Finite Group Databasebeta

← #12970

Subgroups — #12970

Order 256.

236 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 59 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#2379C211yesmaximal, characteristic
2C2×#355C211yesmaximal, characteristic
3#1888C211yesmaximal, characteristic
4#1857C212yesmaximal, complemented
5#1394C212yesmaximal, complemented
6C2×#124C2211yescharacteristic
7#519C2211yescharacteristic
8#428—22no—
9#418—21no—
10C2×#101C2211yescharacteristic
11C2×#101C2212yes—
12#391C2212yes—
13#388—22no—
14#386—21no—
15#385—22no—
16#382—21no—
17#355C414yescomplemented
18#339—22no—
19#339—21no—
20C22×Q8C2311yescharacteristic
21C22×D4D411yescharacteristic
22C23×C4—21no—
23C2×M16—22no—
24C2×M16—21no—
25#124—22no—
26#124C2×C412yes—
27#124Q812yes—
28C4×Q8—41no—
29C4×Q8—41no—
30#117—21no—
31#117—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4D412yes—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C42D411yescharacteristic
39C2×C42D412yes—
40#107—42no—
41#106—42no—
42#102—41no—
43#102—41no—
44#101—22no—
45#101—22no—
46#101C2×C414yescomplemented
47#95—22no—
48#95—22no—
49#95—22no—
50C24—21no—
51C2×Q8—41no—
52C2×Q8—41no—
53C2×Q8C22×C411yescharacteristic
54C2×Q8C2×Q811yescharacteristic
55C2×Q8#4112yes—
56C2×D4—21no—
57C2×D4—41no—
58C2×D4—41no—
59C2×D4C4⋊C412yes—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—82no—
64C22×C4—81no—
65C22×C4C2×D411yescharacteristic
66C22×C4C2×D412yes—
67M16—24no—
68M16—42no—
69M16—41no—
70M16—41no—
71C2×C8—42no—
72C2×C8—41no—
73C4⋊C4—21no—
74C4⋊C4—21no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—42no—
78C4⋊C4—81no—
79C4⋊C4—81no—
80C22⋊C4—24no—
81C22⋊C4—24no—
82C22⋊C4—41no—
83C22⋊C4—82no—
84C22⋊C4—81no—
85C42—21no—
86C42—42no—
87C42—42no—
88C42—41no—
89C42C4⋊C412yes—
90C23—22no—
91C23—42no—
92C23—42no—
93C23—41no—
94C23—41no—
95C23#12011yescharacteristic
96Q8—41no—
97Q8—41no—
98Q8—42no—
99D4—42no—
100D4—81no—
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—22no—
104C2×C4—21no—
105C2×C4—22no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—84no—
117C2×C4C2×C4⋊C411yescharacteristic
118C2×C4C4×D412yes—
119C2×C4C22⋊Q812yes—
120C2×C4#12311yescharacteristic
121C8—44no—
122C8—81no—
123C22—21no—
124C22—21no—
125C22—44no—
126C22—42no—
127C22—82no—
128C22—81no—
129C22—81no—
130C22#38511yescharacteristic
131C22B(4,𝔽2)11yescharacteristic
132C22#45811yescharacteristic
133C4—22no—
134C4—21no—
135C4—21no—
136C4—22no—
137C4—42no—
138C4—82no—
139C4—81no—
140C4—82no—
141C4—81no—
142C2—21no—
143C2—21no—
144C2—82no—
145C2#186711yescharacteristic
146C2#220211yescharacteristic
147C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×Q8C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC22×D4D4
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1857C2semidirect
5. #1394C2semidirect
17. #355C4semidirect
46. #101C2×C4semidirect

Maximal subgroups

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

subgroupindex
#23792
C2×#3552
#18882
#18572
#13942