Finite Group Databasebeta

← #12709

Subgroups — #12709

Order 256.

493 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 73 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#3485C211yesmaximal, characteristic
2#2133C214yesmaximal, complemented
3#1785C212yesmaximal, complemented
4C2×H(5,𝔽2)C411yescharacteristic
5C22×#41C2211yescharacteristic
6#547—24no—
7#546—24no—
8#536C2212yes—
9#534—22no—
10#525C412yes—
11C2×#123—22no—
12#518C411yescharacteristic
13C2×#101C2214yescomplemented
14C2×#100—22no—
15#407—22no—
16#356—24no—
17#355—24no—
18H(5,𝔽2)—44no—
19C2×#41—22no—
20C2×#41—24no—
21C2×#41—44no—
22C2×#41C2×C412yes—
23C22×Q8C2311yescharacteristic
24C22×D4—22no—
25C22×D4—22no—
26C22×D4C2×C411yescharacteristic
27C22×D4D412yes—
28C23×C4C2×C411yescharacteristic
29C23×C4D412yes—
30C2×M16—24no—
31C4⋊Q8D414yes—
32#124D414yes—
33#123—28no—
34#123—24no—
35#123—22no—
36#123—24no—
37C22⋊Q8—22no—
38C22⋊Q8—24no—
39#121—28no—
40#121—22no—
41#120—24no—
42#120—22no—
43C4×D4—24no—
44C4×D4—24no—
45#117C2×C412yes—
46C2×C4⋊C4—24no—
47C2×C22⋊C4—24no—
48C2×C22⋊C4—22no—
49#101—28no—
50#100—44no—
51C22⋊C8—44no—
52C24—22no—
53C24C22⋊C411yescharacteristic
54C24C22⋊C412yes—
55#41—44no—
56#41—416no—
57#41—48no—
58C2×Q8—42no—
59C2×Q8C22⋊C412yes—
60C2×Q8C2×D414yes—
61C2×D4—24no—
62C2×D4—24no—
63C2×D4—24no—
64C2×D4—22no—
65C2×D4—24no—
66C2×D4—22no—
67C2×D4—24no—
68C2×D4—42no—
69C2×D4—48no—
70C2×D4—44no—
71C2×D4—44no—
72C2×D4C22⋊C412yes—
73C22×C4—22no—
74C22×C4—24no—
75C22×C4—24no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—24no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D412yes—
84M16—44no—
85M16—44no—
86C2×C8—44no—
87C4⋊C4—24no—
88C4⋊C4—22no—
89C4⋊C4—28no—
90C22⋊C4—24no—
91C22⋊C4—24no—
92C22⋊C4—24no—
93C22⋊C4—22no—
94C22⋊C4—28no—
95C42C22⋊C414yes—
96C23—22no—
97C23—22no—
98C23—22no—
99C23—21no—
100C23—22no—
101C23—44no—
102C23—82no—
103C23—81no—
104C23—84no—
105C23#9914yes—
106C23C2×C22⋊C411yescharacteristic
107Q8—44no—
108Q8—42no—
109D4—28no—
110D4—48no—
111D4—42no—
112D4—48no—
113D4—44no—
114D4—84no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—48no—
123C2×C4—44no—
124C2×C4—44no—
125C2×C4—44no—
126C2×C4—84no—
127C2×C4C2×C22⋊C412yes—
128C2×C4#12014yes—
129C8—84no—
130C22—24no—
131C22—41no—
132C22—44no—
133C22—44no—
134C22—42no—
135C22—44no—
136C22—41no—
137C22—42no—
138C22—82no—
139C22—82no—
140C22—164no—
141C22#37911yescharacteristic
142C22C2×#9912yes—
143C4—24no—
144C4—22no—
145C4—44no—
146C4—41no—
147C4—42no—
148C2—21no—
149C2—22no—
150C2—44no—
151C2—81no—
152C2—82no—
153C2#178111yescharacteristic
subgroupG/H
center, socleC2#1781
commutatorC22×C4C22×C4
Frattini, cosocleC22×Q8C23
prime radical#3485C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2133C2semidirect
3. #1785C2semidirect
13. C2×#101C22semidirect

Maximal subgroups

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

subgroupindex
#34852
#21332
#17852