Finite Group Databasebeta

← #135613

Subgroups — #135613

Order 512.

318 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 78 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#22706C211yesmaximal, characteristic
2#11487C211yesmaximal, characteristic
3#8622C211yesmaximal, characteristic
4#7299C212yesmaximal, complemented
5#7218C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7#2352—21no—
8#2162C2212yes—
9#1949—21no—
10#1854—21no—
11#1853—21no—
12#1835C2211yescharacteristic
13#1808C411yescharacteristic
14#1807C411yescharacteristic
15C4×#106C2211yescharacteristic
16#1735C411yescharacteristic
17C2×#340C411yescharacteristic
18#1568C2212yes—
19#1466—21no—
20#1314—21no—
21#1314—21no—
22#1290—21no—
23C23×C8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C2×C4×D4D411yescharacteristic
26C2×#117—21no—
27C22×C4⋊C4—21no—
28C8×D4—22no—
29C8×D4—22no—
30#432—22no—
31C2×#107—21no—
32C2×#106—21no—
33C2×#106—21no—
34C2×C4⋊C8D411yescharacteristic
35C2×C22⋊C8D411yescharacteristic
36C2×C22⋊C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#388—41no—
39#387—41no—
40#386—41no—
41#382—21no—
42#380—21no—
43#380—21no—
44C4×C4⋊C4—21no—
45C4×C22⋊C4—41no—
46C4⋊C16—22no—
47C22⋊C16—22no—
48C4×C16—22no—
49#340—21no—
50#340—21no—
51#340—21no—
52#340C2×C411yescharacteristic
53#340C2×C411yescharacteristic
54#340C2×C411yescharacteristic
55#340C2×C411yescharacteristic
56#335—22no—
57#334—22no—
58#323—41no—
59#323—41no—
60C22×D4SD1611yescharacteristic
61C23×C4—41no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—81no—
69C22×C8—81no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4×D4—22no—
75C4×D4—22no—
76#117—42no—
77#117—81no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—81no—
85C2×C4⋊C4SD1611yescharacteristic
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4D811yescharacteristic
90C2×C22⋊C4D811yescharacteristic
91C2×C42—41no—
92C2×C42—41no—
93C2×C42C2×D411yescharacteristic
94C2×C16—42no—
95#107—81no—
96#106—42no—
97#106—81no—
98C4⋊C8—22no—
99C4⋊C8—42no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102C22⋊C8—42no—
103#97—42no—
104C4×C8#4112yes—
105C4×C8#4112yes—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111C24#10211yescharacteristic
112C24#10211yescharacteristic
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—41no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—82no—
129C22×C4—82no—
130C22×C4—161no—
131C22×C4#10211yescharacteristic
132C22×C4#10211yescharacteristic
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C2×D811yescharacteristic
135C22×C4C2×SD1611yescharacteristic
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8—81no—
146C2×C8—82no—
147C2×C8#11711yescharacteristic
148C2×C8#11711yescharacteristic
149C2×C8#12312yes—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—82no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—82no—
163C22⋊C4—82no—
164C22⋊C4—161no—
165C42—82no—
166C42—161no—
167C42#12312yes—
168C42#13712yes—
169C16—82no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23#35912yes—
180C23C2×#10211yescharacteristic
181D4—44no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—164no—
210C2×C4#38111yescharacteristic
211C2×C4#41911yescharacteristic
212C2×C4#48012yes—
213C2×C4#48312yes—
214C8—22no—
215C8—21no—
216C8—21no—
217C8—41no—
218C8—41no—
219C8—162no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—81no—
231C22—81no—
232C22#182311yescharacteristic
233C22C2×#35911yescharacteristic
234C22#214311yescharacteristic
235C4—21no—
236C4—22no—
237C4—22no—
238C4—41no—
239C4—41no—
240C4—82no—
241C4—161no—
242C4—161no—
243C4—161no—
244C4#182711yescharacteristic
245C4#182911yescharacteristic
246C2—21no—
247C2—21no—
248C2—42no—
249C2—81no—
250C2#876411yescharacteristic
251C2#1286911yescharacteristic
252C2#1287311yescharacteristic
subgroupG/H
center, socleC22#1823
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7299C2semidirect
5. #7218C2semidirect

Maximal subgroups

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

subgroupindex
#227062
#114872
#86222
#72992
#72182