Finite Group Databasebeta

← #99720

Subgroups — #99720

Order 512.

407 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 90 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#23656C211yesmaximal, characteristic
2#21921C211yesmaximal, characteristic
3#8709C212yesmaximal
4#7564C211yesmaximal, characteristic
5#7029C212yesmaximal
6C2×#427—21no—
7#2296C2211yescharacteristic
8#2285C2211yescharacteristic
9#2085—22no—
10#2084C412yes—
11#2083C412yes—
12#1917—22no—
13#1916—22no—
14#1818—21no—
15#1797C2212yes—
16#1750—21no—
17C4×C22⋊C8C2212yes—
18#1735—21no—
19C2×#336C2211yescharacteristic
20#1279—22no—
21#1279—22no—
22#1271—22no—
23C23×C8—21no—
24C2×#117—21no—
25C2×#117D411yescharacteristic
26C2×#117D411yescharacteristic
27C22×C42C2311yescharacteristic
28#427—44no—
29C2×#107—22no—
30C2×#106—22no—
31C2×C4⋊C8D412yes—
32C2×C4⋊C8D412yes—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×#97—21no—
36C2×C4×C8—22no—
37C2×C4×C8—21no—
38#389C2×C412yes—
39#389D412yes—
40#385—22no—
41#384C2×C412yes—
42#382—21no—
43#380—22no—
44C4×C4⋊C4—22no—
45C4×C4⋊C4—22no—
46#340—22no—
47#340—22no—
48#336—21no—
49#336—22no—
50#336—21no—
51#336—22no—
52#336—22no—
53#336—21no—
54#336—22no—
55#336C2×C411yescharacteristic
56#336C2×C411yescharacteristic
57C23×C4C2×D411yescharacteristic
58C23×C4#4112yes—
59C22×C8—24no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—42no—
67C22×C8—42no—
68#117—22no—
69#117—22no—
70#117—41no—
71#117—41no—
72#117—42no—
73#117—42no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4C22⋊C412yes—
80C2×C4⋊C4C22⋊C412yes—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—41no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—41no—
87C2×C42—41no—
88C2×C42—41no—
89C2×C42C22×C411yescharacteristic
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92#107—44no—
93#106—44no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C22⋊C8—24no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#97—81no—
101C4×C8—42no—
102C4×C8—42no—
103C4×C8—81no—
104#95—22no—
105#95—22no—
106#95—42no—
107C24#12311yescharacteristic
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C4×D412yes—
120C22×C4#12011yescharacteristic
121C22×C4#12112yes—
122C22×C4#13512yes—
123C22×C4AGL(1,ℤ/8)12yes—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—44no—
129C2×C8—44no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—44no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—82no—
140C2×C8—82no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—82no—
152C22⋊C4—42no—
153C22⋊C4—81no—
154C22⋊C4—82no—
155C42—22no—
156C42—22no—
157C42—41no—
158C42—41no—
159C42—42no—
160C42—41no—
161C42—41no—
162C42—82no—
163C23—22no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23#38611yescharacteristic
168C23#48212yes—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4#41611yescharacteristic
192C2×C4#41811yescharacteristic
193C2×C4#44312yes—
194C2×C4#44412yes—
195C2×C4#44912yes—
196C2×C4#45612yes—
197C2×C4#45811yescharacteristic
198C2×C4#45811yescharacteristic
199C8—42no—
200C8—42no—
201C8—84no—
202C8—82no—
203C8—82no—
204C22—24no—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22#187511yescharacteristic
210C22#187612yes—
211C22#188112yes—
212C22#188611yescharacteristic
213C22#189011yescharacteristic
214C4—21no—
215C4—21no—
216C4—22no—
217C4—22no—
218C4—42no—
219C4—82no—
220C4—82no—
221C4—162no—
222C4#160912yes—
223C4#160912yes—
224C2—42no—
225C2#899612yes—
226C2#900611yescharacteristic
227C2#902611yescharacteristic
228C2#903812yes—
229C2#906811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#236562
#219212
#87092
#75642
#70292