Finite Group Databasebeta

← #18138

Subgroups — #18138

Order 256.

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

normal subgroups only

This group has 527 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×#543C211yesmaximal, characteristic
2#3095C212yesmaximal, complemented
3#2482C211yesmaximal, characteristic
4#1614C214yesmaximal, complemented
5#1608C214yesmaximal, complemented
6#1511C212yesmaximal, complemented
7#1462C211yesmaximal, characteristic
8#543—24no—
9#543—22no—
10#543C2214yes—
11C2×C22⋊Q8—22no—
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8—22no—
14C2×C22⋊Q8C2212yes—
15C2×#117C2211yescharacteristic
16C22×C22⋊C4—21no—
17C22×C22⋊C4C2211yescharacteristic
18#485—24no—
19#484C2214yescomplemented
20#483C2214yescomplemented
21#428—22no—
22#419—22no—
23#407C2212yes—
24#399C2212yes—
25#397C2212yes—
26#388—22no—
27#385—21no—
28#381C2211yescharacteristic
29#379—21no—
30#328C2218yescomplemented
31#323C2214yes—
32C25D411yescharacteristic
33C22×Q8—21no—
34C22×Q8D412yes—
35C23×C4—21no—
36C23×C4—21no—
37C23×C4D411yescharacteristic
38C23×C4C2311yescharacteristic
39C2×M16—22no—
40C22⋊Q8—28no—
41C22⋊Q8—28no—
42C22⋊Q8—24no—
43C22⋊Q8—28no—
44C22⋊Q8—24no—
45C22⋊Q8—22no—
46C22⋊Q8—22no—
47C22⋊Q8—28no—
48C22⋊Q8D414yes—
49C22⋊Q8C2314yes—
50#117—22no—
51#117—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4D412yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C22⋊C4—22no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4D412yes—
70C2×C22⋊C4C2312yes—
71C2×C42—21no—
72#107—24no—
73#106—24no—
74#103—24no—
75#103—24no—
76C22⋊C8—22no—
77C22⋊C8C2314yes—
78#95—21no—
79#95—22no—
80#95C2312yes—
81C24—21no—
82C24—21no—
83C24—24no—
84C24—42no—
85C24—81no—
86C24C2×D411yescharacteristic
87C24C2×D412yes—
88C2×Q8—24no—
89C2×Q8—22no—
90C2×Q8—44no—
91C2×Q8—82no—
92C2×Q8C2×D414yes—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—24no—
97C22×C4—24no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—42no—
103C22×C4—81no—
104C22×C4—82no—
105C22×C4—82no—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4C2411yescharacteristic
111M16—44no—
112C2×C8—24no—
113C4⋊C4—24no—
114C4⋊C4—28no—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—82no—
121C4⋊C4C2×D414yes—
122C22⋊C4—28no—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—42no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—44no—
131C22⋊C4—82no—
132C42—42no—
133C23—24no—
134C23—24no—
135C23—22no—
136C23—22no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23—42no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—81no—
145C23—81no—
146C23—88no—
147C23#12011yescharacteristic
148C23#12011yescharacteristic
149C23#13714yes—
150C23C22×D411yescharacteristic
151Q8—44no—
152Q8—82no—
153Q8—84no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—81no—
173C2×C4—84no—
174C2×C4—84no—
175C2×C4#12012yes—
176C2×C4C22×D412yes—
177C8—44no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—24no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—44no—
188C22—48no—
189C22—42no—
190C22—42no—
191C22—82no—
192C22—84no—
193C22#45812yes—
194C22C2×#12011yescharacteristic
195C22C2×#13712yes—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—44no—
200C4—44no—
201C4—82no—
202C4—81no—
203C4—81no—
204C2—22no—
205C2—22no—
206C2—41no—
207C2—44no—
208C2#300411yescharacteristic
209C2#301911yescharacteristic
210C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
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. #3095C2semidirect
4. #1614C2semidirect
5. #1608C2semidirect
6. #1511C2semidirect
19. #484C22semidirect
20. #483C22semidirect
30. #328C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5432
#30952
#24822
#16142
#16082
#15112
#14622