Finite Group Databasebeta

← #10325

Subgroups — #10325

Order 256.

301 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 61 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#3189C211yesmaximal, characteristic
2#2855C212yesmaximal
3#1817C211yesmaximal, characteristic
4#1520C212yesmaximal
5#1289C211yesmaximal, characteristic
6C2×C22⋊Q8C2212yes—
7C22×C42C2211yescharacteristic
8#484—22no—
9#467—22no—
10#451—22no—
11C2×C22⋊C8C2211yescharacteristic
12C2×C22⋊C8C2211yescharacteristic
13#391—22no—
14#389—22no—
15#387—24no—
16#385—22no—
17#381—22no—
18#380C2212yes—
19#336—22no—
20#328—24no—
21C22×Q8D412yes—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C2×Q16—22no—
26C22×C8—21no—
27C22×C8D412yes—
28C22⋊Q8—22no—
29C22⋊Q8—22no—
30C22⋊Q8—22no—
31C22⋊Q8—22no—
32C2×C4⋊C4—24no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4D412yes—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C42—21no—
39C2×C42—22no—
40C2×C42—22no—
41C2×C42—21no—
42#107—22no—
43#103—24no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46#95—24no—
47#95—24no—
48#95—24no—
49#95D414yes—
50C24C2×D411yescharacteristic
51C2×Q8—22no—
52C2×Q8—22no—
53C2×Q8—44no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—24no—
62C22×C4—22no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4Q1612yescomplemented
66C22×C4C2×D412yes—
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D412yes—
69C22×C4#4111yescharacteristic
70Q16—82no—
71C2×C8—22no—
72C2×C8—42no—
73C2×C8—42no—
74C2×C8—41no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—44no—
78C4⋊C4—48no—
79C4⋊C4—42no—
80C4⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C22⋊C4—42no—
85C42—41no—
86C42—42no—
87C42—41no—
88C23—41no—
89C23—41no—
90C23#12012yes—
91C23#12111yescharacteristic
92C23C2×Q1611yescharacteristic
93C23#13511yescharacteristic
94C23#13712yes—
95Q8—44no—
96Q8—82no—
97C2×C4—24no—
98C2×C4—22no—
99C2×C4—24no—
100C2×C4—22no—
101C2×C4—24no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—44no—
112C2×C4—44no—
113C2×C4—44no—
114C2×C4—44no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—44no—
118C2×C4#12011yescharacteristic
119C2×C4#12112yes—
120C2×C4#12711yescharacteristic
121C8—42no—
122C8—81no—
123C22—22no—
124C22—21no—
125C22—22no—
126C22—21no—
127C22—41no—
128C22—41no—
129C22—42no—
130C22#39211yescharacteristic
131C22#45112yes—
132C22#45212yes—
133C22#46711yescharacteristic
134C22#47011yescharacteristic
135C4—21no—
136C4—22no—
137C4—42no—
138C4—42no—
139C4—84no—
140C4—82no—
141C4—82no—
142C4#45812yes—
143C2—22no—
144C2—22no—
145C2—21no—
146C2—21no—
147C2#200111yescharacteristic
148C2#201011yescharacteristic
149C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
65. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
#31892
#28552
#18172
#15202
#12892