Finite Group Databasebeta

← #135603

Subgroups — #135603

Order 512.

358 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 80 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#22705C211yesmaximal, characteristic
2#11466C211yesmaximal, characteristic
3#8611C211yesmaximal, characteristic
4#7264C212yesmaximal, complemented
5#7235C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7D4×C4⋊C4—21no—
8#2112C2212yes—
9#1947—21no—
10#1852—21no—
11#1852—21no—
12#1834C411yescharacteristic
13#1810C2212yes—
14#1808C411yescharacteristic
15#1808C2211yescharacteristic
16C4×#107C411yescharacteristic
17C2×#340C411yescharacteristic
18C2×#340C2211yescharacteristic
19#1465—21no—
20#1322—21no—
21#1314—21no—
22#1293—21no—
23C23×C8C2×C411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4×D4—41no—
26C2×C4×D4D411yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C2×M32—22no—
30C8×D4—22no—
31C8×D4—22no—
32C2×#108—22no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—21no—
36C2×C4⋊C8D411yescharacteristic
37#407—22no—
38C2×C22⋊C8D411yescharacteristic
39C2×C22⋊C8D411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41#389—41no—
42#385—41no—
43#385—41no—
44#384—21no—
45#380—21no—
46#380—21no—
47C4×C4⋊C4—21no—
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×C4—41no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C2×M16—42no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—81no—
69C22×C8—81no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8#4112yes—
75C22×C8#4112yes—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—84no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4D811yescharacteristic
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4D811yescharacteristic
95C2×C22⋊C4SD1611yescharacteristic
96C2×C42—41no—
97C2×C42C22⋊C411yescharacteristic
98M32—82no—
99C2×C16—42no—
100#108—82no—
101#107—42no—
102#107—81no—
103#107—81no—
104C4⋊C8—22no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C4×C8—21no—
110C4×C8—21no—
111#95—41no—
112#95—41no—
113#95—41no—
114C24#10211yescharacteristic
115C24C2×SD1611yescharacteristic
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—82no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—161no—
134C22×C4—161no—
135C22×C4#10211yescharacteristic
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4#12312yes—
140C22×C4C2×D811yescharacteristic
141C22×C4#13712yes—
142M16—82no—
143M16—82no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—41no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—81no—
153C2×C8—81no—
154C2×C8—82no—
155C2×C8#11711yescharacteristic
156C2×C8#11711yescharacteristic
157C2×C8#12312yes—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—161no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—82no—
174C22⋊C4—82no—
175C42—21no—
176C42—21no—
177C42—82no—
178C16—82no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23#36212yes—
188C23C2×#10211yescharacteristic
189C23#48312yes—
190D4—28no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—164no—
219C2×C4—164no—
220C2×C4#36112yes—
221C2×C4#38111yescharacteristic
222C2×C4#41911yescharacteristic
223C2×C4#48012yes—
224C8—22no—
225C8—21no—
226C8—21no—
227C8—41no—
228C8—41no—
229C8—162no—
230C22—21no—
231C22—21no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22#182311yescharacteristic
243C22C2×#36111yescharacteristic
244C22C2×#36211yescharacteristic
245C4—22no—
246C4—21no—
247C4—22no—
248C4—41no—
249C4—41no—
250C4—82no—
251C4—161no—
252C4—161no—
253C4—161no—
254C4#182711yescharacteristic
255C4#182811yescharacteristic
256C2—21no—
257C2—21no—
258C2—42no—
259C2—42no—
260C2#876011yescharacteristic
261C2#1287111yescharacteristic
262C2#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
#227052
#114662
#86112
#72642
#72352