Finite Group Databasebeta

← #9418

Subgroups — #9418

Order 256.

201 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 57 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#3182C211yesmaximal, characteristic
2#2987C211yesmaximal, characteristic, complemented
3#1536C211yesmaximal, characteristic
4#1515C211yesmaximal, characteristic, complemented
5#1514C211yesmaximal, characteristic
6#1291C211yesmaximal, characteristic, complemented
7#1288C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#500C411yescharacteristic, complemented
11#492C411yescharacteristic, complemented
12#487C411yescharacteristic, complemented
13#486C411yescharacteristic, complemented
14#468—22no—
15#446—22no—
16#446—22no—
17#440—22no—
18#433—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22C4×M16C2211yescharacteristic
23#383C2211yescharacteristic
24#329—22no—
25#328—22no—
26#327—22no—
27C2×#41—21no—
28C2×SD16—22no—
29C2×M16—21no—
30C4⋊Q8C2×C411yescharacteristic
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8C2×C411yescharacteristic
33#127C2×C411yescharacteristic
34#125—21no—
35#124—21no—
36C22⋊Q8—21no—
37C22⋊Q8D412yes—
38#121—21no—
39#121—21no—
40C4×Q8—21no—
41C4×Q8D412yes—
42C4×D4—21no—
43C4×D4—21no—
44#117—21no—
45C2×C42C2311yescharacteristic
46#107—22no—
47C4⋊C8—24no—
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51#103—22no—
52#102—22no—
53C22⋊C8—21no—
54C22⋊C8—21no—
55C22⋊C8D412yes—
56#97—22no—
57#97—21no—
58#97—21no—
59#97—21no—
60#97D412yes—
61C4×C8—22no—
62C4×C8C2×C411yescharacteristic
63C4×C8C2×C411yescharacteristic
64#95—21no—
65#95—21no—
66#95—21no—
67#41—82no—
68C2×Q8—21no—
69C2×Q8—21no—
70C2×Q8C22⋊C412yes—
71C2×D4—21no—
72C2×D4—21no—
73C2×D4—21no—
74C2×D4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4#4111yescharacteristic
80SD16—44no—
81M16—42no—
82C2×C8—21no—
83C2×C8—21no—
84C2×C8—22no—
85C2×C8—22no—
86C2×C8—41no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4C22⋊C412yes—
95C22⋊C4—22no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C42—22no—
99C42C22×C411yescharacteristic
100C42C2×D411yescharacteristic
101C42C2×D411yescharacteristic
102C42#4111yescharacteristic
103C23—41no—
104C23#12111yescharacteristic
105Q8—42no—
106Q8—81no—
107Q8—81no—
108D4—42no—
109D4—81no—
110D4—81no—
111D4—82no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—82no—
119C2×C4—81no—
120C2×C4—81no—
121C2×C4—81no—
122C2×C4C2×C22⋊C411yescharacteristic
123C2×C4C4×D411yescharacteristic
124C2×C4C4×D411yescharacteristic
125C2×C4#12011yescharacteristic
126C2×C4#12111yescharacteristic
127C2×C4#12311yescharacteristic
128C8—22no—
129C8—42no—
130C8—42no—
131C8—41no—
132C8—81no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—81no—
137C22—81no—
138C22—81no—
139C22#38611yescharacteristic
140C4—21no—
141C4—21no—
142C4—21no—
143C4—21no—
144C4—21no—
145C4—21no—
146C4—42no—
147C4—42no—
148C4—81no—
149C4—161no—
150C2—41no—
151C2—161no—
152C2#188411yescharacteristic
153C2#190311yescharacteristic
154C2#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. #2987C2semidirect
4. #1515C2semidirect
6. #1291C2semidirect
7. #1288C2semidirect
10. #500C4semidirect
11. #492C4semidirect
12. #487C4semidirect
13. #486C4semidirect

Maximal subgroups

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

subgroupindex
#31822
#29872
#15362
#15152
#15142
#12912
#12882