Finite Group Databasebeta

← #99274

Subgroups — #99274

Order 512.

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

normal subgroups only

This group has 538 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
1#13953C211yesmaximal, characteristic
2#8559C212yesmaximal
3#7039C214yesmaximal, complemented
4C2×#381C411yescharacteristic
5C2×C4×C22⋊C4C411yescharacteristic
6C2×C4×C22⋊C4C2211yescharacteristic
7#1797C2214yes—
8C2×#323—22no—
9#1439—22no—
10#1436—22no—
11C4⋊#95C2212yes—
12#1434C412yes—
13#1280—22no—
14C24×C4C2×C411yescharacteristic
15C22×C4⋊C4—22no—
16C22×C22⋊C4—21no—
17C22×C22⋊C4—21no—
18C22×C22⋊C4D412yes—
19C22×C42—21no—
20C22×C42C2×C411yescharacteristic
21C22×C42C2311yescharacteristic
22C2×C4⋊C8—24no—
23C2×C22⋊C8—24no—
24#381—42no—
25C4×C22⋊C4—24no—
26C4×C22⋊C4—42no—
27C2×#95—22no—
28C2×#95—21no—
29C2×#95—22no—
30C2×#95—22no—
31C2×#95C2×C412yes—
32C2×#95C2×C412yes—
33C2×#95D412yes—
34#336—24no—
35#323—44no—
36C25C22⋊C411yescharacteristic
37C23×C4—22no—
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—22no—
42C23×C4—81no—
43C23×C4—81no—
44C23×C4C22⋊C412yes—
45C23×C4C22⋊C411yescharacteristic
46C23×C4C22×C411yescharacteristic
47C23×C4C2×D412yes—
48C22×C8—44no—
49C2×C4⋊C4—44no—
50C2×C4⋊C4—44no—
51C2×C4⋊C4—82no—
52C2×C22⋊C4—24no—
53C2×C22⋊C4—24no—
54C2×C22⋊C4—22no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—44no—
57C2×C22⋊C4—42no—
58C2×C22⋊C4—42no—
59C2×C22⋊C4—81no—
60C2×C22⋊C4—81no—
61C2×C42—22no—
62C2×C42—42no—
63C2×C42—81no—
64C2×C42—81no—
65C2×C42—81no—
66C2×C42—81no—
67C2×C42#4114yes—
68C4⋊C8—84no—
69C22⋊C8—44no—
70C22⋊C8—44no—
71#95—24no—
72#95—42no—
73#95—42no—
74#95—44no—
75#95—42no—
76#95—42no—
77#95—42no—
78#95—82no—
79C24—21no—
80C24—21no—
81C24—22no—
82C24—22no—
83C24—42no—
84C24—41no—
85C24—41no—
86C24#10012yes—
87C24C2×C22⋊C411yescharacteristic
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—28no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—42no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—42no—
101C22×C4—41no—
102C22×C4—41no—
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—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—82no—
119C22×C4—82no—
120C22×C4—84no—
121C22×C4—84no—
122C22×C4#10112yes—
123C22×C4#11712yes—
124C22×C4#12314yes—
125C22×C4#13514yes—
126C22×C4AGL(1,ℤ/8)12yes—
127C22×C4#13712yes—
128C2×C8—84no—
129C2×C8—84no—
130C2×C8—84no—
131C4⋊C4—88no—
132C22⋊C4—48no—
133C22⋊C4—44no—
134C22⋊C4—44no—
135C22⋊C4—84no—
136C42—44no—
137C42—82no—
138C42—84no—
139C23—22no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—24no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—41no—
149C23—41no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—82no—
155C23—82no—
156C23—82no—
157C23—82no—
158C23#38111yescharacteristic
159C23C2×#10011yescharacteristic
160C23C2×#10111yescharacteristic
161C23#41612yes—
162C23#41712yes—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—88no—
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—82no—
198C2×C4#48214yes—
199C2×C4#48514yes—
200C8—164no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—84no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22#154411yescharacteristic
218C22#155112yes—
219C22#156111yescharacteristic
220C22#182111yescharacteristic
221C22#182512yes—
222C22#182712yes—
223C22#182811yescharacteristic
224C22#182911yescharacteristic
225C4—24no—
226C4—24no—
227C4—42no—
228C4—42no—
229C4—84no—
230C4—82no—
231C4—162no—
232C2—22no—
233C2—21no—
234C2—21no—
235C2—82no—
236C2#872911yescharacteristic
237C2#874211yescharacteristic
238C2#875611yescharacteristic
239C2#876312yes—
240C2#877012yes—
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#139532
#85592
#70392