Finite Group Databasebeta

← #18187

Subgroups — #18187

Order 256.

583 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 117 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#3497C211yesmaximal, characteristic
2#3112C212yesmaximal, complemented
3#2399C211yesmaximal, characteristic, complemented
4#1610C214yesmaximal, complemented
5#1604C214yesmaximal, complemented
6#1505C212yesmaximal, complemented
7#1467C211yesmaximal, characteristic, complemented
8C2×2−1+4—21no—
9#550—22no—
10D4×Q8—22no—
11#548—24no—
12#537—21no—
13#537C2214yes—
14#532—22no—
15#532C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18#480C2214yes—
19#477—22no—
20#460—22no—
21#448C2214yes—
22#440—24no—
23#432C2212yes—
24#398—21no—
25#398C2212yes—
26#395—21no—
27#395C2212yes—
28C4×C4⋊C4—22no—
29#376C2211yescharacteristic
30#327C2218yes—
31#324C2214yes—
322−1+4—48no—
33C2×#41—24no—
34C2×#41—24no—
35C2×#41—22no—
36C22×Q8—24no—
37C22×Q8D411yescharacteristic
38C22×Q8C2311yescharacteristic
39C2×SD16—24no—
40C4⋊Q8—22no—
41C4⋊Q8D412yes—
42#127—22no—
43#127D412yes—
44#124—24no—
45#124—22no—
46C22⋊Q8—24no—
47C22⋊Q8—28no—
48#121—24no—
49#121—28no—
50#121—22no—
51#121—24no—
52#121C2314yes—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×D4—24no—
58C4×D4—24no—
59C4×D4—22no—
60C4×D4D414yes—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42D412yes—
70C2×C42C2311yescharacteristic
71#107—24no—
72C4⋊C8—22no—
73#103—24no—
74#102—24no—
75C22⋊C8C2314yes—
76#97—22no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95C2312yes—
81#41—48no—
82#41—48no—
83#41—416no—
84#41—48no—
85C2×Q8—24no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—24no—
89C2×Q8—22no—
90C2×Q8—42no—
91C2×Q8—416no—
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4C2×D414yes—
98C22×C4—28no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—24no—
102C22×C4—21no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2411yescharacteristic
108SD16—48no—
109C2×C8—24no—
110C4⋊C4—24no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—24no—
114C4⋊C4—22no—
115C4⋊C4—44no—
116C4⋊C4—42no—
117C4⋊C4—44no—
118C4⋊C4C2×D414yes—
119C22⋊C4—28no—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C42—24no—
123C42—22no—
124C42—22no—
125C42—41no—
126C42—41no—
127C42C2×D412yes—
128C42C2×D412yes—
129C23—24no—
130C23C22×D411yescharacteristic
131Q8—28no—
132Q8—24no—
133Q8—42no—
134Q8—44no—
135Q8—48no—
136D4—44no—
137D4—44no—
138D4—48no—
139D4—48no—
140D4—44no—
141D4—84no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—41no—
151C2×C4—48no—
152C2×C4—416no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4#12011yescharacteristic
161C2×C4#12012yes—
162C2×C4#12011yescharacteristic
163C2×C4AGL(1,ℤ/8)14yes—
164C2×C4C22×D412yes—
165C8—44no—
166C22—21no—
167C22—21no—
168C22—44no—
169C22—44no—
170C22—44no—
171C22C2×#12011yescharacteristic
172C22C2×AGL(1,ℤ/8)12yes—
173C4—22no—
174C4—24no—
175C4—21no—
176C4—22no—
177C4—44no—
178C4—44no—
179C4—42no—
180C4—42no—
181C4—81no—
182C4#45612yes—
183C2—22no—
184C2—84no—
185C2#300311yescharacteristic
186C2C2×#45611yescharacteristic
187C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3497C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3112C2semidirect
3. #2399C2semidirect
4. #1610C2semidirect
5. #1604C2semidirect
6. #1505C2semidirect
7. #1467C2semidirect

Maximal subgroups

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

subgroupindex
#34972
#31122
#23992
#16102
#16042
#15052
#14672