Finite Group Databasebeta

← #124154

Subgroups — #124154

Order 512.

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

normal subgroups only

This group has 542 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#14295C211yesmaximal, characteristic
2C4×#323C212yesmaximal
3#7123C214yesmaximal, complemented
4C2×#381C411yescharacteristic
5C2×#380C411yescharacteristic
6C2×C4×C22⋊C4C2211yescharacteristic
7C4×C22⋊C8—22no—
8C2×#323—22no—
9#1444C2212yes—
10#1442C412yes—
11#1439—22no—
12#1436—22no—
13#1280C2214yes—
14C24×C4C2×C411yescharacteristic
15C22×C4⋊C4—22no—
16C22×C4⋊C4C2×C412yes—
17C22×C22⋊C4—21no—
18C22×C22⋊C4—21no—
19C22×C22⋊C4D412yes—
20C22×C42D412yes—
21C2×C22⋊C8—24no—
22C2×C4×C8—42no—
23#381—22no—
24#381C814yescomplemented
25#380—42no—
26C4×C22⋊C4—42no—
27C2×#95—21no—
28C2×#95—22no—
29C2×#95—21no—
30C2×#95—22no—
31C2×#95—22no—
32C2×#95C2×C412yes—
33C2×#95C2×C411yescharacteristic
34C2×#95C2311yescharacteristic
35#336—42no—
36#323—28no—
37C25C22⋊C411yescharacteristic
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—22no—
42C23×C4—82no—
43C23×C4C22⋊C412yes—
44C23×C4C22⋊C411yescharacteristic
45C23×C4C2×C812yes—
46C23×C4M1612yes—
47C23×C4C22×C411yescharacteristic
48C23×C4C2×D412yes—
49C22×C8—44no—
50C2×C4⋊C4—42no—
51C2×C4⋊C4—44no—
52C2×C4⋊C4—82no—
53C2×C4⋊C4—82no—
54C2×C4⋊C4—82no—
55C2×C22⋊C4—24no—
56C2×C22⋊C4—24no—
57C2×C22⋊C4—24no—
58C2×C22⋊C4—42no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—82no—
61C2×C22⋊C4—81no—
62C2×C22⋊C4—81no—
63C2×C42—22no—
64C2×C42—22no—
65C2×C42—82no—
66C22⋊C8—28no—
67C22⋊C8—44no—
68C4×C8—84no—
69#95—24no—
70#95—22no—
71#95—41no—
72#95—41no—
73#95—42no—
74#95—44no—
75#95—82no—
76#95—82no—
77#95C2×C814yes—
78C24—21no—
79C24—21no—
80C24—22no—
81C24—22no—
82C24—41no—
83C24—42no—
84C24—41no—
85C24C22⋊C812yes—
86C24C2×C22⋊C411yescharacteristic
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—24no—
94C22×C4—24no—
95C22×C4—24no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—82no—
116C22×C4—82no—
117C22×C4—84no—
118C22×C4—82no—
119C22×C4—82no—
120C22×C4C22⋊C812yes—
121C22×C4#9912yes—
122C22×C4#10112yes—
123C22×C4#10414yes—
124C22×C4C22×C811yescharacteristic
125C22×C4C2×M1611yescharacteristic
126C22×C4#13514yes—
127C2×C8—48no—
128C2×C8—84no—
129C2×C8—84no—
130C4⋊C4—84no—
131C4⋊C4—88no—
132C22⋊C4—48no—
133C22⋊C4—84no—
134C22⋊C4—84no—
135C42—44no—
136C42—82no—
137C23—22no—
138C23—21no—
139C23—21no—
140C23—22no—
141C23—24no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—41no—
147C23—42no—
148C23—41no—
149C23—41no—
150C23—42no—
151C23—41no—
152C23—82no—
153C23—82no—
154C23—82no—
155C23—82no—
156C23C2×C22⋊C811yescharacteristic
157C23C2×#9911yescharacteristic
158C23C2×#10111yescharacteristic
159C23#41612yes—
160C23C2×#10412yes—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—84no—
183C2×C4—84no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4#32414yes—
198C8—88no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—84no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22C2×#32411yescharacteristic
216C22#147612yes—
217C22#150112yes—
218C22#152611yescharacteristic
219C22#154411yescharacteristic
220C22#212112yes—
221C22#212811yescharacteristic
222C22#213511yescharacteristic
223C4—24no—
224C4—41no—
225C4—41no—
226C4—44no—
227C4—44no—
228C4—84no—
229C4—162no—
230C2—22no—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#756411yescharacteristic
235C2#1145011yescharacteristic
236C2#1145311yescharacteristic
237C2#1172912yes—
238C2#1173912yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7123C2semidirect
24. #381C8semidirect

Maximal subgroups

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

subgroupindex
#142952
C4×#3232
#71232