Finite Group Databasebeta

← #99258

Subgroups — #99258

Order 512.

664 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 114 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#13965C211yesmaximal, characteristic
2#8559C212yesmaximal
3#7039C214yesmaximal, complemented
4C2×#385—22no—
5C2×#380C411yescharacteristic
6C2×C4×C22⋊C4C2211yescharacteristic
7C22×#95C411yescharacteristic
8#1797C2214yes—
9C2×#323—22no—
10#1444—22no—
11C4⋊#95C412yes—
12C4⋊#95C2212yes—
13#1280—22no—
14C24×C4—21no—
15C24×C4C2×C411yescharacteristic
16C22×C4⋊C4—22no—
17C22×C4⋊C4—22no—
18C22×C4⋊C4—21no—
19C22×C22⋊C4—21no—
20C22×C22⋊C4D412yes—
21C22×C42C2×C411yescharacteristic
22C22×C42C2311yescharacteristic
23C2×C4⋊C8—24no—
24C2×C22⋊C8—24no—
25#385—48no—
26#380—42no—
27C4×C22⋊C4—24no—
28C2×#95—22no—
29C2×#95—21no—
30C2×#95—22no—
31C2×#95—42no—
32C2×#95C2×C412yes—
33C2×#95C2×C412yes—
34C2×#95D412yes—
35#336—24no—
36#323—44no—
37C25C22⋊C411yescharacteristic
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—22no—
42C23×C4—22no—
43C23×C4—44no—
44C23×C4—42no—
45C23×C4—44no—
46C23×C4—81no—
47C23×C4—81no—
48C23×C4—81no—
49C23×C4—81no—
50C23×C4C22⋊C411yescharacteristic
51C23×C4C22⋊C412yes—
52C23×C4C22×C411yescharacteristic
53C23×C4C2×D412yes—
54C22×C8—44no—
55C2×C4⋊C4—44no—
56C2×C4⋊C4—44no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—44no—
59C2×C4⋊C4—44no—
60C2×C4⋊C4—82no—
61C2×C4⋊C4—82no—
62C2×C4⋊C4—81no—
63C2×C4⋊C4—81no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—44no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4SD1614yes—
69C2×C22⋊C4Q1614yes—
70C2×C42—22no—
71C2×C42—81no—
72C2×C42—81no—
73C2×C42#4114yes—
74C4⋊C8—84no—
75C22⋊C8—44no—
76C22⋊C8—44no—
77#95—24no—
78#95—42no—
79#95—42no—
80#95—42no—
81#95—44no—
82#95—44no—
83C24—21no—
84C24—21no—
85C24—22no—
86C24—42no—
87C24—41no—
88C24—41no—
89C24#10012yes—
90C24#10314yes—
91C24C2×C22⋊C411yescharacteristic
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—28no—
99C22×C4—24no—
100C22×C4—42no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—48no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—82no—
123C22×C4—82no—
124C22×C4—84no—
125C22×C4—84no—
126C22×C4—88no—
127C22×C4—88no—
128C22×C4—84no—
129C22×C4—84no—
130C22×C4#10112yes—
131C22×C4#10314yes—
132C22×C4#11712yes—
133C22×C4#12314yes—
134C22×C4C2×SD1612yes—
135C22×C4C2×Q1612yes—
136C22×C4AGL(1,ℤ/8)14yes—
137C2×C8—84no—
138C2×C8—84no—
139C2×C8—84no—
140C4⋊C4—88no—
141C4⋊C4—88no—
142C4⋊C4—84no—
143C22⋊C4—28no—
144C22⋊C4—28no—
145C22⋊C4—84no—
146C42—44no—
147C42—82no—
148C23—22no—
149C23—21no—
150C23—21no—
151C23—24no—
152C23—22no—
153C23—22no—
154C23—24no—
155C23—42no—
156C23—42no—
157C23—44no—
158C23—44no—
159C23—42no—
160C23—44no—
161C23—44no—
162C23—42no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23#33314yes—
167C23#38111yescharacteristic
168C23C2×#10011yescharacteristic
169C23C2×#10111yescharacteristic
170C23C2×#10312yes—
171C23#41812yes—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—88no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—816no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4#48114yes—
209C2×C4#48414yes—
210C8—164no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—24no—
220C22—44no—
221C22—44no—
222C22—48no—
223C22—42no—
224C22—42no—
225C22—44no—
226C22—44no—
227C22C2×#33311yescharacteristic
228C22#154912yes—
229C22#155511yescharacteristic
230C22#182111yescharacteristic
231C22#182412yes—
232C22#182712yes—
233C22#182811yescharacteristic
234C22#182911yescharacteristic
235C4—24no—
236C4—24no—
237C4—42no—
238C4—42no—
239C4—44no—
240C4—84no—
241C4—162no—
242C2—21no—
243C2—21no—
244C2—22no—
245C2—44no—
246C2#872511yescharacteristic
247C2#874211yescharacteristic
248C2#875611yescharacteristic
249C2#876112yes—
250C2#876512yes—
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
#139652
#85592
#70392