Finite Group Databasebeta

← #9301

Subgroups — #9301

Order 256.

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

normal subgroups only

This group has 246 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#3206C211yesmaximal, characteristic
2#2331C211yesmaximal, characteristic, complemented
3#1963C211yesmaximal, characteristic
4#1548C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic
6#1286C211yesmaximal, characteristic, complemented
7#1274C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#492C412yescomplemented
11#487C412yescomplemented
12#461—22no—
13#448—22no—
14#431C2211yescharacteristic
15#431C2211yescharacteristic
16#423C2211yescharacteristic
17C2×#104—21no—
18C2×#104—21no—
19#391—22no—
20#389—21no—
21#387—22no—
22#382C2211yescharacteristic
23C4×C4⋊C4—21no—
24#376C2211yescharacteristic
25#333—22no—
26#327—22no—
27C2×#41—21no—
28C22×Q8D412yes—
29C2×SD16—24no—
30C2×M16—21no—
31C4⋊Q8—21no—
32C4⋊Q8—21no—
33C4⋊Q8C2×C411yescharacteristic
34C4⋊Q8C2×C412yes—
35C4⋊Q8D412yes—
36#127C2×C411yescharacteristic
37#124—21no—
38#121—21no—
39#121—21no—
40C4×D4—21no—
41C4×D4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—22no—
46C2×C42—22no—
47C2×C42D411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42C2311yescharacteristic
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54#104—42no—
55#104—42no—
56#103—24no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C4×C8—21no—
60C4×C8C2×C412yes—
61#95—22no—
62#95—22no—
63#95—21no—
64#95—21no—
65#95—21no—
66#95D412yes—
67#41—41no—
68#41—41no—
69#41—41no—
70#41—41no—
71C2×Q8—22no—
72C2×Q8—21no—
73C2×Q8—82no—
74C2×Q8C22⋊C414yes—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—41no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—41no—
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87SD16—84no—
88M16—41no—
89M16—41no—
90C2×C8—22no—
91C2×C8—41no—
92C2×C8—41no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—82no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C42—21no—
105C42—21no—
106C42—21no—
107C42—82no—
108C42SD1612yescomplemented
109C42C22×C411yescharacteristic
110C42C2×D411yescharacteristic
111C42#4111yescharacteristic
112C42#4111yescharacteristic
113C23—41no—
114C23#12011yescharacteristic
115Q8—44no—
116Q8—41no—
117Q8—41no—
118Q8—82no—
119D4—42no—
120D4—41no—
121D4—41no—
122D4—81no—
123D4—81no—
124D4—81no—
125C2×C4—24no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4C2×C22⋊C411yescharacteristic
143C2×C4C4×D411yescharacteristic
144C2×C4C4×D411yescharacteristic
145C2×C4#12111yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12311yescharacteristic
148C2×C4C2×SD1611yescharacteristic
149C2×C4#13511yescharacteristic
150C2×C4AGL(1,ℤ/8)11yescharacteristic
151C2×C4#13711yescharacteristic
152C8—42no—
153C8—81no—
154C8—81no—
155C22—21no—
156C22—21no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22#38611yescharacteristic
161C22#44811yescharacteristic
162C22#44911yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—42no—
168C4—44no—
169C4—82no—
170C4—81no—
171C4—81no—
172C4C4×SD1611yescharacteristic
173C4#44011yescharacteristic
174C4#46011yescharacteristic
175C4#46411yescharacteristic
176C2—21no—
177C2—21no—
178C2—161no—
179C2#188011yescharacteristic
180C2#190511yescharacteristic
181C2#191111yescharacteristic
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. #2331C2semidirect
4. #1548C2semidirect
6. #1286C2semidirect
7. #1274C2semidirect
10. #492C4semidirect
11. #487C4semidirect
108. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#32062
#23312
#19632
#15482
#15172
#12862
#12742