Finite Group Databasebeta

← #9282

Subgroups — #9282

Order 256.

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

normal subgroups only

This group has 208 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#3210C211yesmaximal, characteristic
2#2985C211yesmaximal, characteristic, complemented
3#1506C211yesmaximal, characteristic, complemented
4#1505C211yesmaximal, characteristic
5#1496C211yesmaximal, characteristic
6#1287C211yesmaximal, characteristic, complemented
7#1274C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#494C411yescharacteristic, complemented
11#492C411yescharacteristic, complemented
12#487C411yescharacteristic, complemented
13#486C411yescharacteristic, complemented
14#464—22no—
15#452—22no—
16#446—22no—
17#446—21no—
18C8×Q8—21no—
19#433—22no—
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#431C2211yescharacteristic
24#376C2211yescharacteristic
25#328—22no—
26#327—22no—
27D4⋊C8—21no—
28D4⋊C8—21no—
29C2×#41—21no—
30C2×Q16—22no—
31C2×SD16—22no—
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C411yescharacteristic
34C4⋊Q8C2×C411yescharacteristic
35#127C2×C411yescharacteristic
36#125—21no—
37#124—21no—
38C22⋊Q8—21no—
39C22⋊Q8D412yes—
40#121—21no—
41#121—21no—
42C4×Q8—21no—
43C4×Q8D412yes—
44C4×D4—21no—
45C4×D4—21no—
46#117—21no—
47C2×C42—21no—
48C2×C42C2311yescharacteristic
49C4⋊C8—22no—
50C4⋊C8—22no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55#103—22no—
56#102—22no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60C22⋊C8D412yes—
61#97—22no—
62#97—21no—
63#97—21no—
64C4×C8—22no—
65C4×C8C2×C411yescharacteristic
66C4×C8C2×C411yescharacteristic
67C4×C8D411yescharacteristic
68C4×C8D411yescharacteristic
69#95—21no—
70#95—21no—
71#41—82no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8C22⋊C412yes—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4#4111yescharacteristic
84Q16—82no—
85SD16—82no—
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—22no—
90C2×C8—21no—
91C2×C8—41no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4C22⋊C412yes—
100C22⋊C4—22no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C42—22no—
104C42—22no—
105C42—41no—
106C42C22×C411yescharacteristic
107C42C2×D411yescharacteristic
108C42C2×D411yescharacteristic
109C42#4111yescharacteristic
110C23—41no—
111C23#12111yescharacteristic
112Q8—42no—
113Q8—42no—
114Q8—41no—
115Q8—41no—
116D4—42no—
117D4—41no—
118D4—41no—
119D4—81no—
120D4—82no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—82no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4C2×C22⋊C411yescharacteristic
133C2×C4C4×D411yescharacteristic
134C2×C4C4×D411yescharacteristic
135C2×C4#12011yescharacteristic
136C2×C4#12111yescharacteristic
137C2×C4#12311yescharacteristic
138C8—41no—
139C8—42no—
140C8—41no—
141C8—41no—
142C8—41no—
143C8—81no—
144C22—21no—
145C22—21no—
146C22—21no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22#38611yescharacteristic
151C4—21no—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—82no—
159C4—81no—
160C4#44311yescharacteristic
161C4#44411yescharacteristic
162C2—41no—
163C2—161no—
164C2#187611yescharacteristic
165C2#190511yescharacteristic
166C2#191011yescharacteristic
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. #2985C2semidirect
3. #1506C2semidirect
6. #1287C2semidirect
7. #1274C2semidirect
10. #494C4semidirect
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
#32102
#29852
#15062
#15052
#14962
#12872
#12742