Finite Group Databasebeta

← #13590

Subgroups — #13590

Order 256.

199 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 45 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#3152C211yesmaximal, characteristic
2C2×#356C212yesmaximal, complemented
3#1411C212yesmaximal, complemented
4#1410C212yesmaximal, complemented
5C2×#137—22no—
6#532C411yescharacteristic
7C2×C4⋊Q8C2211yescharacteristic
8#497—22no—
9#494C411yescharacteristic
10#494C2211yescharacteristic
11#492C411yescharacteristic
12#492C2211yescharacteristic
13C2×#101—22no—
14C4×M16C411yescharacteristic
15#356—22no—
16#356C2214yes—
17C2×#41—21no—
18C22×Q8D412yes—
19#137—44no—
20#137—44no—
21C2×Q16—22no—
22C2×Q16—22no—
23C2×SD16—22no—
24C2×SD16—22no—
25C2×M16—21no—
26C2×M16—42no—
27C4⋊Q8—41no—
28C4⋊Q8C2×C411yescharacteristic
29C4⋊Q8C2×C411yescharacteristic
30C4⋊Q8D412yes—
31C4⋊Q8C2311yescharacteristic
32#127C2×C411yescharacteristic
33#124—21no—
34#121—41no—
35C4×D4—41no—
36C2×C4⋊C4—41no—
37C2×C42C2×C411yescharacteristic
38#101—24no—
39#101—42no—
40#97—21no—
41C4×C8C2×C411yescharacteristic
42C4×C8C2×C411yescharacteristic
43#41—42no—
44#41—42no—
45C2×Q8—22no—
46C2×Q8—21no—
47C2×Q8—42no—
48C2×Q8—42no—
49C2×Q8C22⋊C412yes—
50C2×Q8C2×D412yes—
51C2×D4—21no—
52C2×D4—41no—
53C2×D4—41no—
54C22×C4—21no—
55C22×C4—22no—
56C22×C4—41no—
57C22×C4C22⋊C411yescharacteristic
58Q16—44no—
59Q16—44no—
60SD16—44no—
61SD16—44no—
62M16—24no—
63M16—44no—
64M16—82no—
65C2×C8—21no—
66C2×C8—21no—
67C2×C8—82no—
68C4⋊C4—41no—
69C4⋊C4—42no—
70C4⋊C4—41no—
71C4⋊C4—41no—
72C22⋊C4—42no—
73C42—21no—
74C42C22⋊C411yescharacteristic
75C42C22×C411yescharacteristic
76C23—41no—
77C23#9911yescharacteristic
78Q8—42no—
79Q8—42no—
80Q8—42no—
81Q8—82no—
82D4—42no—
83D4—82no—
84D4—82no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—42no—
91C2×C4—42no—
92C2×C4—42no—
93C2×C4—41no—
94C2×C4—81no—
95C2×C4—81no—
96C2×C4#9911yescharacteristic
97C2×C4C2×C22⋊C411yescharacteristic
98C8—22no—
99C8—22no—
100C8—84no—
101C22—41no—
102C22—82no—
103C22—81no—
104C22#35312yes—
105C22C2×#9911yescharacteristic
106C4—22no—
107C4—21no—
108C4—21no—
109C4—41no—
110C4—42no—
111C4—42no—
112C4—81no—
113C2—21no—
114C2—22no—
115C2—161no—
116C2C2×#35311yescharacteristic
subgroupG/H
center, socleC2C2×#353
commutatorC42C22×C4
Frattini, cosocleC4⋊Q8C23
prime radical#532C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G
LECSG ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#356C2semidirect
3. #1411C2semidirect
4. #1410C2semidirect

Maximal subgroups

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

subgroupindex
#31522
C2×#3562
#14112
#14102