Finite Group Databasebeta

← #8760

Subgroups — #8760

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—42no—
32C4×D4—42no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4D411yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4D411yescharacteristic
43C2×C22⋊C4D411yescharacteristic
44C2×C42—21no—
45C2×C42C2×C411yescharacteristic
46#99—42no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—22no—
53#95C2×C411yescharacteristic
54#95C2×C411yescharacteristic
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C24C2×D411yescharacteristic
59C2×D4—22no—
60C2×D4—22no—
61C2×D4D812yes—
62C2×D4SD1612yes—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—42no—
75C22×C4—81no—
76C22×C4—81no—
77C22×C4—81no—
78C22×C4—81no—
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4#4112yes—
85C22×C4#4112yes—
86C2×C8—42no—
87C2×C8—82no—
88C4⋊C4—22no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—81no—
92C4⋊C4—81no—
93C4⋊C4—82no—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—42no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C42—21no—
102C42—21no—
103C42—42no—
104C23—21no—
105C23—21no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23#10212yes—
111C23C2×C22⋊C411yescharacteristic
112C23#12312yes—
113C23C2×D811yescharacteristic
114C23C2×SD1611yescharacteristic
115C23#13712yes—
116D4—24no—
117D4—24no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—42no—
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—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4—84no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4#10212yes—
149C2×C4#11711yescharacteristic
150C2×C4#11711yescharacteristic
151C2×C4#12312yes—
152C8—82no—
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22—42no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—41no—
165C22#38111yescharacteristic
166C22C2×#10211yescharacteristic
167C22#41911yescharacteristic
168C22#48012yes—
169C22#48312yes—
170C4—22no—
171C4—22no—
172C4—21no—
173C4—21no—
174C4—42no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—81no—
179C4—81no—
180C4—162no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—22no—
185C2—42no—
186C2#182311yescharacteristic
187C2#182711yescharacteristic
188C2#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