Finite Group Databasebeta

← #128455

Subgroups — #128455

Order 512.

427 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 120 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#22208C211yesmaximal, characteristic
2C4×#354C211yesmaximal, characteristic
3#11477C211yesmaximal, characteristic
4C8×#99C212yesmaximal
5#7205C212yesmaximal
6C2×C8×Q8C411yescharacteristic, complemented
7C2×C8×D4C2211yescharacteristic
8C4×#124C2211yescharacteristic
9C2×#354C412yes—
10#1984C411yescharacteristic, complemented
11#1954C2211yescharacteristic
12#1923—22no—
13#1850—22no—
14#1842C411yescharacteristic, complemented
15C4×#99C2212yes—
16C8×C22⋊C4—22no—
17#1744C411yescharacteristic, complemented
18#1459C2212yes—
19#1393C412yes—
20#1324C412yescomplemented
21#1319C412yescomplemented
22#1282—22no—
23C23×C8D412yes—
24C2×#124C2×C411yescharacteristic
25C2×C4×Q8C2×C411yescharacteristic
26C2×C4×D4C2311yescharacteristic
27C8×Q8—42no—
28C8×D4—42no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4⋊C8C2×C411yescharacteristic
33C2×#99C2×C414yes—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8D412yes—
37C2×C4×C8—21no—
38C2×C4×C8—42no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41#391C2×C411yescharacteristic
42#390C2×C411yescharacteristic
43#388—22no—
44#382C2×C411yescharacteristic
45C4×C22⋊C4—22no—
46C4×C22⋊C4—22no—
47C43C2×C411yescharacteristic
48#354C818yescomplemented
49#342—22no—
50#336—22no—
51#336—21no—
52#336—21no—
53#336—42no—
54#324C2×C414yes—
55#323—22no—
56C22×Q8C4211yescharacteristic
57C22×D4C22×C411yescharacteristic
58C23×C4C2×D412yes—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—42no—
64C22×C8—82no—
65C22×C8C22⋊C412yes—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68#124—22no—
69#124C2×C814yes—
70C4×Q8—41no—
71C4×Q8—41no—
72C4×D4—41no—
73C4×D4—41no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C4211yescharacteristic
76C2×C4⋊C4C22×C411yescharacteristic
77C2×C22⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4#4112yes—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—42no—
85C2×C42C4211yescharacteristic
86C2×C42C4211yescharacteristic
87C2×C42C22×C411yescharacteristic
88C4⋊C8—41no—
89C4⋊C8—41no—
90C4⋊C8—42no—
91C4⋊C8—82no—
92#99—24no—
93#99C2×C818yes—
94C22⋊C8—24no—
95C22⋊C8—42no—
96C22⋊C8—42no—
97C22⋊C8—82no—
98C4×C8—22no—
99C4×C8—41no—
100C4×C8—42no—
101C4×C8—84no—
102#95—22no—
103#95—22no—
104#95—42no—
105C24C4×D412yes—
106C2×Q8—21no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8C4×C812yescomplemented
110C2×D4—41no—
111C2×D4—41no—
112C2×D4C22×C812yes—
113C2×D4#13112yes—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4C2×C4211yescharacteristic
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4#11711yescharacteristic
126C22×C4C4×D412yes—
127C2×C8—21no—
128C2×C8—22no—
129C2×C8—41no—
130C2×C8—44no—
131C2×C8—42no—
132C2×C8—41no—
133C2×C8—44no—
134C2×C8—41no—
135C2×C8—42no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—82no—
140C2×C8—82no—
141C2×C8—82no—
142C2×C8#9912yes—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—42no—
146C22⋊C4—28no—
147C22⋊C4—28no—
148C22⋊C4—42no—
149C22⋊C4—44no—
150C22⋊C4—82no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—41no—
155C42—41no—
156C42—41no—
157C42—41no—
158C42—41no—
159C42—81no—
160C42—82no—
161C42—82no—
162C42C4×C812yescomplemented
163C23—22no—
164C23—22no—
165C23—42no—
166C23—42no—
167C23C4×C22⋊C411yescharacteristic
168C23C8×D414yes—
169Q8—42no—
170Q8—42no—
171D4—42no—
172D4—42no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—48no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—82no—
196C2×C4—84no—
197C2×C4—84no—
198C2×C4C2×C4×C811yescharacteristic
199C2×C4#40511yescharacteristic
200C2×C4C2×#9911yescharacteristic
201C2×C4#41011yescharacteristic
202C8—22no—
203C8—42no—
204C8—41no—
205C8—81no—
206C8—84no—
207C8—81no—
208C22—21no—
209C22—21no—
210C22—24no—
211C22—44no—
212C22—42no—
213C22—42no—
214C22—82no—
215C22C8×C22⋊C411yescharacteristic
216C22C4×#9911yescharacteristic
217C22#175711yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—41no—
224C4—42no—
225C4—81no—
226C4—88no—
227C4—81no—
228C4#212711yescharacteristic
229C4#212911yescharacteristic
230C2—21no—
231C2—21no—
232C2—44no—
233C2C8×#9911yescharacteristic
234C2#1262711yescharacteristic
235C2#1263411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×C8×Q8C4semidirect
10. #1984C4semidirect
14. #1842C4semidirect
17. #1744C4semidirect
20. #1324C4semidirect
21. #1319C4semidirect
48. #354C8semidirect
109. C2×Q8C4×C8semidirect
162. C42C4×C8semidirect

Maximal subgroups

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

subgroupindex
#222082
C4×#3542
#114772
C8×#992
#72052