Finite Group Databasebeta

← #150689

Subgroups — #150689

Order 512.

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

normal subgroups only

This group has 287 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#23818C211yesmaximal, characteristic
2#12798C211yesmaximal, characteristic
3#11530C211yesmaximal, characteristic
4#7428C212yesmaximal
5#7412C212yesmaximal
6C4×#125C411yescharacteristic, complemented
7C4×#124C2211yescharacteristic
8C2×#354—21no—
9#2101—21no—
10#2074C411yescharacteristic, complemented
11#2073C2211yescharacteristic
12#2042—21no—
13#2026C411yescharacteristic, complemented
14#2025C2211yescharacteristic
15#1799—21no—
16C4×#103—21no—
17C4×#102—21no—
18#1744C411yescharacteristic, complemented
19#1467—21no—
20#1393C2212yes—
21#1324—21no—
22#1318C2212yes—
23C2×#125C2×C411yescharacteristic
24C2×#124D411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27#432—41no—
28C2×#103—21no—
29C2×#103—21no—
30C2×#102—21no—
31C2×#102—21no—
32C2×#99—22no—
33C2×C4×C8—21no—
34#391C2×C411yescharacteristic
35#390C2311yescharacteristic
36#389—21no—
37#388—41no—
38#387—41no—
39#382C2×C411yescharacteristic
40#382D411yescharacteristic
41#380—41no—
42C4×C4⋊C4—21no—
43C4×C4⋊C4—41no—
44C4×C22⋊C4—41no—
45C43C2×C411yescharacteristic
46#354—42no—
47#342—41no—
48#340—21no—
49#340—21no—
50#340—21no—
51#340—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#324—41no—
57C22×Q8C22⋊C411yescharacteristic
58C22×D4C2×D411yescharacteristic
59C23×C4—21no—
60C22×C8—21no—
61C22×C8—21no—
62#125—42no—
63#124—21no—
64#124—21no—
65#124—41no—
66C4×Q8—41no—
67C4×Q8—41no—
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—81no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—42no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22×C411yescharacteristic
89C4⋊C8—81no—
90#103—42no—
91#103—42no—
92#102—42no—
93#102—41no—
94#102—41no—
95#99—42no—
96#99—42no—
97C22⋊C8—42no—
98#97—81no—
99C4×C8—41no—
100C4×C8—41no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—81no—
107C24—21no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×D4—21no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4AGL(1,ℤ/8)11yescharacteristic
116C2×D4#13711yescharacteristic
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—82no—
127C22×C4—82no—
128C22×C4#9911yescharacteristic
129C22×C4#9911yescharacteristic
130C22×C4C2×C22⋊C411yescharacteristic
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—82no—
138C4⋊C4—22no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—81no—
147C4⋊C4—82no—
148C4⋊C4—162no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—81no—
152C22⋊C4—82no—
153C22⋊C4—82no—
154C22⋊C4—82no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—41no—
160C42—42no—
161C42—41no—
162C42—41no—
163C42—82no—
164C42—81no—
165C23—21no—
166C23—21no—
167C23—41no—
168C23—41no—
169C23—81no—
170C23—81no—
171C23C2×#9911yescharacteristic
172Q8—42no—
173Q8—81no—
174D4—41no—
175D4—41no—
176D4—81no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—161no—
203C2×C4—162no—
204C2×C4—162no—
205C2×C4—162no—
206C2×C4#35411yescharacteristic
207C2×C4#35411yescharacteristic
208C2×C4#41711yescharacteristic
209C2×C4#42111yescharacteristic
210C8—81no—
211C8—81no—
212C8—162no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22—161no—
222C22#151311yescharacteristic
223C22#212011yescharacteristic
224C22C2×#35411yescharacteristic
225C4—21no—
226C4—21no—
227C4—21no—
228C4—41no—
229C4—81no—
230C4—81no—
231C4—82no—
232C4—81no—
233C4—81no—
234C4—81no—
235C4—161no—
236C4—322no—
237C4#141412yes—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—81no—
242C2#1182311yescharacteristic
243C2#1359411yescharacteristic
244C2C2×#141411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C4×#125C4semidirect
10. #2074C4semidirect
13. #2026C4semidirect
18. #1744C4semidirect

Maximal subgroups

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

subgroupindex
#238182
#127982
#115302
#74282
#74122