Finite Group Databasebeta

← #122763

Subgroups — #122763

Order 512.

287 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 64 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#27659C211yesmaximal, characteristic
2#11466C211yesmaximal, characteristic
3#8622C211yesmaximal, characteristic
4#7083C212yesmaximal, complemented
5#7081C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C2×#427C411yescharacteristic
8#2485—21no—
9#2112—21no—
10#2087C2211yescharacteristic
11#2086C2211yescharacteristic
12#2076C411yescharacteristic
13#2075C411yescharacteristic
14#1941—21no—
15#1825—21no—
16#1807—21no—
17#1807C411yescharacteristic
18#1568—21no—
19#1466—21no—
20#1322C2212yes—
21#1314—21no—
22#1290C2212yes—
23C23×C8C2×C411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4×D4D411yescharacteristic
26C2×#117—21no—
27C2×M32—41no—
28C8×D4—41no—
29C8×D4—41no—
30#432—22no—
31#427—41no—
32#427—41no—
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C411yescharacteristic
35C2×#106—21no—
36C2×#106—21no—
37C2×C4⋊C8C2311yescharacteristic
38C2×C22⋊C8C2×C411yescharacteristic
39C2×C22⋊C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#397—41no—
42#394—41no—
43#389—21no—
44#389—21no—
45#388—41no—
46#387—41no—
47#380—21no—
48#380—21no—
49#380—21no—
50C22⋊C16—22no—
51C22⋊C16—41no—
52#340—21no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340C2×C411yescharacteristic
57#340C2×C411yescharacteristic
58#335—41no—
59#334—41no—
60#323—41no—
61#323—41no—
62C22×D4SD1611yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—42no—
70C22×C8—42no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22×C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C4×D4—41no—
76C4×D4—41no—
77#117—42no—
78#117—42no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4SD1611yescharacteristic
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4D811yescharacteristic
91C2×C22⋊C4D811yescharacteristic
92C2×C42—41no—
93C2×C42C2×D411yescharacteristic
94M32—82no—
95C2×C16—42no—
96#107—41no—
97#107—41no—
98#106—42no—
99#106—41no—
100#106—41no—
101C4⋊C8—41no—
102C4⋊C8—42no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—42no—
106#97—42no—
107C4×C8—21no—
108C4×C8—21no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114C24#10211yescharacteristic
115C24#10211yescharacteristic
116C2×D4—41no—
117C2×D4—41no—
118C2×D4—41no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4#10211yescharacteristic
132C22×C4#10211yescharacteristic
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C2×D811yescharacteristic
135C22×C4C2×SD1611yescharacteristic
136C2×C8—22no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—81no—
147C2×C8—82no—
148C2×C8#9911yescharacteristic
149C2×C8#9911yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—82no—
160C4⋊C4—161no—
161C22⋊C4—41no—
162C22⋊C4—41no—
163C22⋊C4—82no—
164C22⋊C4—82no—
165C22⋊C4—161no—
166C22⋊C4—161no—
167C42—21no—
168C42—21no—
169C42—82no—
170C16—82no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23C2×#10211yescharacteristic
182D4—44no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
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—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4C2×#9911yescharacteristic
213C2×C4#42111yescharacteristic
214C8—21no—
215C8—41no—
216C8—41no—
217C8—41no—
218C8—162no—
219C8#35212yes—
220C22—21no—
221C22—21no—
222C22—42no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22#151911yescharacteristic
233C22#214311yescharacteristic
234C22#214811yescharacteristic
235C4—21no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—161no—
241C4—161no—
242C4—161no—
243C4—161no—
244C4C2×#35211yescharacteristic
245C4#213511yescharacteristic
246C2—21no—
247C2—21no—
248C2—81no—
249C2—81no—
250C2#1186711yescharacteristic
251C2#1188111yescharacteristic
252C2#1189411yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7083C2semidirect
5. #7081C2semidirect

Maximal subgroups

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

subgroupindex
#276592
#114662
#86222
#70832
#70812