Finite Group Databasebeta

← #12797

Subgroups — #12797

Order 256.

340 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4×#127C211yesmaximal, characteristic
2C2×#353C212yesmaximal
3#1799C212yesmaximal, complemented
4#1393C212yesmaximal, complemented
5C2×#127C2211yescharacteristic
6C2×C4×D4—22no—
7C2×C4×D4C411yescharacteristic, complemented
8C2×C4×D4C2211yescharacteristic
9C2×#99C2214yes—
10#390—22no—
11#390C411yescharacteristic, complemented
12#390C2211yescharacteristic
13#384C411yescharacteristic, complemented
14#380—22no—
15C43C411yescharacteristic, 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—41no—
27C4×D4—41no—
28C4×D4—48no—
29C4×D4—41no—
30C4×D4—41no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—42no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—24no—
39C2×C22⋊C4D412yes—
40C2×C42—21no—
41C2×C42—21no—
42C2×C42C2×C411yescharacteristic
43C2×C42C2×C411yescharacteristic
44C2×C42C2×C411yescharacteristic
45#99—24no—
46#99—24no—
47#95—42no—
48C24—21no—
49C24C2×D412yes—
50C2×D4—24no—
51C2×D4—24no—
52C2×D4—21no—
53C2×D4—41no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4—44no—
58C2×D4—44no—
59C2×D4C4⋊C412yescomplemented
60C2×D4C2×D411yescharacteristic
61C2×D4C2×Q811yescharacteristic
62C2×D4#4112yes—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—21no—
68C22×C4—44no—
69C22×C4—44no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—44no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C412yes—
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22×C411yescharacteristic
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—44no—
80C4⋊C4—84no—
81C22⋊C4—42no—
82C22⋊C4—44no—
83C22⋊C4—44no—
84C22⋊C4—42no—
85C22⋊C4—44no—
86C22⋊C4—44no—
87C42—24no—
88C42—21no—
89C42—21no—
90C42—21no—
91C42—41no—
92C42—42no—
93C42C4⋊C412yescomplemented
94C23—22no—
95C23—22no—
96C23—22no—
97C23—42no—
98C23—42no—
99C23—42no—
100C23—42no—
101C23—41no—
102C23—41no—
103C23C2×C22⋊C411yescharacteristic
104C23#12112yes—
105C23C22⋊Q812yes—
106D4—41no—
107D4—41no—
108D4—42no—
109D4—42no—
110D4—48no—
111D4—48no—
112D4—42no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—22no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—44no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4—82no—
132C2×C4—84no—
133C2×C4—84no—
134C2×C4—84no—
135C2×C4#9911yescharacteristic
136C2×C4#9911yescharacteristic
137C2×C4C2×C4⋊C411yescharacteristic
138C2×C4#11711yescharacteristic
139C22—21no—
140C22—21no—
141C22—22no—
142C22—22no—
143C22—42no—
144C22—42no—
145C22—44no—
146C22—42no—
147C22—42no—
148C22—42no—
149C22—82no—
150C22—82no—
151C22—81no—
152C22—81no—
153C22#38011yescharacteristic
154C22C2×#9911yescharacteristic
155C22#41011yescharacteristic
156C4—24no—
157C4—21no—
158C4—21no—
159C4—21no—
160C4—42no—
161C4—42no—
162C4—41no—
163C4—81no—
164C4—164no—
165C4#35312yescomplemented
166C2—21no—
167C2—21no—
168C2—42no—
169C2—42no—
170C2—82no—
171C2#179911yescharacteristic
172C2C2×#35311yescharacteristic
173C2#212611yescharacteristic
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
7. C2×C4×D4C4semidirect
11. #390C4semidirect
13. #384C4semidirect
15. C43C4semidirect
59. C2×D4C4⋊C4semidirect
93. C42C4⋊C4semidirect
165. C4#353semidirect

Maximal subgroups

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

subgroupindex
C4×#1272
C2×#3532
#17992
#13932