Finite Group Databasebeta

← #149923

Subgroups — #149923

Order 512.

271 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 56 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#27964C211yesmaximal, characteristic
2#11511C211yesmaximal, characteristic
3C2×#1324C211yesmaximal, characteristic
4#7366C212yesmaximal, complemented
5#7359C212yesmaximal, complemented
6#2851C411yescharacteristic
7C2×#382C2211yescharacteristic
8#2101—21no—
9#2101—21no—
10#1991C411yescharacteristic
11#1991C2211yescharacteristic
12#1835C411yescharacteristic
13#1835C2211yescharacteristic
14#1824—21no—
15#1823—21no—
16C4×C22⋊C8C411yescharacteristic
17#1463—21no—
18C2×#324—21no—
19#1324—21no—
20#1324—21no—
21#1324C2212yes—
22#1324C2212yes—
23C2×#125C2×C411yescharacteristic
24C22×C4⋊C4D411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26C22×C42C2×C411yescharacteristic
27#407—41no—
28C2×C22⋊C8—21no—
29C2×C22⋊C8—41no—
30C2×C4×C8C2×C411yescharacteristic
31C2×C4×C8C2×C411yescharacteristic
32#389—21no—
33#387—41no—
34#385—41no—
35#382—41no—
36#382C2×C411yescharacteristic
37#382C2×C411yescharacteristic
38#382D411yescharacteristic
39#382D411yescharacteristic
40#382C2311yescharacteristic
41#380—21no—
42C2×#95—41no—
43#340—21no—
44#340—21no—
45#340—21no—
46#340—21no—
47#340—21no—
48#340—21no—
49#340—21no—
50#340—21no—
51#336—21no—
52#324—22no—
53#324—22no—
54#324—41no—
55#324—41no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60C2×M16—81no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—81no—
64#125—22no—
65#125—41no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C22⋊C411yescharacteristic
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C22⋊C4—41no—
81C2×C42—22no—
82C2×C42—21no—
83C2×C42—41no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22×C411yescharacteristic
86C22⋊C8—41no—
87C22⋊C8—41no—
88C22⋊C8—42no—
89C22⋊C8—42no—
90C22⋊C8—81no—
91C22⋊C8—81no—
92C4×C8—22no—
93C4×C8—41no—
94#95—41no—
95#95—41no—
96#95—41no—
97#95—41no—
98#95—41no—
99#95—41no—
100#95—41no—
101C24#9911yescharacteristic
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—81no—
122C22×C4—82no—
123C22×C4—82no—
124C22×C4#9911yescharacteristic
125C22×C4C2×C22⋊C411yescharacteristic
126M16—162no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—82no—
134C2×C8—82no—
135C2×C8—162no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—42no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—81no—
144C4⋊C4—81no—
145C4⋊C4—82no—
146C4⋊C4—82no—
147C22⋊C4—82no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—42no—
152C42#13112yes—
153C23—21no—
154C23—22no—
155C23—21no—
156C23—41no—
157C23#35411yescharacteristic
158C23#35411yescharacteristic
159C23C2×#9911yescharacteristic
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—84no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—161no—
193C2×C4—161no—
194C2×C4#40811yescharacteristic
195C2×C4#41311yescharacteristic
196C8—81no—
197C8—81no—
198C8—162no—
199C8—162no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—42no—
208C22—42no—
209C22#141412yes—
210C22#146811yescharacteristic
211C22C2×#35411yescharacteristic
212C22#213311yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—41no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4—81no—
221C4—81no—
222C4—161no—
223C4—161no—
224C2—21no—
225C2—21no—
226C2—22no—
227C2—41no—
228C2#1155111yescharacteristic
229C2C2×#141411yescharacteristic
230C2#1360311yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7366C2semidirect
5. #7359C2semidirect

Maximal subgroups

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

subgroupindex
#279642
#115112
C2×#13242
#73662
#73592