Finite Group Databasebeta

← #18147

Subgroups — #18147

Order 256.

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

normal subgroups only

This group has 605 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
1C2×#537C211yesmaximal, characteristic
2#3094C212yesmaximal, complemented
3#2484C211yesmaximal, characteristic, complemented
4#1616C214yesmaximal, complemented
5#1610C214yesmaximal, complemented
6#1510C212yesmaximal, complemented
7#1463C211yesmaximal, characteristic, complemented
8C22×#41—21no—
9#537—24no—
10#537—22no—
11#537C2214yes—
12C2×C22⋊Q8—22no—
13C2×#121—22no—
14C2×#121—21no—
15C2×#121C2212yes—
16C2×#117C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#482—24no—
19#481C2214yescomplemented
20#480C2214yescomplemented
21#428—22no—
22#418—22no—
23#407C2212yes—
24#400C2212yescomplemented
25#398C2212yescomplemented
26#387—22no—
27#385—21no—
28#381—21no—
29#381C2211yescharacteristic
30#327C2218yescomplemented
31#324C2214yescomplemented
32C2×#41—24no—
33C2×#41—24no—
34C2×#41—48no—
35C22×Q8—21no—
36C22×D4—21no—
37C22×D4—22no—
38C22×D4—21no—
39C22×D4D412yes—
40C23×C4—22no—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C2×M16—22no—
45C22⋊Q8—28no—
46C22⋊Q8—28no—
47#121—24no—
48#121—28no—
49#121—28no—
50#121—24no—
51#121—22no—
52#121—22no—
53#121D414yes—
54#121C2314yes—
55#117—22no—
56#117—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4D412yes—
62C2×C4⋊C4D412yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42—21no—
71#107—24no—
72#106—24no—
73#102—24no—
74#102—24no—
75C22⋊C8—22no—
76C22⋊C8C2314yes—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—22no—
81#95C2312yes—
82C24—22no—
83C24C2×D411yescharacteristic
84#41—416no—
85#41—48no—
86#41—48no—
87C2×Q8—24no—
88C2×Q8—44no—
89C2×D4—24no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4—22no—
94C2×D4—44no—
95C2×D4—44no—
96C2×D4—48no—
97C2×D4—82no—
98C2×D4C2×D414yes—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—24no—
102C22×C4—28no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—42no—
110C22×C4—48no—
111C22×C4—42no—
112C22×C4C2×D412yes—
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2411yescharacteristic
118M16—44no—
119C2×C8—24no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—44no—
127C4⋊C4C2×D414yes—
128C22⋊C4—28no—
129C22⋊C4—24no—
130C22⋊C4—24no—
131C22⋊C4—42no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C42—42no—
135C23—22no—
136C23—22no—
137C23—24no—
138C23—41no—
139C23—44no—
140C23—44no—
141C23—44no—
142C23#12011yescharacteristic
143C23#12011yescharacteristic
144C23C22×D411yescharacteristic
145Q8—48no—
146D4—44no—
147D4—48no—
148D4—48no—
149D4—48no—
150D4—82no—
151D4—84no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—416no—
163C2×C4—44no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—81no—
174C2×C4#12012yes—
175C2×C4AGL(1,ℤ/8)14yes—
176C2×C4C22×D412yes—
177C8—44no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—44no—
186C22—44no—
187C22—44no—
188C22—44no—
189C22—88no—
190C22#45812yes—
191C22C2×#12011yescharacteristic
192C22C2×AGL(1,ℤ/8)12yes—
193C4—24no—
194C4—22no—
195C4—22no—
196C4—44no—
197C4—42no—
198C4—82no—
199C4—81no—
200C4—81no—
201C2—22no—
202C2—22no—
203C2—41no—
204C2—84no—
205C2#300311yescharacteristic
206C2#302011yescharacteristic
207C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3094C2semidirect
3. #2484C2semidirect
4. #1616C2semidirect
5. #1610C2semidirect
6. #1510C2semidirect
7. #1463C2semidirect
19. #481C22semidirect
20. #480C22semidirect
24. #400C22semidirect
25. #398C22semidirect
30. #327C22semidirect
31. #324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
#30942
#24842
#16162
#16102
#15102
#14632