Finite Group Databasebeta

← #12798

Subgroups — #12798

Order 256.

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

normal subgroups only

This group has 272 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×#124C211yesmaximal, characteristic
2C2×#354C212yesmaximal
3#1799C212yesmaximal
4#1393C212yesmaximal
5C2×#124C2211yescharacteristic
6C2×C4×Q8C411yescharacteristic, complemented
7C2×C4×D4C2211yescharacteristic
8C2×#99C2214yes—
9#391C411yescharacteristic, complemented
10#390C2211yescharacteristic
11#388—22no—
12#382C411yescharacteristic, complemented
13#380—22no—
14C4×C22⋊C4—22no—
15C43C411yescharacteristic, complemented
16#354—24no—
17#342—22no—
18C22×Q8C2×C411yescharacteristic
19C22×D4C2311yescharacteristic
20C23×C4D412yes—
21#124—22no—
22#124D412yes—
23#124Q812yes—
24C4×Q8—41no—
25C4×Q8—41no—
26C4×D4—41no—
27C4×D4—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—42no—
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—24no—
35C2×C22⋊C4D412yes—
36C2×C42—21no—
37C2×C42—21no—
38C2×C42—21no—
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42C2×C411yescharacteristic
42#99—24no—
43#99—24no—
44#95—22no—
45#95—22no—
46#95—42no—
47C24C2×D412yes—
48C2×Q8—21no—
49C2×Q8—41no—
50C2×Q8—41no—
51C2×Q8C4⋊C412yescomplemented
52C2×D4—41no—
53C2×D4—41no—
54C2×D4C2×D411yescharacteristic
55C2×D4C2×Q811yescharacteristic
56C2×D4#4112yes—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—21no—
63C22×C4—44no—
64C22×C4—44no—
65C22×C4C22⋊C412yes—
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22×C411yescharacteristic
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—42no—
72C4⋊C4—84no—
73C22⋊C4—24no—
74C22⋊C4—24no—
75C22⋊C4—42no—
76C22⋊C4—44no—
77C22⋊C4—44no—
78C22⋊C4—44no—
79C42—21no—
80C42—21no—
81C42—21no—
82C42—41no—
83C42—42no—
84C42—42no—
85C42—41no—
86C42—41no—
87C42C4⋊C412yescomplemented
88C23—22no—
89C23—22no—
90C23—42no—
91C23—42no—
92C23C2×C22⋊C411yescharacteristic
93C23#12112yes—
94C23C22⋊Q812yes—
95Q8—42no—
96Q8—42no—
97D4—41no—
98D4—41no—
99D4—42no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—44no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—82no—
120C2×C4—84no—
121C2×C4—84no—
122C2×C4—84no—
123C2×C4#9911yescharacteristic
124C2×C4#9911yescharacteristic
125C2×C4C2×C4⋊C411yescharacteristic
126C2×C4#11711yescharacteristic
127C22—21no—
128C22—21no—
129C22—22no—
130C22—22no—
131C22—42no—
132C22—42no—
133C22—42no—
134C22—42no—
135C22—82no—
136C22#38011yescharacteristic
137C22C2×#9911yescharacteristic
138C22#41011yescharacteristic
139C4—21no—
140C4—21no—
141C4—21no—
142C4—42no—
143C4—42no—
144C4—42no—
145C4—42no—
146C4—41no—
147C4—81no—
148C4—164no—
149C4#35412yescomplemented
150C2—21no—
151C2—21no—
152C2—42no—
153C2—42no—
154C2#179911yescharacteristic
155C2C2×#35411yescharacteristic
156C2#212611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, prime radical, cosocleC22×D4C23
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
6. C2×C4×Q8C4semidirect
9. #391C4semidirect
12. #382C4semidirect
15. C43C4semidirect
51. C2×Q8C4⋊C4semidirect
87. C42C4⋊C4semidirect
149. C4#354semidirect

Maximal subgroups

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

subgroupindex
C4×#1242
C2×#3542
#17992
#13932