Finite Group Databasebeta

← #18294

Subgroups — #18294

Order 256.

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

normal subgroups only

This group has 501 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
1C2×#544C211yesmaximal, characteristic, complemented
2#3187C212yesmaximal
3#2618C211yesmaximal, characteristic
4#1611C214yesmaximal, complemented
5#1599C214yesmaximal, complemented
6#1520C212yesmaximal, complemented
7C2×#324C211yesmaximal, characteristic, complemented
8#544—24no—
9#544—22no—
10#544C2214yes—
11C2×C4⋊Q8—22no—
12C2×#125—22no—
13C2×#123—21no—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8C2212yes—
16C2×#121C2211yescharacteristic
17C2×#117—21no—
18C22×C4⋊C4C2211yescharacteristic
19#483C2214yes—
20#481—22no—
21#465—24no—
22#450C2214yes—
23#448—22no—
24C2×C22⋊C8C2212yes—
25#398C2212yes—
26#396—22no—
27#394C2212yes—
28#386—21no—
29#380C2211yescharacteristic
30#328C2218yes—
31#324C2214yes—
32C22×Q8—21no—
33C22×Q8D412yes—
34C22×D4—21no—
35C22×D4C2311yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4C2311yescharacteristic
39C2×SD16—24no—
40C22×C8—22no—
41C4⋊Q8—28no—
42C4⋊Q8—28no—
43#125—28no—
44#125—28no—
45#123—44no—
46C22⋊Q8—24no—
47C22⋊Q8—24no—
48C22⋊Q8—22no—
49C22⋊Q8—22no—
50C22⋊Q8D414yes—
51C22⋊Q8C2314yes—
52#121—22no—
53#121—22no—
54#117—24no—
55#117—24no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4D412yes—
66C2×C4⋊C4D412yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—22no—
75#106—24no—
76#103—24no—
77#102—24no—
78C22⋊C8—22no—
79C22⋊C8C2314yes—
80#95—22no—
81#95C2312yes—
82C24—21no—
83C24C2×D411yescharacteristic
84C2×Q8—24no—
85C2×Q8—22no—
86C2×Q8—82no—
87C2×Q8—82no—
88C2×Q8C2×D414yes—
89C2×D4—22no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—42no—
93C2×D4—42no—
94C2×D4—81no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—81no—
105C22×C4—81no—
106C22×C4SD1614yescomplemented
107C22×C4C2×D412yes—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2411yescharacteristic
113SD16—84no—
114C2×C8—24no—
115C2×C8—82no—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—28no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—44no—
127C4⋊C4—81no—
128C4⋊C4C2×D414yes—
129C22⋊C4—24no—
130C22⋊C4—24no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—44no—
134C22⋊C4—82no—
135C42—28no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—42no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23#12011yescharacteristic
144C23#12011yescharacteristic
145C23C2×SD1612yes—
146C23#13712yes—
147C23C22×D411yescharacteristic
148Q8—44no—
149Q8—82no—
150Q8—84no—
151D4—44no—
152D4—82no—
153D4—82no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—81no—
176C2×C4#12012yes—
177C2×C4C2×SD1614yes—
178C2×C4C22×D412yes—
179C8—44no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—42no—
187C22—42no—
188C22—82no—
189C22—82no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22#44812yes—
194C22#44812yes—
195C22C2×#12011yescharacteristic
196C22C22×SD1611yescharacteristic
197C22C2×#13711yescharacteristic
198C4—24no—
199C4—22no—
200C4—22no—
201C4—44no—
202C4—42no—
203C4—44no—
204C4—82no—
205C4—81no—
206C2—21no—
207C2—21no—
208C2—22no—
209C2—22no—
210C2—161no—
211C2C2×#44811yescharacteristic
212C2#302011yescharacteristic
213C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
1. C2×#544C2semidirect
4. #1611C2semidirect
5. #1599C2semidirect
6. #1520C2semidirect
7. C2×#324C2semidirect
106. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#31872
#26182
#16112
#15992
#15202
C2×#3242