Finite Group Databasebeta

← #12967

Subgroups — #12967

Order 256.

258 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 59 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#2379C211yesmaximal, characteristic
2C2×#354C211yesmaximal, characteristic
3#1885C211yesmaximal, characteristic
4#1867C212yesmaximal
5#1390C212yesmaximal
6C2×#124C2211yescharacteristic
7#518C2211yescharacteristic
8#419—21no—
9C2×#100C2211yescharacteristic
10C2×#99C2212yes—
11#391—21no—
12#388—22no—
13#386C2212yes—
14#385—22no—
15#385—22no—
16#382—21no—
17#354C414yescomplemented
18#342—22no—
19#339—21no—
20C22×Q8D411yescharacteristic
21C22×D4C2311yescharacteristic
22C23×C4D412yes—
23C2×M16—21no—
24#124—22no—
25#124C2×C412yes—
26#124Q812yes—
27C4×D4—41no—
28C4×D4—41no—
29#117—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—42no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4D412yes—
39C2×C42—21no—
40C2×C42D411yescharacteristic
41#103—41no—
42#103—41no—
43#100—22no—
44#99—22no—
45#99C2×C414yes—
46#95—22no—
47#95—22no—
48#95—22no—
49#95—42no—
50C24C2×D412yes—
51C2×Q8—21no—
52C2×Q8—41no—
53C2×Q8—41no—
54C2×Q8C4⋊C412yescomplemented
55C2×D4—41no—
56C2×D4—41no—
57C2×D4C22×C411yescharacteristic
58C2×D4C2×Q811yescharacteristic
59C2×D4#4112yes—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—82no—
69C22×C4C2×D411yescharacteristic
70M16—41no—
71M16—41no—
72C2×C8—41no—
73C4⋊C4—21no—
74C4⋊C4—21no—
75C4⋊C4—81no—
76C4⋊C4—81no—
77C4⋊C4—81no—
78C4⋊C4—84no—
79C22⋊C4—24no—
80C22⋊C4—24no—
81C22⋊C4—24no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C22⋊C4—41no—
85C22⋊C4—42no—
86C22⋊C4—82no—
87C42—21no—
88C42—41no—
89C42—81no—
90C42C4⋊C412yescomplemented
91C23—22no—
92C23—22no—
93C23—42no—
94C23—42no—
95C23C4×D412yes—
96C23#12011yescharacteristic
97C23C22⋊Q812yes—
98Q8—42no—
99Q8—81no—
100D4—41no—
101D4—41no—
102D4—42no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—44no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—44no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—82no—
122C2×C4—82no—
123C2×C4—82no—
124C2×C4—82no—
125C2×C4—82no—
126C2×C4C2×C4⋊C411yescharacteristic
127C2×C4#12311yescharacteristic
128C8—81no—
129C22—21no—
130C22—21no—
131C22—22no—
132C22—22no—
133C22—42no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—82no—
138C22#38511yescharacteristic
139C22B(4,𝔽2)11yescharacteristic
140C22#45811yescharacteristic
141C4—21no—
142C4—21no—
143C4—42no—
144C4—42no—
145C4—82no—
146C4—84no—
147C4—82no—
148C4—81no—
149C4—81no—
150C2—21no—
151C2—21no—
152C2—42no—
153C2—42no—
154C2#186711yescharacteristic
155C2#219911yescharacteristic
156C2#220111yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×D4C22×C4
Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #354C4semidirect
54. C2×Q8C4⋊C4semidirect
90. C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23792
C2×#3542
#18852
#18672
#13902