Finite Group Databasebeta

← #99721

Subgroups — #99721

Order 512.

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

normal subgroups only

This group has 431 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#23656C211yesmaximal, characteristic
2#21957C211yesmaximal, characteristic
3#8709C212yesmaximal
4#7564C211yesmaximal, characteristic
5#7029C212yesmaximal
6C2×#433—21no—
7C2×#427—21no—
8#2296C2211yescharacteristic
9#2285C2211yescharacteristic
10#2085—22no—
11#2084C412yes—
12#2083C412yes—
13#1940—22no—
14#1939—22no—
15#1798—21no—
16#1797C2212yes—
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#433—42no—
29#433—42no—
30#427—42no—
31#427—42no—
32C2×#107—22no—
33C2×#106—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8D412yes—
37C2×C4⋊C8D412yes—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×#97—21no—
41C2×C4×C8—21no—
42C2×C4×C8—22no—
43#389C2×C412yes—
44#389D412yes—
45#385—22no—
46#384C2×C412yes—
47#382—21no—
48#380—22no—
49C4×C4⋊C4—22no—
50C4×C4⋊C4—22no—
51#340—22no—
52#340—22no—
53#336—21no—
54#336—21no—
55#336—22no—
56#336—21no—
57#336—22no—
58#336C2×C411yescharacteristic
59#336C2×C411yescharacteristic
60C23×C4C2×D411yescharacteristic
61C23×C4#4112yes—
62C22×C8—24no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—42no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8—42no—
71#117—21no—
72#117—21no—
73#117—42no—
74#117—41no—
75#117—41no—
76#117—41no—
77#117—41no—
78#117D812yes—
79#117SD1612yes—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4C22⋊C412yes—
86C2×C4⋊C4C22⋊C412yes—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—41no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—42no—
93C2×C42—41no—
94C2×C42C22×C411yescharacteristic
95C2×C42C2×D412yes—
96C2×C42C2×D411yescharacteristic
97#107—44no—
98#106—44no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—44no—
102C4⋊C8—82no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107#97—42no—
108C4×C8—42no—
109C4×C8—42no—
110C4×C8—42no—
111#95—22no—
112#95—22no—
113#95—42no—
114C24#12311yescharacteristic
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C4×D412yes—
126C22×C4#12011yescharacteristic
127C22×C4#12112yes—
128C22×C4C2×D811yescharacteristic
129C22×C4C2×SD1611yescharacteristic
130C22×C4AGL(1,ℤ/8)11yescharacteristic
131C22×C4#13711yescharacteristic
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—44no—
137C2×C8—44no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—44no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—82no—
148C2×C8—82no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4—44no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—82no—
159C4⋊C4#10214yes—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—81no—
163C22⋊C4—81no—
164C22⋊C4—81no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42—41no—
170C42—41no—
171C42—42no—
172C42—81no—
173C42—81no—
174C23—22no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23#38611yescharacteristic
179C23#48011yescharacteristic
180C23#48111yescharacteristic
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4C2×#10211yescharacteristic
204C2×C4#41711yescharacteristic
205C2×C4#44312yes—
206C2×C4#44412yes—
207C2×C4#44711yescharacteristic
208C2×C4#44811yescharacteristic
209C2×C4#45612yes—
210C2×C4#45812yes—
211C8—42no—
212C8—42no—
213C8—84no—
214C8—82no—
215C8—82no—
216C22—22no—
217C22—22no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22#187411yescharacteristic
224C22#187612yes—
225C22#188112yes—
226C22#188611yescharacteristic
227C22#189011yescharacteristic
228C4—21no—
229C4—21no—
230C4—22no—
231C4—22no—
232C4—44no—
233C4—42no—
234C4—82no—
235C4—162no—
236C4#161012yes—
237C4#161112yes—
238C2—41no—
239C2—41no—
240C2#899512yes—
241C2#900611yescharacteristic
242C2#902711yescharacteristic
243C2#903812yes—
244C2#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
#219572
#87092
#75642
#70292