Finite Group Databasebeta

← #135604

Subgroups — #135604

Order 512.

329 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 76 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#11466C211yesmaximal, characteristic
3#8611C211yesmaximal, characteristic
4#7264C212yesmaximal, complemented
5#7235C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7#2352—21no—
8#2112C2212yes—
9#1949—21no—
10#1854—21no—
11#1853—21no—
12#1835C411yescharacteristic
13#1810C2212yes—
14#1808C2211yescharacteristic
15#1807C411yescharacteristic
16C4×#106C411yescharacteristic
17#1735C411yescharacteristic
18C2×#340C2211yescharacteristic
19#1465—21no—
20#1322—21no—
21#1314—21no—
22#1293—21no—
23C23×C8C2×C411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4×D4D411yescharacteristic
26C2×#117—21no—
27C22×C4⋊C4—21no—
28C2×M32—22no—
29C8×D4—22no—
30C8×D4—22no—
31C2×#108—22no—
32C2×#107—21no—
33C2×#106—21no—
34C2×#106—21no—
35C2×C4⋊C8D411yescharacteristic
36#407—22no—
37C2×C22⋊C8D411yescharacteristic
38C2×C22⋊C8D411yescharacteristic
39C2×C4×C8C2×C411yescharacteristic
40#388—41no—
41#387—41no—
42#386—41no—
43#382—21no—
44#380—21no—
45#380—21no—
46C4×C4⋊C4—21no—
47C4×C22⋊C4—41no—
48C22⋊C16—22no—
49C22⋊C16—22no—
50#341—22no—
51#340—21no—
52#340—21no—
53#340—21no—
54#340C2×C411yescharacteristic
55#340C2×C411yescharacteristic
56#340C2×C411yescharacteristic
57#340C2×C411yescharacteristic
58#324—41no—
59#323—41no—
60C22×D4SD1611yescharacteristic
61C23×C4—41no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C2×M16—42no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—81no—
68C22×C8—81no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22×C411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C22×C8#4112yes—
74C22×C8#4112yes—
75C4×D4—22no—
76C4×D4—22no—
77#117—42no—
78#117—81no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4D811yescharacteristic
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4D811yescharacteristic
91C2×C22⋊C4SD1611yescharacteristic
92C2×C42—41no—
93C2×C42—41no—
94C2×C42C22⋊C411yescharacteristic
95M32—82no—
96C2×C16—42no—
97#108—82no—
98#107—81no—
99#106—42no—
100#106—81no—
101C4⋊C8—22no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C4×C8—21no—
107C4×C8—21no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113C24#10211yescharacteristic
114C24C2×SD1611yescharacteristic
115C2×D4—22no—
116C2×D4—41no—
117C2×D4—41no—
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×C4#10211yescharacteristic
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4#12312yes—
136C22×C4C2×D811yescharacteristic
137C22×C4#13712yes—
138M16—82no—
139M16—82no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—41no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—82no—
151C2×C8#11711yescharacteristic
152C2×C8#11711yescharacteristic
153C2×C8#12312yes—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—82no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—82no—
167C22⋊C4—82no—
168C22⋊C4—161no—
169C42—21no—
170C42—21no—
171C42—82no—
172C42—161no—
173C16—82no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23C2×#10211yescharacteristic
184C23#48312yes—
185D4—44no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—164no—
215C2×C4#38111yescharacteristic
216C2×C4#41911yescharacteristic
217C2×C4#48012yes—
218C8—22no—
219C8—21no—
220C8—21no—
221C8—41no—
222C8—41no—
223C8—162no—
224C22—21no—
225C22—21no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—42no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—81no—
235C22—81no—
236C22#182311yescharacteristic
237C22#214811yescharacteristic
238C22#214811yescharacteristic
239C4—22no—
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—82no—
245C4—161no—
246C4—161no—
247C4—161no—
248C4#182711yescharacteristic
249C4#182811yescharacteristic
250C2—21no—
251C2—21no—
252C2—42no—
253C2—81no—
254C2#876011yescharacteristic
255C2#1287211yescharacteristic
256C2#1287411yescharacteristic
subgroupG/H
center, socleC22#1823
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7264C2semidirect
5. #7235C2semidirect

Maximal subgroups

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

subgroupindex
#227062
#114662
#86112
#72642
#72352