Finite Group Databasebeta

← #10320

Subgroups — #10320

Order 256.

533 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 61 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#3186C211yesmaximal, characteristic, complemented
2#2847C212yesmaximal, complemented
3#1779C211yesmaximal, characteristic, complemented
4#1519C212yesmaximal, complemented
5#1289C211yesmaximal, characteristic, complemented
6C24×C4C2211yescharacteristic
7C2×#121—22no—
8C2×#121C2212yes—
9C2×#120—22no—
10#480—22no—
11#466—22no—
12#447—22no—
13C2×C22⋊C8—22no—
14C2×C22⋊C8C2211yescharacteristic
15C2×C22⋊C8C2211yescharacteristic
16#386—22no—
17#386—24no—
18#380—22no—
19#380C2212yes—
20#327—24no—
21C25D411yescharacteristic
22C22×D4—24no—
23C22×D4—22no—
24C22×D4D412yescomplemented
25C23×C4—22no—
26C23×C4—21no—
27C23×C4—22no—
28C23×C4—21no—
29C23×C4—44no—
30C23×C4D411yescharacteristic
31C23×C4C2311yescharacteristic
32C2×D8—22no—
33C22×C8—21no—
34C22×C8D412yescomplemented
35#121—22no—
36#121—22no—
37#121—22no—
38#121—22no—
39#121—48no—
40#120—44no—
41#120—44no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4D412yescomplemented
44C2×C22⋊C4—24no—
45C2×C22⋊C4—24no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—22no—
48#107—22no—
49#102—24no—
50C22⋊C8—22no—
51C22⋊C8—22no—
52C22⋊C8—44no—
53#95—24no—
54#95D414yescomplemented
55C24—22no—
56C24—21no—
57C24—21no—
58C24—22no—
59C24—22no—
60C24—42no—
61C24—42no—
62C24D812yescomplemented
63C24C2×D411yescharacteristic
64C2×D4—22no—
65C2×D4—22no—
66C2×D4—44no—
67C2×D4—48no—
68C2×D4—48no—
69C2×D4—48no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—42no—
75C2×D4—42no—
76C22×C4—24no—
77C22×C4—28no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—24no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—44no—
85C22×C4—42no—
86C22×C4—44no—
87C22×C4—42no—
88C22×C4—44no—
89C22×C4—42no—
90C22×C4—41no—
91C22×C4—42no—
92C22×C4—41no—
93C22×C4C2×D412yes—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D412yes—
96C22×C4#4111yescharacteristic
97D8—82no—
98C2×C8—22no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—41no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—44no—
105C22⋊C4—48no—
106C22⋊C4—48no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C23—24no—
112C23—22no—
113C23—24no—
114C23—22no—
115C23—24no—
116C23—22no—
117C23—41no—
118C23—44no—
119C23—41no—
120C23—42no—
121C23—41no—
122C23—42no—
123C23—41no—
124C23—48no—
125C23—44no—
126C23—44no—
127C23—44no—
128C23—41no—
129C23—42no—
130C23—41no—
131C23—42no—
132C23—42no—
133C23—42no—
134C23#12012yes—
135C23#12111yescharacteristic
136C23C2×D811yescharacteristic
137C23#13511yescharacteristic
138C23AGL(1,ℤ/8)12yes—
139D4—44no—
140D4—82no—
141D4—816no—
142D4—82no—
143D4—82no—
144D4—82no—
145D4—82no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—44no—
153C2×C4—48no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—42no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4#12011yescharacteristic
165C2×C4#12112yes—
166C2×C4#12711yescharacteristic
167C8—42no—
168C8—81no—
169C22—22no—
170C22—21no—
171C22—22no—
172C22—21no—
173C22—28no—
174C22—24no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—44no—
179C22—42no—
180C22—44no—
181C22—44no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—84no—
188C22—84no—
189C22—82no—
190C22—82no—
191C22—82no—
192C22#39211yescharacteristic
193C22#44712yes—
194C22#44912yes—
195C22#46611yescharacteristic
196C22#46911yescharacteristic
197C4—21no—
198C4—22no—
199C4—42no—
200C4—84no—
201C4—82no—
202C4B(4,𝔽2)12yes—
203C2—22no—
204C2—21no—
205C2—22no—
206C2—21no—
207C2—44no—
208C2—162no—
209C2#199911yescharacteristic
210C2#201011yescharacteristic
211C2#202111yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3186C2semidirect
2. #2847C2semidirect
3. #1779C2semidirect
4. #1519C2semidirect
5. #1289C2semidirect
24. C22×D4D4semidirect
34. C22×C8D4semidirect
43. C2×C4⋊C4D4semidirect
54. #95D4semidirect
62. C24D8semidirect

Maximal subgroups

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

subgroupindex
#31862
#28472
#17792
#15192
#12892