Finite Group Databasebeta

← #9419

Subgroups — #9419

Order 256.

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

normal subgroups only

This group has 212 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#2980C211yesmaximal, characteristic
3#1558C211yesmaximal, characteristic, complemented
4#1556C211yesmaximal, characteristic, complemented
5#1536C211yesmaximal, characteristic
6#1291C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#489—22no—
11#487C412yes—
12#486C412yes—
13#435—22no—
14#435—22no—
15#432C2211yescharacteristic
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18#432C2211yescharacteristic
19#417—22no—
20#408—22no—
21C4×M16C2211yescharacteristic
22#333—22no—
23#332—22no—
24#329—22no—
25C2×#41D412yes—
26C22×Q8—21no—
27C2×M16—21no—
28C2×M16—22no—
29C22×C8—22no—
30C4⋊Q8—21no—
31C4⋊Q8—21no—
32C4⋊Q8—21no—
33C4⋊Q8C2×C412yes—
34C4⋊Q8C2×C411yescharacteristic
35#127C2×C411yescharacteristic
36#124D412yes—
37#121—21no—
38#121—21no—
39C4×D4—21no—
40C4×D4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C42C2311yescharacteristic
44C4⋊C8—21no—
45C4⋊C8—21no—
46C4⋊C8D412yes—
47C4⋊C8D412yes—
48#103—24no—
49#102—24no—
50C22⋊C8—24no—
51C22⋊C8—21no—
52C22⋊C8—21no—
53C22⋊C8—21no—
54C22⋊C8—21no—
55#97—21no—
56#97—21no—
57#97—21no—
58#97—21no—
59#97—21no—
60C4×C8C2×C412yes—
61#41—42no—
62#41—42no—
63C2×Q8—22no—
64C2×Q8—41no—
65C2×Q8—82no—
66C2×Q8C22⋊C412yes—
67C2×D4—22no—
68C2×D4—21no—
69C2×D4—21no—
70C2×D4C22⋊C412yes—
71C22×C4—22no—
72C22×C4—41no—
73C22×C4C2×D411yescharacteristic
74C22×C4#4111yescharacteristic
75C22×C4#4111yescharacteristic
76M16—42no—
77M16—82no—
78C2×C8—22no—
79C2×C8—22no—
80C2×C8—22no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—82no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C22⋊C4—22no—
91C22⋊C4—22no—
92C42C22×C411yescharacteristic
93C42C2×D411yescharacteristic
94C42C2×D411yescharacteristic
95C42C2×D411yescharacteristic
96C23—22no—
97C23#12311yescharacteristic
98Q8—42no—
99Q8—82no—
100Q8—82no—
101D4—42no—
102D4—81no—
103D4—81no—
104D4—82no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—82no—
113C2×C4C2×C22⋊C411yescharacteristic
114C2×C4C4×D411yescharacteristic
115C2×C4C4×D411yescharacteristic
116C2×C4#12011yescharacteristic
117C2×C4#12111yescharacteristic
118C2×C4#12111yescharacteristic
119C8—42no—
120C8—42no—
121C8—42no—
122C8—81no—
123C8—81no—
124C22—21no—
125C22—21no—
126C22—21no—
127C22—42no—
128C22—42no—
129C22—42no—
130C22#38611yescharacteristic
131C4—21no—
132C4—21no—
133C4—21no—
134C4—21no—
135C4—21no—
136C4—42no—
137C4—82no—
138C4#45512yes—
139C2—41no—
140C2—82no—
141C2#188511yescharacteristic
142C2#190211yescharacteristic
143C2#190311yescharacteristic
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
3. #1558C2semidirect
4. #1556C2semidirect
6. #1291C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#31412
#29802
#15582
#15562
#15362
#12912