Finite Group Databasebeta

← #18219

Subgroups — #18219

Order 256.

577 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 577 conjugacy classes of subgroups — too many to draw. Only its 127 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#3103C212yesmaximal, complemented
3#2581C211yesmaximal, characteristic, complemented
4#1616C214yesmaximal, complemented
5#1604C214yesmaximal, complemented
6#1517C212yesmaximal, complemented
7#1461C211yesmaximal, 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#481C2214yes—
19#475—22no—
20#460—22no—
21#448C2214yes—
22C4×SD16—24no—
23#431C2212yes—
24#400C2212yes—
25#395C2212yes—
26#391—21no—
27#389—21no—
28#387—22no—
29#382C2211yescharacteristic
30#327C2218yes—
31#324C2214yes—
322−1+4—44no—
332−1+4—44no—
34C2×#41—24no—
35C2×#41—24no—
36C2×#41—22no—
37C22×Q8—24no—
38C22×Q8D411yescharacteristic
39C22×Q8C2311yescharacteristic
40C2×SD16—24no—
41C4⋊Q8—22no—
42C4⋊Q8D412yes—
43#127—22no—
44#127D412yes—
45#124—24no—
46#124—22no—
47C22⋊Q8—24no—
48C22⋊Q8—28no—
49#121—24no—
50#121—28no—
51#121—22no—
52#121—24no—
53#121C2314yes—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×D4—24no—
59C4×D4—24no—
60C4×D4—22no—
61C4×D4D414yes—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C42—21no—
69C2×C42D412yes—
70C2×C42C2311yescharacteristic
71#106—24no—
72C4⋊C8—22no—
73#103—24no—
74#102—24no—
75C22⋊C8C2314yes—
76C4×C8—22no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95—22no—
81#95C2312yes—
82#41—48no—
83#41—416no—
84#41—48no—
85#41—44no—
86#41—44no—
87C2×Q8—24no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—42no—
93C2×Q8—416no—
94C2×Q8SD1614yescomplemented
95C2×D4—24no—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4C2×D414yes—
101C22×C4—28no—
102C22×C4—21no—
103C22×C4—24no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111SD16—48no—
112C2×C8—24no—
113C4⋊C4—24no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—44no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—42no—
122C4⋊C4C2×D414yes—
123C22⋊C4—28no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C42—22no—
127C42—22no—
128C42—42no—
129C42C2×D412yes—
130C42C2×D412yes—
131C23—24no—
132C23C22×D411yescharacteristic
133Q8—28no—
134Q8—24no—
135Q8—42no—
136Q8—48no—
137Q8—44no—
138D4—44no—
139D4—44no—
140D4—48no—
141D4—44no—
142D4—44no—
143D4—44no—
144D4—84no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—416no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—48no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4#12011yescharacteristic
163C2×C4#12011yescharacteristic
164C2×C4#12012yes—
165C2×C4C2×SD1614yes—
166C2×C4C2×SD1612yes—
167C2×C4AGL(1,ℤ/8)12yes—
168C2×C4C22×D412yes—
169C8—44no—
170C22—21no—
171C22—21no—
172C22—44no—
173C22—44no—
174C22—44no—
175C22C2×#12011yescharacteristic
176C22C22×SD1611yescharacteristic
177C22C2×AGL(1,ℤ/8)11yescharacteristic
178C4—24no—
179C4—21no—
180C4—21no—
181C4—22no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4—81no—
187C4#45014yes—
188C2—21no—
189C2—21no—
190C2—84no—
191C2C2×#45011yescharacteristic
192C2#302011yescharacteristic
193C2#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. #3103C2semidirect
3. #2581C2semidirect
4. #1616C2semidirect
5. #1604C2semidirect
6. #1517C2semidirect
7. #1461C2semidirect
94. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#34972
#31032
#25812
#16162
#16042
#15172
#14612