Finite Group Databasebeta

← #18324

Subgroups — #18324

Order 256.

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

normal subgroups only

This group has 628 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×#537C211yesmaximal, characteristic, complemented
2#3192C212yesmaximal, complemented
3#2591C211yesmaximal, characteristic, complemented
4#1616C214yesmaximal, complemented
5#1598C214yesmaximal, complemented
6#1507C212yesmaximal, complemented
7#1463C211yesmaximal, characteristic, complemented
8C22×#41—21no—
9#537—24no—
10#537—22no—
11#537C2214yes—
12C2×C22⋊Q8—22no—
13C2×#121—22no—
14C2×#121—21no—
15C2×#121C2212yes—
16C2×#121C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#481—22no—
19#481C2214yescomplemented
20#469—24no—
21#447—22no—
22#447C2214yescomplemented
23#407C2212yes—
24#400C2212yescomplemented
25#394C2212yescomplemented
26#388—22no—
27#386—21no—
28#386—21no—
29C4×C22⋊C4C2211yescharacteristic
30#327C2218yescomplemented
31#324C2214yescomplemented
32C2×#41—24no—
33C2×#41—24no—
34C2×#41—48no—
35C22×Q8—21no—
36C22×D4—21no—
37C22×D4—22no—
38C22×D4—21no—
39C22×D4—21no—
40C22×D4D412yes—
41C22×D4C2311yescharacteristic
42C23×C4—22no—
43C23×C4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C2×D8—24no—
47C2×M16—22no—
48C22⋊Q8—28no—
49C22⋊Q8—28no—
50#121—24no—
51#121—28no—
52#121—28no—
53#121—24no—
54#121—22no—
55#121—22no—
56#121—22no—
57#121—22no—
58#121D414yes—
59#121C2314yes—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D412yes—
64C2×C4⋊C4D412yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74#106—24no—
75#102—24no—
76#102—24no—
77C22⋊C8—22no—
78C22⋊C8C2314yes—
79#95—22no—
80#95—22no—
81#95C2312yes—
82C24—22no—
83C24—21no—
84C24C2×D411yescharacteristic
85#41—416no—
86#41—48no—
87#41—48no—
88C2×Q8—24no—
89C2×Q8—44no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—44no—
97C2×D4—48no—
98C2×D4—44no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—81no—
102C2×D4—82no—
103C2×D4—82no—
104C2×D4C2×D414yes—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—24no—
108C22×C4—28no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—48no—
115C22×C4—81no—
116C22×C4—81no—
117C22×C4C2×D412yes—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2411yescharacteristic
123D8—84no—
124M16—82no—
125C2×C8—24no—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4C2×D414yes—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—44no—
138C22⋊C4—82no—
139C42—81no—
140C23—22no—
141C23—22no—
142C23—24no—
143C23—41no—
144C23—44no—
145C23—44no—
146C23—42no—
147C23—42no—
148C23—44no—
149C23—41no—
150C23—42no—
151C23#12011yescharacteristic
152C23#12011yescharacteristic
153C23C22×D411yescharacteristic
154Q8—48no—
155D4—44no—
156D4—48no—
157D4—44no—
158D4—48no—
159D4—48no—
160D4—82no—
161D4—84no—
162D4—82no—
163D4—82no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—416no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—81no—
184C2×C4#12012yes—
185C2×C4AGL(1,ℤ/8)14yes—
186C2×C4C22×D412yes—
187C8—44no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—44no—
196C22—44no—
197C22—44no—
198C22—44no—
199C22—88no—
200C22—84no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22#45412yes—
205C22C2×#12011yescharacteristic
206C22C2×AGL(1,ℤ/8)12yes—
207C4—24no—
208C4—22no—
209C4—22no—
210C4—44no—
211C4—42no—
212C4—82no—
213C4—81no—
214C2—22no—
215C2—22no—
216C2—41no—
217C2—84no—
218C2—161no—
219C2#300311yescharacteristic
220C2C2×#45411yescharacteristic
221C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
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×#537C2semidirect
2. #3192C2semidirect
3. #2591C2semidirect
4. #1616C2semidirect
5. #1598C2semidirect
6. #1507C2semidirect
7. #1463C2semidirect
19. #481C22semidirect
22. #447C22semidirect
24. #400C22semidirect
25. #394C22semidirect
30. #327C22semidirect
31. #324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
#31922
#25912
#16162
#15982
#15072
#14632