Finite Group Databasebeta

← #12799

Subgroups — #12799

Order 256.

294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 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#2328C211yesmaximal, characteristic
2C2×#353C212yesmaximal
3#1799C212yesmaximal, complemented
4#1393C212yesmaximal, complemented
5C2×#127C2211yescharacteristic
6C2×C4×D4C411yescharacteristic, complemented
7C2×C4×D4C2211yescharacteristic
8C2×#99C2214yes—
9#390C411yescharacteristic, complemented
10#390C2211yescharacteristic
11#386—22no—
12#386—22no—
13#382C411yescharacteristic, complemented
14#380—22no—
15#376C411yescharacteristic, complemented
16#353—24no—
17#342—22no—
18C22×D4—22no—
19C22×D4C2×C411yescharacteristic
20C22×D4C2311yescharacteristic
21C23×C4—21no—
22C23×C4D412yes—
23#127—22no—
24#127D412yes—
25#127Q812yes—
26C4×D4—42no—
27C4×D4—42no—
28C2×C4⋊C4—42no—
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C22⋊C4—21no—
32C2×C22⋊C4—24no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4D412yes—
36C2×C42C2×C411yescharacteristic
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39#99—24no—
40#99—24no—
41#95—21no—
42#95—22no—
43#95—21no—
44#95—42no—
45C24—21no—
46C24C2×D412yes—
47C2×D4—24no—
48C2×D4—24no—
49C2×D4—21no—
50C2×D4—41no—
51C2×D4—41no—
52C2×D4—41no—
53C2×D4—41no—
54C2×D4—82no—
55C2×D4—82no—
56C2×D4C4⋊C412yescomplemented
57C2×D4C2×D411yescharacteristic
58C2×D4C2×Q811yescharacteristic
59C2×D4#4112yes—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—42no—
66C22×C4—44no—
67C22×C4—82no—
68C22×C4—82no—
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C412yes—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22×C411yescharacteristic
73C4⋊C4—41no—
74C4⋊C4—41no—
75C4⋊C4—84no—
76C22⋊C4—42no—
77C22⋊C4—42no—
78C22⋊C4—44no—
79C22⋊C4—44no—
80C22⋊C4—82no—
81C22⋊C4—82no—
82C42—22no—
83C42—21no—
84C42—41no—
85C42C4⋊C412yescomplemented
86C23—22no—
87C23—22no—
88C23—22no—
89C23—42no—
90C23—42no—
91C23—42no—
92C23—42no—
93C23—41no—
94C23—41no—
95C23C2×C22⋊C411yescharacteristic
96C23#12112yes—
97C23C22⋊Q812yes—
98D4—41no—
99D4—41no—
100D4—42no—
101D4—42no—
102D4—42no—
103D4—84no—
104D4—84no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—21no—
109C2×C4—22no—
110C2×C4—41no—
111C2×C4—44no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—82no—
122C2×C4—84no—
123C2×C4—84no—
124C2×C4—84no—
125C2×C4—84no—
126C2×C4#9912yes—
127C2×C4C2×C4⋊C411yescharacteristic
128C2×C4#11711yescharacteristic
129C22—21no—
130C22—21no—
131C22—22no—
132C22—22no—
133C22—42no—
134C22—42no—
135C22—44no—
136C22—42no—
137C22—42no—
138C22—42no—
139C22—82no—
140C22—82no—
141C22—81no—
142C22—81no—
143C22#38011yescharacteristic
144C22C2×#9911yescharacteristic
145C22#41011yescharacteristic
146C4—21no—
147C4—21no—
148C4—22no—
149C4—42no—
150C4—42no—
151C4—41no—
152C4—81no—
153C4—81no—
154C4—164no—
155C2—21no—
156C2—21no—
157C2—42no—
158C2—42no—
159C2—82no—
160C2#179911yescharacteristic
161C2#212711yescharacteristic
162C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC22×D4C23
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1799C2semidirect
4. #1393C2semidirect
6. C2×C4×D4C4semidirect
9. #390C4semidirect
13. #382C4semidirect
15. #376C4semidirect
56. C2×D4C4⋊C4semidirect
85. C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23282
C2×#3532
#17992
#13932