Finite Group Databasebeta

← #139593

Subgroups — #139593

Order 512.

367 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 84 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#33489C211yesmaximal, characteristic
2#27662C211yesmaximal, characteristic
3#12871C212yesmaximal, complemented
4#8612C211yesmaximal, characteristic
5#7235C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×#107C2211yescharacteristic
8#2487—21no—
9#2169—24no—
10#2162—21no—
11#2112—21no—
12#2112C2212yes—
13#2087C412yes—
14#2076C412yes—
15#1942—21no—
16#1824—21no—
17#1810C2212yes—
18#1808—21no—
19#1808C2211yescharacteristic
20#1465—21no—
21#1387—24no—
22#1293—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4×D4D411yescharacteristic
26C22×C4⋊C4—21no—
27C2×M32—22no—
28C2×M32D414yes—
29C8×D4—22no—
30C8×D4—22no—
31C2×#108—22no—
32C2×#107—22no—
33C2×#107—22no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37C2×C4⋊C8C2×C411yescharacteristic
38#407—22no—
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C22⋊C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#400—41no—
43#399—41no—
44#389—22no—
45#386—41no—
46#385—41no—
47#380—22no—
48#380—21no—
49C4⋊C16—22no—
50C22⋊C16—22no—
51C22⋊C16—22no—
52C22⋊C16—22no—
53C4×C16—22no—
54#341—22no—
55#340—22no—
56#340—22no—
57#340C2×C412yes—
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—42no—
68C22×C8—42no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22×C411yescharacteristic
72C22×C8C2×D412yes—
73C22×C8C2×D411yescharacteristic
74C22×C8#4112yes—
75C4×D4—22no—
76C4×D4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4D811yescharacteristic
84C2×C22⋊C4—41no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4D811yescharacteristic
87C2×C22⋊C4SD1611yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89M32—44no—
90M32—82no—
91C2×C16—24no—
92#108—44no—
93#107—24no—
94#107—42no—
95#107—41no—
96#107—41no—
97C4⋊C8—22no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102C4×C8—21no—
103C4×C8—21no—
104#95—42no—
105#95—42no—
106#95—41no—
107C24#10211yescharacteristic
108C24C2×SD1611yescharacteristic
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—41no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—82no—
121C22×C4—162no—
122C22×C4#10211yescharacteristic
123C22×C4#10211yescharacteristic
124C22×C4#10211yescharacteristic
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4#12112yes—
127C22×C4C2×D811yescharacteristic
128C22×C4#13512yes—
129M16—82no—
130M16—82no—
131C2×C8—22no—
132C2×C8—24no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—41no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—81no—
141C2×C8—82no—
142C2×C8C4×D412yes—
143C2×C8#12011yescharacteristic
144C2×C8#12311yescharacteristic
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—82no—
151C4⋊C4—162no—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—82no—
155C22⋊C4—162no—
156C42—21no—
157C42—21no—
158C16—44no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23C2×#10211yescharacteristic
169C23#46512yes—
170D4—44no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—164no—
193C2×C4#36212yes—
194C2×C4#38611yescharacteristic
195C2×C4#41611yescharacteristic
196C2×C4#46612yes—
197C8—21no—
198C8—22no—
199C8—41no—
200C8—41no—
201C8—162no—
202C8#45812yes—
203C22—21no—
204C22—21no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—81no—
215C22#189311yescharacteristic
216C22C2×#36211yescharacteristic
217C22#214811yescharacteristic
218C4—22no—
219C4—22no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—162no—
224C4—162no—
225C4#189011yescharacteristic
226C4#190911yescharacteristic
227C2—21no—
228C2—21no—
229C2—42no—
230C2—81no—
231C2#947011yescharacteristic
232C2#1306611yescharacteristic
233C2#1307911yescharacteristic
subgroupG/H
center, socleC22#1893
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
3. #12871C2semidirect
5. #7235C2semidirect

Maximal subgroups

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

subgroupindex
#334892
#276622
#128712
#86122
#72352