex
C2×#19572
#127882
#71072