Finite Group Databasebeta

← #12315

Subgroups — #12315

Order 256.

181 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 39 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#3291C211yesmaximal, characteristic
2#2214C212yesmaximal, complemented
3#1722C211yesmaximal, characteristic, complemented
4#1359C211yesmaximal, characteristic, complemented
5#1336C212yesmaximal, complemented
6#577—22no—
7#510—22no—
8#500—21no—
9#494C2211yescharacteristic, complemented
10#492C2211yescharacteristic
11#486—21no—
12#471—22no—
13#456—21no—
14#453—21no—
15#443C2212yes—
16#364C2212yescomplemented
17#362—22no—
18#322C2211yescharacteristic, complemented
192−1+4—21no—
20H(5,𝔽2)—21no—
21#137—22no—
22#137—42no—
23AGL(1,ℤ/8)—22no—
24AGL(1,ℤ/8)—42no—
25#135—42no—
26#135D412yes—
27C2×Q16—22no—
28C2×SD16—22no—
29C2×SD16—42no—
30#131—22no—
31C4⋊Q8—21no—
32C4⋊Q8D411yescharacteristic
33#127D411yescharacteristic
34Q32—42no—
35SD32—42no—
36M32—22no—
37#108D412yes—
38#107—42no—
39#104—22no—
40#102—42no—
41#101—21no—
42#100—21no—
43C4×C8—21no—
44C4×C8C2311yescharacteristic
45#41—22no—
46#41—42no—
47#41—42no—
48#41—81no—
49C2×Q8—21no—
50C2×Q8—42no—
51C2×Q8—41no—
52C2×D4—21no—
53C2×D4—41no—
54C2×D4—42no—
55C2×D4—81no—
56Q16—42no—
57Q16—42no—
58Q16—42no—
59Q16D812yes—
60SD16—22no—
61SD16—42no—
62SD16—42no—
63SD16—42no—
64SD16—42no—
65D8—42no—
66D8D812yes—
67M16—22no—
68M16—42no—
69C2×C8—22no—
70C2×C8—42no—
71C2×C8C2×D412yes—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—41no—
75C42C2×D411yescharacteristic
76C16—42no—
77C23—41no—
78C23—81no—
79Q8—22no—
80Q8—42no—
81Q8—81no—
82Q8—82no—
83D4—22no—
84D4—42no—
85D4—42no—
86D4—82no—
87D4—82no—
88D4—81no—
89C2×C4—21no—
90C2×C4—42no—
91C2×C4—41no—
92C2×C4—81no—
93C2×C4—82no—
94C2×C4—81no—
95C2×C4#12011yescharacteristic
96C8—42no—
97C8—42no—
98C8C2×D812yes—
99C8AGL(1,ℤ/8)12yes—
100C22—42no—
101C22—81no—
102C22—161no—
103C22—321no—
104C22#45311yescharacteristic
105C4—22no—
106C4—42no—
107C4—81no—
108C4—161no—
109C4#44712yes—
110C2—21no—
111C2—82no—
112C2—161no—
113C2#161911yescharacteristic
subgroupG/H
center, socleC2#1619
commutator, Frattini, cosocleC4×C8C23
prime radical#3291C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2214C2semidirect
3. #1722C2semidirect
4. #1359C2semidirect
5. #1336C2semidirect
9. #494C22semidirect
16. #364C22semidirect
18. #322C22semidirect

Maximal subgroups

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

subgroupindex
#32912
#22142
#17222
#13592
#13362