Finite Group Databasebeta

← #23561

Subgroups — #23561

Order 256.

537 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 183 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
2#2911C212yesmaximal
3#2317C211yesmaximal, characteristic
4#2061C214yesmaximal, complemented
5#2060C214yesmaximal, complemented
6C2×#343C211yesmaximal, characteristic
7#1733C212yesmaximal
8C22×M16C2212yes—
9#544—24no—
10#544—22no—
11#544C2214yes—
12C2×C4⋊Q8—22no—
13C2×#125—22no—
14C2×C22⋊Q8—22no—
15C2×#117C2212yes—
16C2×#117C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19#428—22no—
20#428—22no—
21#428C2214yes—
22#428C2214yes—
23#427—24no—
24#389—22no—
25#385—22no—
26#384C2212yes—
27#382C2212yes—
28#343C2214yes—
29#339C2218yes—
30C22×Q8—22no—
31C23×C4D412yes—
32C23×C4C2311yescharacteristic
33C2×M16—22no—
34C2×M16—24no—
35C2×M16Q818yescomplemented
36C2×M16C2314yes—
37C22×C8—22no—
38C4⋊Q8—28no—
39C4⋊Q8—28no—
40#125—28no—
41#125—28no—
42C22⋊Q8—28no—
43C22⋊Q8—28no—
44#117—22no—
45#117—22no—
46#117—22no—
47#117—22no—
48#117—22no—
49#117D414yes—
50#117Q814yescomplemented
51#117C2314yes—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4D412yes—
59C2×C4⋊C4D412yes—
60C2×C4⋊C4D412yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C42—22no—
68C2×C42—21no—
69C2×C42C2312yes—
70#107—24no—
71#107—24no—
72#106—24no—
73#106—24no—
74#95—24no—
75C24C2×D411yescharacteristic
76C2×Q8—28no—
77C2×Q8—84no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—82no—
86C22×C4C2×D414yes—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D412yes—
89C22×C4C2×D412yes—
90C22×C4C2×Q814yes—
91C22×C4C2×Q812yes—
92C22×C4#4112yes—
93C22×C4C2411yescharacteristic
94M16—28no—
95M16—24no—
96M16C2×Q818yescomplemented
97C2×C8—28no—
98C2×C8—24no—
99C4⋊C4—28no—
100C4⋊C4—216no—
101C4⋊C4—24no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—84no—
109C4⋊C4C2×D414yes—
110C4⋊C4C2×D414yes—
111C22⋊C4—24no—
112C22⋊C4—24no—
113C22⋊C4—24no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C42—24no—
117C42—24no—
118C42—22no—
119C42—42no—
120C42C2×Q814yescomplemented
121C23—41no—
122C23—41no—
123C23#12011yescharacteristic
124C23#12011yescharacteristic
125C23C22⋊Q812yes—
126C23C4⋊Q812yes—
127C23C22×D411yescharacteristic
128Q8—88no—
129C2×C4—24no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—24no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—44no—
136C2×C4—44no—
137C2×C4—44no—
138C2×C4—44no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—82no—
144C2×C4#12012yes—
145C2×C4C22⋊Q814yes—
146C2×C4C22⋊Q814yes—
147C2×C4C22⋊Q812yes—
148C2×C4C4⋊Q814yes—
149C2×C4C4⋊Q814yes—
150C2×C4C4⋊Q812yes—
151C2×C4C22×D412yes—
152C2×C4C22×Q812yes—
153C2×C4C22×Q811yescharacteristic
154C2×C4C2×#4111yescharacteristic
155C8—28no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—42no—
163C22—42no—
164C22#39512yes—
165C22#39512yes—
166C22C2×#12011yescharacteristic
167C22C2×C22⋊Q811yescharacteristic
168C22C2×C4⋊Q811yescharacteristic
169C4—44no—
170C4—44no—
171C4—44no—
172C4—44no—
173C4—81no—
174C4—81no—
175C4#39514yes—
176C4C2×C22⋊Q812yes—
177C4C2×C4⋊Q812yes—
178C2—21no—
179C2—21no—
180C2—22no—
181C2—22no—
182C2C2×#39511yescharacteristic
183C2#301911yescharacteristic
184C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2061C2semidirect
5. #2060C2semidirect
35. C2×M16Q8semidirect
50. #117Q8semidirect
96. M16C2×Q8semidirect
120. C42C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#29112
#23172
#20612
#20602
C2×#3432
#17332