Finite Group Databasebeta

← #142004

Subgroups — #142004

Order 512.

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

normal subgroups only

This group has 402 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#33208C211yesmaximal, characteristic
2C4×C22⋊C16C211yesmaximal, characteristic
3#10864C211yesmaximal, characteristic
4#7259C214yesmaximal
5C2×#499C411yescharacteristic
6C22×#108—21no—
7C22×C4×C8C2211yescharacteristic
8#2851—21no—
9C2×C22⋊C16C2212yes—
10C2×C4×C16—21no—
11#1991—21no—
12#1942C412yes—
13#1941—21no—
14#1846C411yescharacteristic
15#1808C2212yes—
16#1807—21no—
17C2×#341C2212yes—
18#1728—21no—
19#1374—21no—
20C22×M16—22no—
21C23×C8D411yescharacteristic
22C23×C8C2311yescharacteristic
23C2×#125—21no—
24C22×C42D411yescharacteristic
25C22×C16—22no—
26#499—42no—
27C2×#108—24no—
28C2×#108—42no—
29C2×#108—42no—
30C2×#107C2×C412yes—
31C2×#107C2×C412yes—
32C2×#106—21no—
33C2×#106—21no—
34C2×C4×C8—42no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#389—22no—
40#387—42no—
41#385—42no—
42#382—21no—
43#380—22no—
44C22⋊C16—22no—
45C22⋊C16Q814yes—
46C4×C16—42no—
47#341—24no—
48#340—22no—
49#340—22no—
50#336—42no—
51C23×C4SD1612yes—
52C23×C4C2×D411yescharacteristic
53C2×M16—24no—
54C2×M16—42no—
55C2×M16—42no—
56C2×M16—42no—
57C2×M16—42no—
58C2×M16—42no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—42no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8SD1612yes—
67C22×C8Q1612yes—
68C22×C8C22×C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C22×C8C2×Q812yes—
71C22×C8#4112yes—
72#125—42no—
73#125—42no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C42—41no—
80C2×C42—41no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42D812yes—
84C2×C16—24no—
85C2×C16—42no—
86C2×C16—42no—
87#108—48no—
88#107—42no—
89#107—42no—
90#106—42no—
91#106—42no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8—41no—
95C4×C8—41no—
96#95—42no—
97#95—42no—
98#95—42no—
99C24C2×SD1611yescharacteristic
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4#10212yes—
110C22×C4#10212yes—
111C22×C4C2×C22⋊C411yescharacteristic
112C22×C4#12411yescharacteristic
113C22×C4C4⋊Q811yescharacteristic
114C22×C4C2×D811yescharacteristic
115C22×C4C2×SD1611yescharacteristic
116C22×C4C2×Q1611yescharacteristic
117M16—44no—
118M16—44no—
119M16—82no—
120M16—82no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—24no—
128C2×C8—24no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—44no—
132C2×C8—82no—
133C2×C8—82no—
134C2×C8#9912yes—
135C2×C8#10312yes—
136C2×C8#10312yes—
137C2×C8#10412yes—
138C2×C8C4×Q812yes—
139C2×C8C22⋊Q812yes—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—84no—
145C4⋊C4—84no—
146C22⋊C4—84no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—42no—
152C16—44no—
153C23—22no—
154C23—22no—
155C23—21no—
156C23—21no—
157C23C2×#10211yescharacteristic
158C23#48711yescharacteristic
159C23#49811yescharacteristic
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—48no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4#35914yes—
183C2×C4#39111yescharacteristic
184C2×C4C2×#9911yescharacteristic
185C2×C4#41011yescharacteristic
186C2×C4C2×#10311yescharacteristic
187C2×C4C2×#10411yescharacteristic
188C2×C4#42111yescharacteristic
189C2×C4#48611yescharacteristic
190C2×C4#50011yescharacteristic
191C8—22no—
192C8—22no—
193C8—22no—
194C8—84no—
195C8#32814yes—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—42no—
203C22—42no—
204C22#150211yescharacteristic
205C22#151911yescharacteristic
206C22#199011yescharacteristic
207C22C2×#35912yes—
208C22#214812yes—
209C4—24no—
210C4—21no—
211C4—21no—
212C4—42no—
213C4—162no—
214C4—162no—
215C4C2×#32811yescharacteristic
216C4#151811yescharacteristic
217C4#198611yescharacteristic
218C4#199611yescharacteristic
219C2—41no—
220C2—41no—
221C2#988411yescharacteristic
222C2#1174912yes—
223C2#1189312yes—
224C2#1328511yescharacteristic
225C2#1328911yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332082
C4×C22⋊C162
#108642
#72592