Finite Group Databasebeta

← #99717

Subgroups — #99717

Order 512.

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

normal subgroups only

This group has 409 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#23744C211yesmaximal, characteristic
2#21957C211yesmaximal, characteristic
3#8710C212yesmaximal
4#7564C211yesmaximal, characteristic
5#7029C212yesmaximal
6C2×#433—21no—
7#2296C2211yescharacteristic
8#2285C2211yescharacteristic
9#2101—22no—
10#2100C412yes—
11#2099C412yes—
12#1940—22no—
13#1939—22no—
14#1798—21no—
15#1797C2212yes—
16C4×C22⋊C8C2212yes—
17#1735—21no—
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—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8D412yes—
33C2×C4⋊C8D412yes—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×#97—21no—
37C2×C4×C8—21no—
38C2×C4×C8—22no—
39#389C2×C412yes—
40#389D412yes—
41#385—22no—
42#384C2×C411yescharacteristic
43#384C2×C411yescharacteristic
44#382—21no—
45#380—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—22no—
48#340—22no—
49#340—22no—
50#340—22no—
51#340—22no—
52#336—21no—
53#336—21no—
54#336—22no—
55#336—21no—
56#336—22no—
57#336C2×C412yes—
58C23×C4C2×D411yescharacteristic
59C23×C4#4112yes—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—42no—
64C22×C8—42no—
65C22×C8—42no—
66C22×C8—41no—
67C22×C8—41no—
68#117—21no—
69#117—21no—
70#117—42no—
71#117—44no—
72#117SD1612yes—
73#117Q1612yes—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4C22⋊C412yes—
81C2×C4⋊C4C22⋊C412yes—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—41no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—42no—
88C2×C42—41no—
89C2×C42C22×C411yescharacteristic
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—44no—
95C4⋊C8—82no—
96C22⋊C8—24no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#97—42no—
101C4×C8—42no—
102C4×C8—42no—
103C4×C8—42no—
104#95—22no—
105#95—22no—
106#95—42no—
107C24#12311yescharacteristic
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C4×D412yes—
120C22×C4#12011yescharacteristic
121C22×C4#12112yes—
122C22×C4C2×SD1611yescharacteristic
123C22×C4C2×Q1611yescharacteristic
124C22×C4AGL(1,ℤ/8)11yescharacteristic
125C22×C4#13711yescharacteristic
126C2×C8—44no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—82no—
137C2×C8—82no—
138C2×C8—82no—
139C2×C8—82no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—82no—
150C4⋊C4#10314yes—
151C22⋊C4—22no—
152C22⋊C4—22no—
153C22⋊C4—81no—
154C22⋊C4—82no—
155C42—22no—
156C42—22no—
157C42—22no—
158C42—22no—
159C42—41no—
160C42—41no—
161C42—42no—
162C42—82no—
163C23—22no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23#38611yescharacteristic
168C23#48311yescharacteristic
169C23#48411yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4C2×#10311yescharacteristic
194C2×C4#41711yescharacteristic
195C2×C4#44312yes—
196C2×C4#44412yes—
197C2×C4#45011yescharacteristic
198C2×C4#45111yescharacteristic
199C2×C4#45412yes—
200C2×C4#45812yes—
201C8—82no—
202C8—84no—
203C8—82no—
204C8—82no—
205C22—22no—
206C22—22no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22#187211yescharacteristic
213C22#187612yes—
214C22#188112yes—
215C22#188811yescharacteristic
216C22#189011yescharacteristic
217C4—21no—
218C4—21no—
219C4—22no—
220C4—22no—
221C4—44no—
222C4—42no—
223C4—82no—
224C4—161no—
225C4—161no—
226C4#161612yes—
227C4#161712yes—
228C2—41no—
229C2—41no—
230C2#899312yes—
231C2#900811yescharacteristic
232C2#902311yescharacteristic
233C2#903812yes—
234C2#907011yescharacteristic
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
#237442
#219572
#87102
#75642
#70292