Finite Group Databasebeta

← #9430

Subgroups — #9430

Order 256.

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

normal subgroups only

This group has 222 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#3141C211yesmaximal, characteristic
2#2381C211yesmaximal, characteristic, complemented
3#1556C211yesmaximal, characteristic, complemented
4#1514C211yesmaximal, characteristic
5#1514C211yesmaximal, characteristic
6#1288C211yesmaximal, characteristic, complemented
7#1288C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#489—22no—
11#487C412yescomplemented
12#486C412yescomplemented
13#432C2211yescharacteristic
14#432C2211yescharacteristic
15#417—22no—
16C4×M16C2211yescharacteristic
17#391—22no—
18#387—22no—
19#387—22no—
20#383C2211yescharacteristic
21#383C2211yescharacteristic
22#332—22no—
23#327—22no—
24#327—22no—
25C2×#41—21no—
26C22×Q8—21no—
27C2×M16—21no—
28C4⋊Q8—21no—
29C4⋊Q8—21no—
30C4⋊Q8C2×C411yescharacteristic
31C4⋊Q8C2×C412yes—
32C4⋊Q8D412yes—
33#127C2×C411yescharacteristic
34#124—21no—
35#121—21no—
36#121—21no—
37C4×D4—21no—
38C4×D4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4D412yes—
42C2×C42—22no—
43C2×C42C2311yescharacteristic
44C4⋊C8—21no—
45C4⋊C8—21no—
46#103—24no—
47#102—24no—
48C22⋊C8—21no—
49C22⋊C8—21no—
50#97—21no—
51#97—21no—
52#97—21no—
53C4×C8C2×C412yes—
54#95—24no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95D412yes—
60#95D412yes—
61#41—44no—
62C2×Q8—22no—
63C2×Q8—22no—
64C2×Q8—21no—
65C2×Q8—81no—
66C2×Q8—81no—
67C2×D4—21no—
68C2×D4—21no—
69C2×D4—21no—
70C2×D4—41no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—41no—
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78M16—42no—
79C2×C8—22no—
80C2×C8—41no—
81C2×C8—41no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—82no—
88C4⋊C4C22⋊C414yes—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C42—82no—
92C42C22×C411yescharacteristic
93C42C2×D411yescharacteristic
94C42#4111yescharacteristic
95C42#4111yescharacteristic
96C23—41no—
97C23#12011yescharacteristic
98Q8—42no—
99Q8—82no—
100Q8—82no—
101D4—42no—
102D4—81no—
103D4—81no—
104D4—82no—
105C2×C4—22no—
106C2×C4—24no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—82no—
119C2×C4C2×C22⋊C411yescharacteristic
120C2×C4C4×D411yescharacteristic
121C2×C4C4×D411yescharacteristic
122C2×C4#12111yescharacteristic
123C2×C4#12111yescharacteristic
124C2×C4#12311yescharacteristic
125C8—42no—
126C8—81no—
127C8—81no—
128C22—21no—
129C22—21no—
130C22—81no—
131C22—81no—
132C22—81no—
133C22#38611yescharacteristic
134C22#45412yes—
135C4—21no—
136C4—21no—
137C4—21no—
138C4—21no—
139C4—21no—
140C4—21no—
141C4—44no—
142C4—82no—
143C4—82no—
144C4—81no—
145C2—22no—
146C2—161no—
147C2#188811yescharacteristic
148C2#190411yescharacteristic
149C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2381C2semidirect
3. #1556C2semidirect
6. #1288C2semidirect
7. #1288C2semidirect
11. #487C4semidirect
12. #486C4semidirect

Maximal subgroups

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

subgroupindex
#31412
#23812
#15562
#15142
#12882