Finite Group Databasebeta

← #18359

Subgroups — #18359

Order 256.

502 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 115 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#3215C212yesmaximal, complemented
3#2631C211yesmaximal, characteristic
4#1602C214yesmaximal, complemented
5#1596C214yesmaximal, complemented
6#1515C212yesmaximal, complemented
7#1466C211yesmaximal, characteristic, complemented
8#554—22no—
9#554—24no—
10D4×Q8—22no—
11#544—21no—
12#543C2214yes—
13#533C2212yes—
14C2×#124C2211yescharacteristic
15C2×C22⋊Q8—22no—
16#518—21no—
17C2×C4×D4C2211yescharacteristic
18#497—22no—
19#489—22no—
20#464—24no—
21#451C2214yes—
22#450C2214yes—
23#432C2212yes—
24#397—21no—
25#393C2212yes—
26#392—21no—
27#392C2212yes—
28#386—22no—
29#383C2211yescharacteristic
30#328C2218yes—
31#323C2214yes—
32C22×Q8—22no—
33C22×Q8C2311yescharacteristic
34C22×D4—22no—
35C22×D4D411yescharacteristic
36C22×D4C2311yescharacteristic
37C23×C4D412yes—
38C2×Q16—24no—
39C2×SD16—24no—
40C4⋊Q8—24no—
41C4⋊Q8D412yescomplemented
42C4⋊Q8D412yescomplemented
43#125—24no—
44#125—22no—
45#124—22no—
46#124—22no—
47C22⋊Q8—24no—
48C22⋊Q8—24no—
49C22⋊Q8—28no—
50C22⋊Q8—22no—
51C22⋊Q8—24no—
52C22⋊Q8—28no—
53C22⋊Q8C2314yes—
54C4×Q8—22no—
55C4×Q8D414yescomplemented
56C4×D4—24no—
57C4×D4—24no—
58C4×D4—22no—
59C4×D4—22no—
60#117—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4D411yescharacteristic
64C2×C22⋊C4—24no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4C2312yes—
70C2×C42C2311yescharacteristic
71C4⋊C8—22no—
72#103—24no—
73#102—24no—
74C22⋊C8C2314yes—
75#97—22no—
76#95—21no—
77#95—21no—
78#95—22no—
79#95C2312yes—
80C24—21no—
81C24C2×D412yes—
82C2×Q8—24no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—81no—
86C2×Q8—84no—
87C2×Q8C2×D414yes—
88C2×D4—24no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—44no—
92C2×D4—81no—
93C2×D4—82no—
94C2×D4—82no—
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—82no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2411yescharacteristic
106Q16—84no—
107SD16—84no—
108C2×C8—24no—
109C4⋊C4—24no—
110C4⋊C4—28no—
111C4⋊C4—22no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—24no—
117C4⋊C4C2×D414yes—
118C22⋊C4—28no—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C22⋊C4—42no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—82no—
125C22⋊C4—82no—
126C42—24no—
127C42C2×D412yes—
128C42C2×D412yes—
129C23—22no—
130C23—22no—
131C23—42no—
132C23—42no—
133C23—41no—
134C23—41no—
135C23—42no—
136C23—41no—
137C23—42no—
138C23#13714yes—
139C23C22×D411yescharacteristic
140Q8—44no—
141Q8—44no—
142Q8—44no—
143Q8—84no—
144D4—28no—
145D4—44no—
146D4—88no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—44no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—82no—
161C2×C4—84no—
162C2×C4—84no—
163C2×C4#12012yes—
164C2×C4#12011yescharacteristic
165C2×C4#12011yescharacteristic
166C2×C4C22×D412yes—
167C8—44no—
168C22—21no—
169C22—21no—
170C22—24no—
171C22—44no—
172C22—42no—
173C22—42no—
174C22—82no—
175C22—82no—
176C22—82no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22C2×#12011yescharacteristic
181C22C2×#13712yes—
182C4—22no—
183C4—21no—
184C4—21no—
185C4—22no—
186C4—42no—
187C4—44no—
188C4—44no—
189C4—82no—
190C4—81no—
191C2—22no—
192C2—44no—
193C2—161no—
194C2#300411yescharacteristic
195C2#301911yescharacteristic
196C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2631C2
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. #3215C2semidirect
4. #1602C2semidirect
5. #1596C2semidirect
6. #1515C2semidirect
7. #1466C2semidirect
41. C4⋊Q8D4semidirect
42. C4⋊Q8D4semidirect
55. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
#35012
#32152
#26312
#16022
#15962
#15152
#14662