Finite Group Databasebeta

← #21423

Subgroups — #21423

Order 256.

581 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 191 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×#539C211yesmaximal, characteristic
2#2313C211yesmaximal, characteristic, complemented
3#2274C211yesmaximal, characteristic, complemented
4#1996C212yesmaximal, complemented
5#1964C212yesmaximal
6#1899C214yesmaximal, complemented
7C4×#99C214yesmaximal, complemented
8#539—24no—
9#539—24no—
10C2×#126C412yescomplemented
11C2×#125C412yescomplemented
12C2×#123C412yescomplemented
13C2×#123C2212yes—
14C2×C22⋊Q8C412yescomplemented
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C2×C4×D4C2212yes—
18C2×#117C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20C2×#99C2218yes—
21#387—24no—
22#381C2212yes—
23#380C2212yes—
24C4×C4⋊C4—24no—
25C4×C22⋊C4C2212yes—
26C4×C22⋊C4C2212yes—
27C4×C22⋊C4C2212yes—
28#342C2218yes—
29C22×Q8C2×C411yescharacteristic
30C22×D4C2×C411yescharacteristic
31C22×D4C2312yes—
32C23×C4C2×C412yes—
33C23×C4C2311yescharacteristic
34C23×C4C2311yescharacteristic
35C23×C4C2311yescharacteristic
36#126—28no—
37#125—28no—
38#123—24no—
39#123—24no—
40#123—28no—
41C22⋊Q8—22no—
42C22⋊Q8—22no—
43C22⋊Q8—28no—
44#121—22no—
45#121—22no—
46C4×D4—24no—
47C4×D4—24no—
48#117—28no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4C2×C412yes—
52C2×C4⋊C4C2×C411yescharacteristic
53C2×C4⋊C4C2×C412yes—
54C2×C4⋊C4C2×C412yes—
55C2×C4⋊C4C2×C411yescharacteristic
56C2×C4⋊C4C2×C412yes—
57C2×C4⋊C4C2×C412yes—
58C2×C4⋊C4C2×C412yes—
59C2×C4⋊C4D414yes—
60C2×C4⋊C4D414yes—
61C2×C22⋊C4C2×C412yes—
62C2×C22⋊C4C2×C412yes—
63C2×C22⋊C4C2×C412yes—
64C2×C22⋊C4C2×C412yes—
65C2×C22⋊C4C2314yes—
66C2×C22⋊C4C2312yes—
67C2×C22⋊C4C2314yes—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42C2×C412yes—
73#99—216no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—24no—
79C24C22×C412yes—
80C24C2411yescharacteristic
81C2×Q8—22no—
82C2×Q8—81no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—28no—
86C2×D4—22no—
87C2×D4—81no—
88C22×C4—24no—
89C22×C4—24no—
90C22×C4—24no—
91C22×C4—24no—
92C22×C4—81no—
93C22×C4—81no—
94C22×C4—82no—
95C22×C4C22×C412yes—
96C22×C4C22×C412yes—
97C22×C4C22×C412yes—
98C22×C4C22×C412yes—
99C22×C4C22×C412yes—
100C22×C4C22×C411yescharacteristic
101C22×C4C22×C411yescharacteristic
102C22×C4C2×D412yes—
103C22×C4C2×D412yes—
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C4⋊C4—28no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—24no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—82no—
116C4⋊C4C22⋊C418yes—
117C4⋊C4C22⋊C418yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C22⋊C4—24no—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—28no—
126C22⋊C4#4118yes—
127C42—28no—
128C42—24no—
129C42—82no—
130C42—82no—
131C42—82no—
132C23—22no—
133C23—24no—
134C23—41no—
135C23—41no—
136C23—42no—
137C23—42no—
138C23C23×C411yescharacteristic
139C23C22×D411yescharacteristic
140C23C22×D411yescharacteristic
141C23C2×#4114yes—
142Q8—82no—
143D4—44no—
144D4—44no—
145D4—82no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—28no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—84no—
170C2×C4C2×C22⋊C414yes—
171C2×C4C2×C22⋊C414yes—
172C2×C4C2×C22⋊C412yes—
173C2×C4C2×C22⋊C412yes—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22C22×C22⋊C411yescharacteristic
186C22C4×#4112yes—
187C22#52512yes—
188C22#53212yes—
189C4—24no—
190C4—22no—
191C4—22no—
192C4—24no—
193C4—48no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4—84no—
198C2—22no—
199C2—22no—
200C2—22no—
201C2—82no—
202C2#229711yescharacteristic
203C2#288311yescharacteristic
204C2#288411yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC24C24
prime radicalC2×#121C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2313C2semidirect
3. #2274C2semidirect
4. #1996C2semidirect
6. #1899C2semidirect
7. C4×#99C2semidirect
10. C2×#126C4semidirect
11. C2×#125C4semidirect
12. C2×#123C4semidirect
14. C2×C22⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5392
#23132
#22742
#19962
#19642
#18992
C4×#992