Finite Group Databasebeta

← #9078

Subgroups — #9078

Order 256.

377 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×C4×M16C211yesmaximal, characteristic, complemented
2C4×#121C211yesmaximal, characteristic
3#1797C211yesmaximal, characteristic, complemented
4C2×D4⋊C8C212yesmaximal, complemented
5#1279C212yesmaximal, complemented
6C22×M16—21no—
7C2×#121C411yescharacteristic
8C2×C4×D4—21no—
9C2×C4×D4C2211yescharacteristic
10C22×C42C2211yescharacteristic
11C2×C4⋊C8C2212yes—
12C2×C22⋊C8—21no—
13C4×M16—24no—
14C2×#97—21no—
15C2×C4×C8C2212yes—
16#390C411yescharacteristic
17#389C411yescharacteristic
18#388—21no—
19#386—21no—
20#380C411yescharacteristic
21C4×C4⋊C4C2211yescharacteristic
22C4×C22⋊C4—21no—
23#336—21no—
24D4⋊C8—24no—
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C23×C4—21no—
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4D411yescharacteristic
31C2×M16—24no—
32C2×M16—24no—
33C2×M16—24no—
34C22×C8—22no—
35C22×C8—22no—
36#121—22no—
37#121—22no—
38C4×D4—22no—
39C4×D4—22no—
40C4×D4—44no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C4⋊C4C2×C411yescharacteristic
44C2×C4⋊C4C2×C411yescharacteristic
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C42—22no—
51C2×C42—22no—
52C2×C42—21no—
53C2×C42C2×C411yescharacteristic
54C2×C42D411yescharacteristic
55C2×C42D411yescharacteristic
56C2×C42C2311yescharacteristic
57C4⋊C8—22no—
58C4⋊C8—22no—
59C22⋊C8—42no—
60#97—24no—
61C4×C8—22no—
62C4×C8D414yescomplemented
63#95—21no—
64#95—21no—
65#95—21no—
66C24—21no—
67C24C22⋊C411yescharacteristic
68C2×D4—24no—
69C2×D4—22no—
70C2×D4—21no—
71C2×D4—22no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4M1612yescomplemented
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22×C411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100M16—216no—
101C2×C8—24no—
102C2×C8—24no—
103C2×C8—24no—
104C2×C8—42no—
105C2×C8—42no—
106C2×C8—42no—
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—42no—
111C4⋊C4M1612yescomplemented
112C22⋊C4—24no—
113C22⋊C4—42no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C42—22no—
119C42—21no—
120C42—22no—
121C42—21no—
122C42—42no—
123C42C2×D412yes—
124C42#4112yes—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—21no—
130C23—21no—
131C23—22no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23#10412yes—
139C23C2×C22⋊C411yescharacteristic
140D4—24no—
141D4—48no—
142D4—42no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4#10412yes—
171C2×C4C4×D412yes—
172C2×C4#12011yescharacteristic
173C2×C4#12112yes—
174C2×C4#12311yescharacteristic
175C2×C4C2×M1611yescharacteristic
176C2×C4C2×M1611yescharacteristic
177C2×C4#13112yes—
178C2×C4AGL(1,ℤ/8)12yes—
179C2×C4AGL(1,ℤ/8)11yescharacteristic
180C2×C4#13711yescharacteristic
181C8—28no—
182C8—82no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22#38611yescharacteristic
198C22#40711yescharacteristic
199C22#40811yescharacteristic
200C22#41711yescharacteristic
201C22#41811yescharacteristic
202C22C2×#10411yescharacteristic
203C22C2×#10411yescharacteristic
204C4—21no—
205C4—22no—
206C4—21no—
207C4—41no—
208C4—41no—
209C4—42no—
210C4—81no—
211C4—81no—
212C4#43512yes—
213C4#43612yes—
214C4#46812yes—
215C4#46912yes—
216C2—42no—
217C2—82no—
218C2#148311yescharacteristic
219C2#148411yescharacteristic
220C2#148911yescharacteristic
221C2#149011yescharacteristic
222C2#187011yescharacteristic
223C2#189511yescharacteristic
224C2#189611yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC2×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×M16C2semidirect
3. #1797C2semidirect
4. C2×D4⋊C8C2semidirect
5. #1279C2semidirect
62. C4×C8D4semidirect
74. C2×D4M16semidirect
111. C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×M162
C4×#1212
#17972
C2×D4⋊C82
#12792