Finite Group Databasebeta

← #142020

Subgroups — #142020

Order 512.

476 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 122 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
2#12599C211yesmaximal, characteristic
3#10841C211yesmaximal, characteristic
4#7258C214yesmaximal
5C2×#499C411yescharacteristic, complemented
6C22×C4×C8C2211yescharacteristic
7#2851—21no—
8C2×#384—21no—
9C2×C22⋊C16C2212yes—
10C2×#346—21no—
11#1991—21no—
12#1942C412yescomplemented
13#1941—21no—
14#1846C411yescharacteristic, complemented
15#1808C2212yes—
16#1807—21no—
17C2×#340—21no—
18C2×#340C2212yes—
19#1374—21no—
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#125—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—42no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27#499—42no—
28C2×#107C2×C412yes—
29C2×#107C2×C412yes—
30C2×#106—21no—
31C2×#106—21no—
32C2×C4×C8—42no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37#389—22no—
38#387—42no—
39#385—42no—
40#384—24no—
41#384—42no—
42#382—21no—
43#380—22no—
44C22⋊C16—22no—
45C22⋊C16Q814yes—
46#346—42no—
47#340—22no—
48#340—22no—
49#340—24no—
50#340—42no—
51C23×C4—42no—
52C23×C4D812yes—
53C23×C4C2×D411yescharacteristic
54C22×C8—41no—
55C22×C8—41no—
56C22×C8—41no—
57C22×C8—41no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8SD1612yes—
61C22×C8Q1612yes—
62C22×C8C22×C411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C22×C8C2×Q812yes—
65C22×C8#4112yes—
66#125—42no—
67#125—42no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—48no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—84no—
80C2×C4⋊C4—84no—
81C2×C22⋊C4—42no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42SD1612yes—
87C2×C16—24no—
88C2×C16—42no—
89C2×C16—42no—
90#107—42no—
91#107—42no—
92#106—42no—
93#106—42no—
94C4×C8—22no—
95C4×C8—22no—
96C4×C8—41no—
97C4×C8—41no—
98#95—42no—
99#95—42no—
100#95—42no—
101C24C2×D811yescharacteristic
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—44no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—82no—
114C22×C4—84no—
115C22×C4—82no—
116C22×C4—84no—
117C22×C4#10212yes—
118C22×C4#10212yes—
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4#12411yescharacteristic
121C22×C4C4⋊Q811yescharacteristic
122C22×C4C2×SD1611yescharacteristic
123C22×C4C2×SD1611yescharacteristic
124C22×C4C2×Q1611yescharacteristic
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8#10312yes—
138C2×C8#10312yes—
139C2×C8C4×Q812yes—
140C2×C8C22⋊Q812yes—
141C4⋊C4—44no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—82no—
148C4⋊C4—82no—
149C4⋊C4—816no—
150C4⋊C4—84no—
151C4⋊C4—84no—
152C22⋊C4—82no—
153C22⋊C4—82no—
154C42—24no—
155C42—21no—
156C42—21no—
157C42#9912yes—
158C42#10412yes—
159C16—44no—
160C23—22no—
161C23—21no—
162C23—21no—
163C23C2×#10211yescharacteristic
164C23#48611yescharacteristic
165C23#50011yescharacteristic
166C23#50912yes—
167C23#51012yes—
168C2×C4—28no—
169C2×C4—28no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—84no—
185C2×C4—88no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—164no—
193C2×C4—162no—
194C2×C4—162no—
195C2×C4#32714yes—
196C2×C4#36014yes—
197C2×C4#39111yescharacteristic
198C2×C4C2×#9911yescharacteristic
199C2×C4#41011yescharacteristic
200C2×C4C2×#10311yescharacteristic
201C2×C4C2×#10411yescharacteristic
202C2×C4#42111yescharacteristic
203C2×C4#48711yescharacteristic
204C2×C4#49811yescharacteristic
205C8—22no—
206C8—22no—
207C8—22no—
208C8—22no—
209C22—22no—
210C22—22no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—22no—
215C22—21no—
216C22—21no—
217C22C2×#32711yescharacteristic
218C22#151711yescharacteristic
219C22#199011yescharacteristic
220C22#213912yes—
221C22C2×#36012yes—
222C22#224411yescharacteristic
223C22#224511yescharacteristic
224C22#225512yes—
225C4—21no—
226C4—21no—
227C4—24no—
228C4—24no—
229C4—164no—
230C4—162no—
231C4—162no—
232C4#150311yescharacteristic
233C4#152011yescharacteristic
234C4#198611yescharacteristic
235C4#199611yescharacteristic
236C2—21no—
237C2—21no—
238C2—22no—
239C2#988511yescharacteristic
240C2#1170412yes—
241C2#1186012yes—
242C2#1328411yescharacteristic
243C2#1328711yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
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
5. C2×#499C4semidirect
12. #1942C4semidirect
14. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332082
#125992
#108412
#72582