Finite Group Databasebeta

← #139583

Subgroups — #139583

Order 512.

405 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 98 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#33486C211yesmaximal, characteristic
2#27661C211yesmaximal, characteristic
3#12866C212yesmaximal, complemented
4#8609C211yesmaximal, characteristic
5#7230C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×#107C2211yescharacteristic
8#2486—21no—
9D4×C16—24no—
10#2162—21no—
11C2×C22⋊C16—21no—
12C2×C22⋊C16C2212yes—
13#2086C412yescomplemented
14#2075C412yescomplemented
15#1942—21no—
16#1823—21no—
17#1808—21no—
18#1808C2212yes—
19#1808C2211yescharacteristic
20#1799—21no—
21#1380—24no—
22#1289—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#123—41no—
26C2×C4×D4D411yescharacteristic
27C22×C4⋊C4—21no—
28C22×C16—22no—
29C22×C16D414yes—
30C8×D4—22no—
31C8×D4—22no—
32C2×#107—22no—
33C2×#107—22no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107—22no—
37C2×#107C2×C412yes—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×#99—41no—
40C2×C22⋊C8C2×C411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#398—41no—
44#397—41no—
45#389—22no—
46#381—41no—
47#380—22no—
48#380—21no—
49#380—22no—
50C4⋊C16—22no—
51C22⋊C16—22no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54C4×C16—22no—
55#342—41no—
56#340—22no—
57#340—22no—
58#340—22no—
59#340C2×C412yes—
60C22×D4D811yescharacteristic
61C23×C4—41no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—42no—
67C22×C8—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D412yes—
72C22×C8C2×D411yescharacteristic
73C22×C8#4112yes—
74#123—82no—
75#123—82no—
76C4×D4—22no—
77C4×D4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4SD1611yescharacteristic
86C2×C22⋊C4—42no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4D811yescharacteristic
90C2×C22⋊C4SD1611yescharacteristic
91C2×C42C22⋊C411yescharacteristic
92C2×C16—24no—
93C2×C16—44no—
94C2×C16—82no—
95#107—24no—
96#107—42no—
97#107—41no—
98#107—41no—
99#107—44no—
100C4⋊C8—22no—
101#99—82no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104C4×C8—21no—
105C4×C8—21no—
106#95—42no—
107#95—42no—
108#95—41no—
109#95—42no—
110C24#10211yescharacteristic
111C24C2×D811yescharacteristic
112C2×D4—22no—
113C2×D4—22no—
114C2×D4D1612yes—
115C2×D4SD3212yes—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—82no—
125C22×C4—81no—
126C22×C4—81no—
127C22×C4—82no—
128C22×C4—82no—
129C22×C4#10211yescharacteristic
130C22×C4#10211yescharacteristic
131C22×C4#10211yescharacteristic
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4#12112yes—
134C22×C4C2×SD1611yescharacteristic
135C22×C4#13512yes—
136C2×C8—24no—
137C2×C8—22no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—41no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—81no—
146C2×C8C4×D412yes—
147C2×C8#12011yescharacteristic
148C2×C8#12311yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—82no—
155C4⋊C4—82no—
156C4⋊C4—84no—
157C4⋊C4—82no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—84no—
161C22⋊C4—81no—
162C22⋊C4—81no—
163C22⋊C4—82no—
164C22⋊C4—82no—
165C42—21no—
166C42—21no—
167C16—44no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23#35712yes—
175C23C2×#10211yescharacteristic
176C23#46612yes—
177C23C2×D1611yescharacteristic
178C23C2×SD3211yescharacteristic
179C23#50812yes—
180D4—24no—
181D4—24no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—162no—
204C2×C4—162no—
205C2×C4—162no—
206C2×C4—162no—
207C2×C4—162no—
208C2×C4#35712yes—
209C2×C4#38611yescharacteristic
210C2×C4#41611yescharacteristic
211C2×C4#46512yes—
212C8—21no—
213C8—22no—
214C8—41no—
215C8—41no—
216C8#45812yes—
217C22—21no—
218C22—22no—
219C22—21no—
220C22—22no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22—41no—
227C22—41no—
228C22—81no—
229C22#189311yescharacteristic
230C22C2×#35711yescharacteristic
231C22#213811yescharacteristic
232C22#221512yes—
233C22#221712yes—
234C4—22no—
235C4—21no—
236C4—22no—
237C4—41no—
238C4—41no—
239C4—162no—
240C4—162no—
241C4—322no—
242C4#189011yescharacteristic
243C4#190911yescharacteristic
244C2—21no—
245C2—22no—
246C2—21no—
247C2—22no—
248C2—42no—
249C2#947011yescharacteristic
250C2#1305911yescharacteristic
251C2#1307511yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4D8
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
3. #12866C2semidirect
5. #7230C2semidirect
13. #2086C4semidirect
14. #2075C4semidirect

Maximal subgroups

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

subgroupindex
#334862
#276612
#128662
#86092
#72302