Finite Group Databasebeta

← #149705

Subgroups — #149705

Order 512.

304 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C4×#488C211yesmaximal, characteristic
2#11477C211yesmaximal, characteristic
3#11476C211yesmaximal, characteristic
4#7366C212yesmaximal, complemented
5#7359C212yesmaximal, complemented
6C2×#488C411yescharacteristic
7C4×#125C411yescharacteristic
8C4×#124C2211yescharacteristic
9#1991C2211yescharacteristic
10#1958C411yescharacteristic
11#1934—21no—
12#1933—21no—
13#1835C2211yescharacteristic
14C4×#103—21no—
15C4×#102—21no—
16C42×C8C411yescharacteristic
17#1461—21no—
18#1459—21no—
19#1324C2212yes—
20#1324C2212yes—
21#1319—21no—
22#1318—21no—
23C2×#125C2×C411yescharacteristic
24C2×#124C2×C411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27#488—22no—
28#488C814yescomplemented
29#431—41no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#102—21no—
33C2×#102—21no—
34C2×C22⋊C8—41no—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39#391D411yescharacteristic
40#390D411yescharacteristic
41#389—21no—
42#388—41no—
43#387—41no—
44#382C2×C411yescharacteristic
45#382C2311yescharacteristic
46C4×C4⋊C4—21no—
47C4×C4⋊C4—41no—
48C4×C22⋊C4—41no—
49C43C2×C411yescharacteristic
50#340—21no—
51#340—21no—
52#340—21no—
53#340—21no—
54#324—22no—
55#324—22no—
56#324—41no—
57#323—41no—
58C22×Q8C22⋊C411yescharacteristic
59C22×D4C22⋊C411yescharacteristic
60C23×C4—21no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—81no—
64#125—21no—
65#125—41no—
66#125C2×C812yescomplemented
67#124—41no—
68#124C2×C812yes—
69#124M1612yes—
70C4×Q8—41no—
71C4×Q8—41no—
72C4×D4—41no—
73C4×D4—41no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C4⋊C4C2×D411yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—41no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22×C411yescharacteristic
91C4⋊C8—81no—
92#103—24no—
93#103—42no—
94#102—24no—
95#102—42no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98C22⋊C8—81no—
99C4×C8—21no—
100C4×C8—41no—
101C4×C8—41no—
102C4×C8—41no—
103C4×C8—42no—
104C4×C8—81no—
105C4×C8C2×C812yescomplemented
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111C24—21no—
112C2×Q8—21no—
113C2×Q8—41no—
114C2×Q8—41no—
115C2×Q8C22⋊C812yes—
116C2×D4—21no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4C22⋊C812yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—82no—
130C22×C4#9911yescharacteristic
131C22×C4#9911yescharacteristic
132C22×C4C2×C22⋊C411yescharacteristic
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—82no—
140C2×C8—82no—
141C2×C8—162no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—42no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—81no—
151C4⋊C4—82no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—81no—
155C22⋊C4—82no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—41no—
160C42—42no—
161C42—42no—
162C42—41no—
163C42—41no—
164C42—82no—
165C42C22×C811yescharacteristic
166C42C2×M1611yescharacteristic
167C23—22no—
168C23—41no—
169C23—41no—
170C23—81no—
171C23—81no—
172C23C2×#9911yescharacteristic
173Q8—42no—
174Q8—81no—
175D4—42no—
176D4—81no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—161no—
204C2×C4#35411yescharacteristic
205C2×C4#35411yescharacteristic
206C2×C4C2×C22⋊C811yescharacteristic
207C2×C4#41311yescharacteristic
208C8—42no—
209C8—42no—
210C8—162no—
211C8—162no—
212C22—21no—
213C22—21no—
214C22—42no—
215C22—82no—
216C22—81no—
217C22—81no—
218C22—161no—
219C22#145911yescharacteristic
220C22C2×#35411yescharacteristic
221C22#213211yescharacteristic
222C4—21no—
223C4—21no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—42no—
228C4—41no—
229C4—42no—
230C4—82no—
231C4—81no—
232C4—81no—
233C4—161no—
234C4#141412yes—
235C2—21no—
236C2—21no—
237C2—82no—
238C2#1147711yescharacteristic
239C2C2×#141411yescharacteristic
240C2#1360211yescharacteristic
subgroupG/H
centerC2×C4#354
commutatorC42C22×C8
Frattini, cosocle#382C23
socleC22C2×#354
prime radicalC22×D4C22⋊C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ 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
28. #488C8semidirect
66. #125C2×C8semidirect
105. C4×C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#4882
#114772
#114762
#73662
#73592