Finite Group Databasebeta

← #9307

Subgroups — #9307

Order 256.

232 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 67 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#3077C211yesmaximal, characteristic
2#2975C211yesmaximal, characteristic
3#1804C211yesmaximal, characteristic
4#1518C211yesmaximal, characteristic, complemented
5#1481C211yesmaximal, characteristic, complemented
6#1301C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#478—22no—
11#476C412yescomplemented
12#474C412yescomplemented
13#436—22no—
14#436—22no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#428—21no—
18#423C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20#418—21no—
21C2×#102—21no—
22#407—22no—
23#382C2211yescharacteristic
24#340—21no—
25#328—22no—
26Q8⋊C8—22no—
27C22×D4D411yescharacteristic
28C23×C4—21no—
29C2×M16—21no—
30C2×M16—24no—
31C22×C8—21no—
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C411yescharacteristic
34#125—21no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37C4×Q8—21no—
38C4×Q8—21no—
39C4×D4—21no—
40C4×D4—21no—
41C4×D4C2×C412yes—
42C4×D4D412yes—
43#117—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4D411yescharacteristic
46C2×C22⋊C4—21no—
47C2×C42C2311yescharacteristic
48#107—22no—
49#106—22no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8C2×C412yes—
56#102—22no—
57#102—22no—
58#102—42no—
59C22⋊C8—24no—
60C22⋊C8—21no—
61C22⋊C8D412yes—
62C4×C8—21no—
63C4×C8D412yes—
64#95—21no—
65#95—21no—
66C24—21no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×D4—21no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—21no—
73C2×D4—21no—
74C2×D4C22⋊C412yes—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—41no—
79C22×C4—82no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4#4111yescharacteristic
83M16—41no—
84M16—41no—
85M16—84no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4C22⋊C412yes—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C22⋊C4—41no—
105C42—41no—
106C42C22×C411yescharacteristic
107C42C2×D411yescharacteristic
108C42C2×D411yescharacteristic
109C42#4111yescharacteristic
110C23—22no—
111C23—41no—
112C23—41no—
113C23—81no—
114C23—81no—
115C23#12111yescharacteristic
116Q8—42no—
117Q8—81no—
118D4—22no—
119D4—22no—
120D4—41no—
121D4—41no—
122C2×C4—22no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—44no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—82no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4C2×C22⋊C411yescharacteristic
134C2×C4C4×D411yescharacteristic
135C2×C4C4×D411yescharacteristic
136C2×C4#12011yescharacteristic
137C2×C4#12111yescharacteristic
138C2×C4#12311yescharacteristic
139C2×C4AGL(1,ℤ/8)11yescharacteristic
140C2×C4#13711yescharacteristic
141C2×C4#13711yescharacteristic
142C2×C4#13711yescharacteristic
143C8—42no—
144C8—42no—
145C8—42no—
146C8—81no—
147C22—21no—
148C22—21no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22—82no—
153C22—82no—
154C22#38611yescharacteristic
155C22#46811yescharacteristic
156C22#47011yescharacteristic
157C4—21no—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—42no—
162C4—81no—
163C4—81no—
164C4—161no—
165C4#41711yescharacteristic
166C4#41911yescharacteristic
167C4#46811yescharacteristic
168C4#47011yescharacteristic
169C2—21no—
170C2—21no—
171C2—82no—
172C2#188111yescharacteristic
173C2#189411yescharacteristic
174C2#190411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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
4. #1518C2semidirect
5. #1481C2semidirect
6. #1301C2semidirect
7. #1286C2semidirect
11. #476C4semidirect
12. #474C4semidirect

Maximal subgroups

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

subgroupindex
#30772
#29752
#18042
#15182
#14812
#13012
#12862