Finite Group Databasebeta

← #12710

Subgroups — #12710

Order 256.

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

normal subgroups only

This group has 476 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#3486C211yesmaximal, characteristic
2#2133C214yesmaximal, complemented
3#1784C212yesmaximal, complemented
4C2×2−1+4C411yescharacteristic
5C23×Q8C2211yescharacteristic
6D4×Q8—24no—
7#548—24no—
8#536—22no—
9#536C2212yes—
10#525—22no—
11#525C412yes—
12#519C411yescharacteristic
13C2×#101—22no—
14C2×#101C2214yes—
15#407—22no—
16#356—24no—
17#355—24no—
182−1+4—44no—
19C2×#41—22no—
20C2×#41—22no—
21C2×#41C2×C412yes—
22C22×Q8—22no—
23C22×Q8—24no—
24C22×Q8—44no—
25C22×Q8C2×C411yescharacteristic
26C22×Q8D412yes—
27C22×Q8C2311yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D412yes—
30C2×M16—24no—
31C4⋊Q8—24no—
32C4⋊Q8D414yes—
33#124—24no—
34#124D414yes—
35#123—22no—
36#123—24no—
37C22⋊Q8—24no—
38C22⋊Q8—28no—
39C22⋊Q8—22no—
40C22⋊Q8—24no—
41#121—24no—
42#121—22no—
43#120—22no—
44C4×Q8—24no—
45C4×D4—28no—
46C4×D4—24no—
47#117—22no—
48#117C2×C412yes—
49#101—28no—
50#101—44no—
51C22⋊C8—44no—
52C24C22⋊C411yescharacteristic
53#41—44no—
54#41—44no—
55#41—48no—
56#41—44no—
57C2×Q8—22no—
58C2×Q8—24no—
59C2×Q8—24no—
60C2×Q8—22no—
61C2×Q8—24no—
62C2×Q8—42no—
63C2×Q8—42no—
64C2×Q8—416no—
65C2×Q8—48no—
66C2×Q8C22⋊C412yes—
67C2×Q8C2×D414yes—
68C2×D4—24no—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4C22⋊C412yes—
72C22×C4—24no—
73C22×C4—24no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—24no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C412yes—
81C22×C4C22×C411yescharacteristic
82C22×C4C2×D412yes—
83M16—44no—
84M16—44no—
85C2×C8—44no—
86C4⋊C4—24no—
87C4⋊C4—24no—
88C4⋊C4—22no—
89C4⋊C4—28no—
90C22⋊C4—24no—
91C22⋊C4—24no—
92C22⋊C4—22no—
93C22⋊C4—28no—
94C42—24no—
95C42C22⋊C414yes—
96C23—21no—
97C23—22no—
98C23—22no—
99C23—42no—
100C23C2×C22⋊C411yescharacteristic
101Q8—28no—
102Q8—44no—
103Q8—48no—
104Q8—42no—
105Q8—44no—
106D4—42no—
107D4—44no—
108D4—48no—
109D4—44no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—24no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—48no—
120C2×C4—44no—
121C2×C4—48no—
122C2×C4—44no—
123C2×C4#9914yes—
124C2×C4C2×C22⋊C412yes—
125C2×C4#12014yes—
126C8—84no—
127C22—22no—
128C22—24no—
129C22—41no—
130C22—41no—
131C22—42no—
132C22—44no—
133C22—41no—
134C22—42no—
135C22#37911yescharacteristic
136C22C2×#9912yes—
137C4—24no—
138C4—24no—
139C4—22no—
140C4—44no—
141C4—41no—
142C4—42no—
143C2—21no—
144C2—22no—
145C2—42no—
146C2—82no—
147C2#178111yescharacteristic
subgroupG/H
center, socleC2#1781
commutatorC22×C4C22×C4
Frattini, cosocleC22×Q8C23
prime radical#3486C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2133C2semidirect
3. #1784C2semidirect

Maximal subgroups

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

subgroupindex
#34862
#21332
#17842