Finite Group Databasebeta

← #9291

Subgroups — #9291

Order 256.

247 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 69 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#3226C211yesmaximal, characteristic
2#2932C211yesmaximal, characteristic
3#1960C211yesmaximal, characteristic, complemented
4#1546C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic
6#1286C211yesmaximal, characteristic, complemented
7#1275C211yesmaximal, characteristic, complemented
8C2×#131—21no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#500C412yescomplemented
12#486C412yescomplemented
13#480—22no—
14#474—22no—
15#435—22no—
16#435—22no—
17#431C2211yescharacteristic
18#431C2211yescharacteristic
19#423C2211yescharacteristic
20C2×#103—21no—
21C2×#103—21no—
22#408—21no—
23C2×#97C2211yescharacteristic
24#382C2211yescharacteristic
25#332—22no—
26#327—22no—
27C2×#41—21no—
28C22×Q8—21no—
29#131—42no—
30#131—42no—
31C2×M16—22no—
32C2×M16—21no—
33C22×C8—22no—
34C22×C8D411yescharacteristic
35C22×C8D411yescharacteristic
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8C2×C412yes—
40C4⋊Q8C2×C411yescharacteristic
41#127C2×C411yescharacteristic
42#124—21no—
43#121—21no—
44#121D412yes—
45C4×D4—21no—
46C4×D4D412yes—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C42C2311yescharacteristic
51#107—24no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8D412yes—
56#103—42no—
57#103—42no—
58#102—24no—
59C22⋊C8—24no—
60C22⋊C8—21no—
61C22⋊C8—21no—
62#97—21no—
63#97—21no—
64C4×C8—21no—
65C4×C8C2×C412yes—
66#95—21no—
67#95—21no—
68#41—41no—
69#41—41no—
70#41—41no—
71#41—41no—
72C2×Q8—21no—
73C2×Q8—22no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×Q8—41no—
79C2×D4—22no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4C22⋊C412yes—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4#4111yescharacteristic
89M16—44no—
90M16—41no—
91M16—41no—
92C2×C8—21no—
93C2×C8—21no—
94C2×C8—22no—
95C2×C8—21no—
96C2×C8—22no—
97C2×C8—21no—
98C2×C8—21no—
99C2×C8—44no—
100C2×C8—41no—
101C2×C8D812yes—
102C4⋊C4—21no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4C22⋊C412yes—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42C22×C411yescharacteristic
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42#4111yescharacteristic
116C23—22no—
117C23#12111yescharacteristic
118Q8—41no—
119Q8—41no—
120Q8—42no—
121Q8—42no—
122Q8—81no—
123Q8—81no—
124D4—42no—
125D4—42no—
126D4—42no—
127D4—41no—
128D4—41no—
129D4—81no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4C2×C22⋊C411yescharacteristic
142C2×C4C4×D411yescharacteristic
143C2×C4C4×D411yescharacteristic
144C2×C4#12011yescharacteristic
145C2×C4#12111yescharacteristic
146C2×C4#12311yescharacteristic
147C2×C4C2×D811yescharacteristic
148C2×C4#13511yescharacteristic
149C2×C4AGL(1,ℤ/8)11yescharacteristic
150C2×C4AGL(1,ℤ/8)11yescharacteristic
151C8—22no—
152C8—42no—
153C8—42no—
154C8—41no—
155C8—81no—
156C22—21no—
157C22—21no—
158C22—42no—
159C22—42no—
160C22—42no—
161C22#38611yescharacteristic
162C22#45911yescharacteristic
163C22#46311yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—42no—
169C4—82no—
170C4—161no—
171C4C4×D811yescharacteristic
172C4#44211yescharacteristic
173C4#44711yescharacteristic
174C4#44911yescharacteristic
175C2—21no—
176C2—21no—
177C2—82no—
178C2#187911yescharacteristic
179C2#190211yescharacteristic
180C2#190911yescharacteristic
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. #1960C2semidirect
4. #1546C2semidirect
6. #1286C2semidirect
7. #1275C2semidirect
11. #500C4semidirect
12. #486C4semidirect

Maximal subgroups

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

subgroupindex
#32262
#29322
#19602
#15462
#15172
#12862
#12752