Finite Group Databasebeta

← #11475

Subgroups — #11475

Order 256.

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

normal subgroups only

This group has 308 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
2#1459C212yesmaximal, complemented
3#1325C212yesmaximal, complemented
4#1318C212yesmaximal, complemented
5C2×#127C411yescharacteristic
6C2×C4×D4—22no—
7C2×C4×D4C411yescharacteristic
8C2×C4×D4C2211yescharacteristic
9C2×C22⋊C8—22no—
10#390—22no—
11#390C411yescharacteristic
12#390C2211yescharacteristic
13#384C2211yescharacteristic
14C43C411yescharacteristic
15#324C2214yes—
16#323—22no—
17C22×D4—22no—
18C22×D4C2×C411yescharacteristic
19C22×D4C2×C411yescharacteristic
20C23×C4—21no—
21C23×C4D412yes—
22C22×C8—42no—
23#127—22no—
24#127C814yescomplemented
25C4×D4—41no—
26C4×D4—41no—
27C4×D4—48no—
28C4×D4—41no—
29C4×D4—41no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4C2311yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4D412yes—
37C2×C42—21no—
38C2×C42—21no—
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42C2×C411yescharacteristic
42C22⋊C8—24no—
43C22⋊C8—42no—
44C24—21no—
45C24C22⋊C412yes—
46C2×D4—28no—
47C2×D4—21no—
48C2×D4—41no—
49C2×D4—41no—
50C2×D4—41no—
51C2×D4—41no—
52C2×D4—44no—
53C2×D4—44no—
54C2×D4C2×C812yes—
55C2×D4C2×C812yescomplemented
56C2×D4M1612yes—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—21no—
62C22×C4—44no—
63C22×C4—44no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D412yes—
70C2×C8—44no—
71C2×C8—84no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—44no—
75C22⋊C4—42no—
76C22⋊C4—44no—
77C22⋊C4—44no—
78C22⋊C4—42no—
79C42—24no—
80C42—21no—
81C42—21no—
82C42—21no—
83C42—41no—
84C42—42no—
85C42C2×C812yescomplemented
86C23—22no—
87C23—22no—
88C23—22no—
89C23—42no—
90C23—42no—
91C23—42no—
92C23—42no—
93C23—41no—
94C23—41no—
95C23C22⋊C814yes—
96C23C2×C22⋊C411yescharacteristic
97D4—42no—
98D4—42no—
99D4—42no—
100D4—416no—
101D4—42no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—44no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—41no—
118C2×C4—82no—
119C2×C4—82no—
120C2×C4—82no—
121C2×C4#9911yescharacteristic
122C2×C4#9911yescharacteristic
123C2×C4C22×C811yescharacteristic
124C2×C4C2×M1611yescharacteristic
125C8—84no—
126C22—21no—
127C22—21no—
128C22—24no—
129C22—44no—
130C22—44no—
131C22—42no—
132C22—42no—
133C22—42no—
134C22—82no—
135C22—82no—
136C22—81no—
137C22—81no—
138C22C2×C22⋊C811yescharacteristic
139C22C2×#9911yescharacteristic
140C22#41311yescharacteristic
141C4—24no—
142C4—21no—
143C4—21no—
144C4—21no—
145C4—42no—
146C4—42no—
147C4—41no—
148C4—81no—
149C4#35312yes—
150C2—21no—
151C2—21no—
152C2—44no—
153C2—82no—
154C2#145911yescharacteristic
155C2C2×#35311yescharacteristic
156C2#213211yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC2×C4C22×C8
Frattini, cosocleC2×C4⋊C4C23
socleC22C2×#99
prime radicalC2×#127C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1459C2semidirect
3. #1325C2semidirect
4. #1318C2semidirect
24. #127C8semidirect
55. C2×D4C2×C8semidirect
85. C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#1272
#14592
#13252
#13182