Finite Group Databasebeta

← #8761

Subgroups — #8761

Order 256.

265 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 71 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#2349C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1294C212yesmaximal, complemented
5#1289C212yesmaximal, complemented
6C2×C4×D4—21no—
7C2×C4×D4C2211yescharacteristic
8C2×#99—21no—
9C2×C22⋊C8C2212yes—
10#389—21no—
11#385—21no—
12#385—21no—
13#382C411yescharacteristic, complemented
14#380C411yescharacteristic, complemented
15#380C2212yes—
16#380C2211yescharacteristic
17C4×C4⋊C4C411yescharacteristic, complemented
18C2×#95C411yescharacteristic, complemented
19C2×#95C2211yescharacteristic
20#342—21no—
21#324—21no—
22#323—21no—
23C22×D4D411yescharacteristic
24C23×C4—21no—
25C23×C4—21no—
26C23×C4C2×C411yescharacteristic
27C23×C4C2311yescharacteristic
28C22×C8—22no—
29C4×D4—22no—
30C4×D4—22no—
31C4×D4—44no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4D411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C42—21no—
44C2×C42C2×C411yescharacteristic
45#99—42no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—22no—
52#95C2×C411yescharacteristic
53#95C2×C411yescharacteristic
54#95C2×C411yescharacteristic
55#95C2×C411yescharacteristic
56C24C22⋊C411yescharacteristic
57C24C2×D411yescharacteristic
58C2×D4—22no—
59C2×D4—22no—
60C2×D4SD1612yes—
61C2×D4Q1612yes—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—81no—
74C22×C4—81no—
75C22×C4—81no—
76C22×C4—81no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4#4112yes—
83C22×C4#4112yes—
84C2×C8—42no—
85C2×C8—82no—
86C4⋊C4—22no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—81no—
90C4⋊C4—81no—
91C4⋊C4—82no—
92C22⋊C4—22no—
93C22⋊C4—22no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C42—21no—
99C42—21no—
100C42—42no—
101C23—21no—
102C23—21no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23#10312yes—
108C23C2×C22⋊C411yescharacteristic
109C23#12312yes—
110C23C2×SD1611yescharacteristic
111C23C2×Q1611yescharacteristic
112C23AGL(1,ℤ/8)12yes—
113D4—24no—
114D4—24no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—84no—
140C2×C4—84no—
141C2×C4—82no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4#10312yes—
145C2×C4#11711yescharacteristic
146C2×C4#11711yescharacteristic
147C2×C4#12312yes—
148C8—82no—
149C22—21no—
150C22—22no—
151C22—21no—
152C22—22no—
153C22—22no—
154C22—42no—
155C22—41no—
156C22—41no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22#38111yescharacteristic
162C22C2×#10311yescharacteristic
163C22#41811yescharacteristic
164C22#48112yes—
165C22#48412yes—
166C4—22no—
167C4—22no—
168C4—21no—
169C4—21no—
170C4—42no—
171C4—41no—
172C4—41no—
173C4—81no—
174C4—81no—
175C4—81no—
176C4—162no—
177C2—21no—
178C2—22no—
179C2—21no—
180C2—22no—
181C2—42no—
182C2#182411yescharacteristic
183C2#182711yescharacteristic
184C2#182811yescharacteristic
subgroupG/H
center, socleC22#381
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1294C2semidirect
5. #1289C2semidirect
13. #382C4semidirect
14. #380C4semidirect
17. C4×C4⋊C4C4semidirect
18. C2×#95C4semidirect

Maximal subgroups

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

subgroupindex
#23492
#17992
#14592
#12942
#12892