Finite Group Databasebeta

← #12170

Subgroups — #12170

Order 256.

178 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 35 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#3259C211yesmaximal, characteristic, complemented
2#2190C212yesmaximal, complemented
3#2053C211yesmaximal, characteristic, complemented
4#1339C212yesmaximal, complemented
5#1328C211yesmaximal, characteristic, complemented
6#575C2211yescharacteristic
7#540—21no—
8#509—22no—
9#478—22no—
10#469C2212yes—
11#466—21no—
12#459—22no—
13#424C2211yescharacteristic
14#416—21no—
15#410—21no—
16#360—22no—
17#349C2212yes—
18H(3,ℤ/4)C2211yescharacteristic
19C2×#41—21no—
20#137—41no—
21AGL(1,ℤ/8)—41no—
22#135—22no—
23C2×D8—21no—
24C2×D8D412yes—
25C2×M16C2311yescharacteristic
26C22×C8D411yescharacteristic
27#127—21no—
28#125—21no—
29#123—42no—
30#121—22no—
31#121—42no—
32C4×D4—22no—
33#117—21no—
34#117D411yescharacteristic
35SD32—42no—
36D16—42no—
37M32—22no—
38#107—21no—
39#106D412yes—
40C4⋊C8—22no—
41#103—41no—
42#102—22no—
43#102—22no—
44#102—41no—
45#99—42no—
46#41—41no—
47#41—41no—
48#41—41no—
49#41—41no—
50C2×Q8—21no—
51C2×D4—21no—
52C2×D4—22no—
53C2×D4—42no—
54C2×D4—41no—
55C2×D4—42no—
56C22×C4—41no—
57C22×C4—42no—
58C22×C4C2×D411yescharacteristic
59Q16—22no—
60SD16—42no—
61D8—22no—
62D8—42no—
63D8—81no—
64M16—41no—
65C2×C8—21no—
66C2×C8—41no—
67C2×C8C2×D412yes—
68C4⋊C4—21no—
69C4⋊C4—22no—
70C4⋊C4—42no—
71C4⋊C4—41no—
72C22⋊C4—42no—
73C22⋊C4—42no—
74C22⋊C4—41no—
75C22⋊C4D812yes—
76C42—21no—
77C42—41no—
78C16—42no—
79C23—41no—
80C23—42no—
81C23C2×D811yescharacteristic
82Q8—41no—
83Q8—41no—
84D4—41no—
85D4—42no—
86D4—41no—
87D4—42no—
88D4—81no—
89D4—81no—
90D4—162no—
91D4—162no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—22no—
95C2×C4—41no—
96C2×C4—42no—
97C2×C4—81no—
98C2×C4—81no—
99C2×C4—81no—
100C2×C4—162no—
101C2×C4#12011yescharacteristic
102C2×C4AGL(1,ℤ/8)11yescharacteristic
103C8—22no—
104C8—41no—
105C22—21no—
106C22—41no—
107C22—82no—
108C22—82no—
109C22—81no—
110C22—82no—
111C22—161no—
112C22#44711yescharacteristic
113C4—21no—
114C4—42no—
115C4—82no—
116C4—81no—
117C4—161no—
118C4#45411yescharacteristic
119C4B(4,𝔽2)11yescharacteristic
120C2—21no—
121C2—41no—
122C2—162no—
123C2—161no—
124C2#159811yescharacteristic
subgroupG/H
center, socleC2#1598
commutator, Frattini, cosocleC2×M16C23
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3259C2semidirect
2. #2190C2semidirect
3. #2053C2semidirect
4. #1339C2semidirect
5. #1328C2semidirect

Maximal subgroups

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

subgroupindex
#32592
#21902
#20532
#13392
#13282