Finite Group Databasebeta

← #142158

Subgroups — #142158

Order 512.

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

normal subgroups only

This group has 438 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#33209C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C4×#341C211yesmaximal, characteristic
4#7259C214yesmaximal
5C2×#500C411yescharacteristic
6C2×C4×M16—21no—
7C22×C4×C8C2211yescharacteristic
8#2857—21no—
9C2×C4⋊C16—21no—
10C2×C22⋊C16C2212yes—
11#1990—21no—
12#1942—21no—
13#1942C412yes—
14#1845C411yescharacteristic
15#1808—21no—
16#1808C2212yes—
17C2×#341—21no—
18C2×#341C2212yes—
19#1374—21no—
20C22×M16—22no—
21C23×C8D411yescharacteristic
22C23×C8C2311yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C22⋊Q8—41no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27#500—42no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#107C2×C412yes—
32C4×M16—24no—
33C4×M16—42no—
34C2×#97—41no—
35C2×C4×C8—42no—
36C2×C4×C8—41no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#391—41no—
42#389—22no—
43#389—41no—
44#384—21no—
45#380—22no—
46#380—41no—
47C4⋊C16—42no—
48C22⋊C16—24no—
49#341—24no—
50#341—22no—
51#341Q814yescomplemented
52#340—22no—
53#340—22no—
54C22×Q8—41no—
55C23×C4SD1611yescharacteristic
56C23×C4SD1611yescharacteristic
57C23×C4C2×D411yescharacteristic
58C2×M16—24no—
59C2×M16—24no—
60C2×M16—24no—
61C2×M16—42no—
62C2×M16—42no—
63C2×M16—42no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×Q812yes—
75C22×C8#4112yes—
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78C22⋊Q8—84no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C22⋊C4—42no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42D811yescharacteristic
91C2×C42D811yescharacteristic
92C2×C16—42no—
93C2×C16—42no—
94C2×C16—42no—
95#107—42no—
96#107—42no—
97#107—42no—
98#107—42no—
99#97—44no—
100C4×C8—22no—
101C4×C8—22no—
102C4×C8—41no—
103C4×C8—41no—
104C4×C8—44no—
105#95—42no—
106#95—42no—
107C24C2×SD1611yescharacteristic
108C2×Q8—82no—
109C2×Q8—82no—
110C2×Q8—82no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4#10211yescharacteristic
124C22×C4#10211yescharacteristic
125C22×C4#10211yescharacteristic
126C22×C4#10211yescharacteristic
127C22×C4C2×C22⋊C411yescharacteristic
128C22×C4C22⋊Q812yes—
129C22×C4C2×D811yescharacteristic
130C22×C4#13712yes—
131M16—28no—
132M16—44no—
133M16—44no—
134M16—44no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8#10412yes—
151C2×C8C4×Q812yes—
152C2×C8#12411yescharacteristic
153C2×C8C4⋊Q811yescharacteristic
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—84no—
162C22⋊C4—84no—
163C42—22no—
164C42—22no—
165C42—21no—
166C42—21no—
167C42#9912yes—
168C42#10412yes—
169C16—82no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23C2×#10211yescharacteristic
177C23#47512yes—
178Q8—164no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#32712yes—
201C2×C4#32712yes—
202C2×C4#35912yes—
203C2×C4#36112yes—
204C2×C4#36212yes—
205C2×C4#39111yescharacteristic
206C2×C4C2×#9911yescharacteristic
207C2×C4#41011yescharacteristic
208C2×C4#41911yescharacteristic
209C2×C4C2×#10411yescharacteristic
210C2×C4C2×#10411yescharacteristic
211C2×C4#47412yes—
212C8—21no—
213C8—22no—
214C8—21no—
215C8—22no—
216C8—48no—
217C8—41no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22C2×#32711yescharacteristic
226C22#150211yescharacteristic
227C22#198811yescharacteristic
228C22C2×#35911yescharacteristic
229C22#214311yescharacteristic
230C22C2×#36111yescharacteristic
231C22C2×#36211yescharacteristic
232C4—22no—
233C4—22no—
234C4—22no—
235C4—41no—
236C4—162no—
237C4—162no—
238C4#135014yes—
239C4#150611yescharacteristic
240C4#150811yescharacteristic
241C4#199411yescharacteristic
242C4#199611yescharacteristic
243C2—42no—
244C2#990911yescharacteristic
245C2C2×#135011yescharacteristic
246C2#1171111yescharacteristic
247C2#1174711yescharacteristic
248C2#1175211yescharacteristic
249C2#1327411yescharacteristic
250C2#1327911yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
51. #341Q8semidirect

Maximal subgroups

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

subgroupindex
#332092
#127832
C4×#3412
#72592