Finite Group Databasebeta

← C8×#354 #128454

Subgroups — C8×#354 #128454

Order 512.

442 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C8×#124C211yesmaximal, 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#1831C411yescharacteristic, complemented
14C4×#99C2212yes—
15C8×C22⋊C4—22no—
16C8×C22⋊C4—22no—
17C42×C8C411yescharacteristic, 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—41no—
28C8×Q8—41no—
29C8×D4—41no—
30C8×D4—41no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×#99C2×C414yes—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8D412yes—
38C2×C4×C8—21no—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8—42no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44#391C2×C411yescharacteristic
45#390C2×C411yescharacteristic
46#388—22no—
47#382C2×C411yescharacteristic
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C43C2×C411yescharacteristic
51#354C818yesdirect product factor
52#342—22no—
53#336—22no—
54#336—22no—
55#336—42no—
56#324C2×C414yes—
57#323—22no—
58C22×Q8C4211yescharacteristic
59C22×D4C22×C411yescharacteristic
60C23×C4C2×D412yes—
61C22×C8—21no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—21no—
65C22×C8—44no—
66C22×C8—42no—
67C22×C8C22⋊C412yes—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70#124—22no—
71#124C2×C814yes—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×D4—41no—
75C4×D4—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4C4211yescharacteristic
78C2×C4⋊C4C22×C411yescharacteristic
79C2×C22⋊C4—22no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4#4112yes—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—42no—
87C2×C42C4211yescharacteristic
88C2×C42C4211yescharacteristic
89C2×C42C22×C411yescharacteristic
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—42no—
93#99—24no—
94#99C2×C818yes—
95C22⋊C8—24no—
96C22⋊C8—42no—
97C22⋊C8—44no—
98C22⋊C8—42no—
99C4×C8—21no—
100C4×C8—21no—
101C4×C8—41no—
102C4×C8—42no—
103C4×C8—42no—
104C4×C8—41no—
105C4×C8—41no—
106C4×C8—84no—
107#95—22no—
108#95—22no—
109#95—42no—
110C24C4×D412yes—
111C2×Q8—21no—
112C2×Q8—41no—
113C2×Q8—41no—
114C2×Q8C4×C812yescomplemented
115C2×D4—41no—
116C2×D4—41no—
117C2×D4C22×C812yes—
118C2×D4#13112yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—44no—
128C22×C4C2×C4211yescharacteristic
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4#11711yescharacteristic
131C22×C4C4×D412yes—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—44no—
139C2×C8—41no—
140C2×C8—42no—
141C2×C8—44no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8#9911yescharacteristic
148C2×C8#9911yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C22⋊C4—28no—
153C22⋊C4—28no—
154C22⋊C4—42no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—41no—
161C42—41no—
162C42—41no—
163C42—42no—
164C42—41no—
165C42—41no—
166C42—82no—
167C42—82no—
168C42C4×C812yescomplemented
169C23—22no—
170C23—22no—
171C23—42no—
172C23—42no—
173C23C4×C22⋊C411yescharacteristic
174C23C8×D414yes—
175Q8—42no—
176Q8—42no—
177D4—42no—
178D4—42no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—82no—
202C2×C4—84no—
203C2×C4—84no—
204C2×C4C2×C4×C811yescharacteristic
205C2×C4#40511yescharacteristic
206C2×C4C2×#9911yescharacteristic
207C2×C4#41011yescharacteristic
208C8—21no—
209C8—42no—
210C8—42no—
211C8—41no—
212C8—84no—
213C8—81no—
214C8#35412yesdirect product factor
215C22—21no—
216C22—21no—
217C22—24no—
218C22—44no—
219C22—42no—
220C22—42no—
221C22—82no—
222C22C8×C22⋊C411yescharacteristic
223C22C4×#9911yescharacteristic
224C22#175711yescharacteristic
225C4—21no—
226C4—21no—
227C4—21no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—41no—
232C4—42no—
233C4—88no—
234C4—81no—
235C4C2×#35411yescharacteristic
236C4#212611yescharacteristic
237C2—21no—
238C2—21no—
239C2—44no—
240C2C8×#9911yescharacteristic
241C2C4×#35411yescharacteristic
242C2#1263311yescharacteristic
subgroupG/H
centerC2×C8#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ 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
13. #1831C4semidirect
17. C42×C8C4semidirect
20. #1324C4semidirect
21. #1319C4semidirect
51. #354C8direct (N × H)
114. C2×Q8C4×C8semidirect
168. C42C4×C8semidirect
214. C8#354direct (N × H)

Maximal subgroups

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

subgroupindex
C8×#1242
C4×#3542
#114772
C8×#992
#72052