Finite Group Databasebeta

← #18212

Subgroups — #18212

Order 256.

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

normal subgroups only

This group has 529 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#3501C211yesmaximal, characteristic, complemented
2#3101C212yesmaximal, complemented
3#2535C211yesmaximal, characteristic
4#1608C214yesmaximal, complemented
5#1596C214yesmaximal, complemented
6#1518C212yesmaximal, complemented
7#1460C211yesmaximal, characteristic, complemented
8#554—22no—
9#554—24no—
10D4×Q8—22no—
11#544—21no—
12#543C2214yes—
13#533C2212yes—
14C2×C22⋊Q8—22no—
15C2×#121—21no—
16#518—21no—
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#483C2214yes—
20#476—22no—
21#461—22no—
22#450C2214yes—
23C4×SD16—24no—
24#431C2212yes—
25#397C2212yes—
26#392C2212yes—
27#386—22no—
28#382C2211yescharacteristic
29#380—21no—
30#328C2218yes—
31#323C2214yes—
32C22×Q8—22no—
33C22×D4—22no—
34C22×D4D411yescharacteristic
35C22×D4C2311yescharacteristic
36C23×C4—21no—
37C23×C4D412yes—
38C2×SD16—24no—
39C4⋊Q8—24no—
40C4⋊Q8D412yescomplemented
41C4⋊Q8D412yescomplemented
42#125—24no—
43#125—22no—
44C22⋊Q8—24no—
45C22⋊Q8—24no—
46C22⋊Q8—28no—
47C22⋊Q8—22no—
48C22⋊Q8—24no—
49C22⋊Q8—28no—
50C22⋊Q8C2314yes—
51#121—44no—
52C4×Q8—22no—
53C4×Q8D414yescomplemented
54C4×D4—24no—
55C4×D4—24no—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4—22no—
59C4×D4—22no—
60#117—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—24no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4C2312yes—
71C2×C42C2311yescharacteristic
72#106—24no—
73C4⋊C8—22no—
74#103—24no—
75#102—24no—
76C22⋊C8C2314yes—
77C4×C8—22no—
78#95—22no—
79#95—21no—
80#95C2312yes—
81C24—21no—
82C24C2×D412yes—
83C2×Q8—24no—
84C2×Q8—22no—
85C2×Q8—84no—
86C2×Q8C2×D414yes—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4—42no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4—82no—
94C2×D4SD1614yescomplemented
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—82no—
105C22×C4C2×D412yes—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2411yescharacteristic
109SD16—48no—
110C2×C8—24no—
111C4⋊C4—24no—
112C4⋊C4—28no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4C2×D414yes—
122C22⋊C4—28no—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—44no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—82no—
129C42—24no—
130C42C2×D412yes—
131C42C2×D412yes—
132C23—22no—
133C23—22no—
134C23—22no—
135C23—42no—
136C23—42no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23C2×SD1614yes—
144C23C22×D411yescharacteristic
145Q8—44no—
146Q8—44no—
147Q8—84no—
148D4—24no—
149D4—28no—
150D4—42no—
151D4—88no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4—84no—
167C2×C4—84no—
168C2×C4—84no—
169C2×C4#12011yescharacteristic
170C2×C4#12011yescharacteristic
171C2×C4#12012yes—
172C2×C4C2×SD1612yes—
173C2×C4#13712yes—
174C2×C4C22×D412yes—
175C8—44no—
176C22—21no—
177C22—21no—
178C22—24no—
179C22—44no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22C2×#12011yescharacteristic
191C22C22×SD1611yescharacteristic
192C22C2×#13711yescharacteristic
193C4—21no—
194C4—21no—
195C4—22no—
196C4—42no—
197C4—44no—
198C4—44no—
199C4—82no—
200C4—81no—
201C4#44814yes—
202C2—21no—
203C2—21no—
204C2—44no—
205C2—82no—
206C2C2×#44811yescharacteristic
207C2#301911yescharacteristic
208C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2535C2
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
1. #3501C2semidirect
2. #3101C2semidirect
4. #1608C2semidirect
5. #1596C2semidirect
6. #1518C2semidirect
7. #1460C2semidirect
40. C4⋊Q8D4semidirect
41. C4⋊Q8D4semidirect
53. C4×Q8D4semidirect
94. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#35012
#31012
#25352
#16082
#15962
#15182
#14602