Finite Group Databasebeta

← #7778

Subgroups — #7778

Order 256.

287 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 51 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#2682C211yesmaximal, characteristic
2#2058C211yesmaximal, characteristic, complemented
3#2023C211yesmaximal, characteristic, complemented
4#2013C211yesmaximal, characteristic, complemented
5#1271C211yesmaximal, characteristic, complemented
6#1270C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic, complemented
8C22×SD16—21no—
9C2×C4⋊Q8C2211yescharacteristic
10C2×#123—21no—
11C2×#121—21no—
12C2×#106—21no—
13C2×#103—21no—
14C2×#102—21no—
15#396—21no—
16#394—21no—
17#392—21no—
18#390C2211yescharacteristic
19#390C2211yescharacteristic
20#389C2211yescharacteristic
21#385—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8—21no—
28C22×D4—21no—
29C22×D4—21no—
30C22×D4D411yescharacteristic
31C22×D4D411yescharacteristic
32C23×C4—21no—
33C2×SD16—42no—
34C2×SD16—42no—
35C2×SD16—42no—
36C22×C8—21no—
37C22×C8—21no—
38C22×C8—21no—
39C4⋊Q8—42no—
40#123—42no—
41#123—42no—
42#121—42no—
43#121—42no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C42C2311yescharacteristic
60#106—42no—
61#103—42no—
62#102—42no—
63#95—21no—
64#95—21no—
65C24—21no—
66C24—21no—
67C2×Q8—22no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8—41no—
72C2×Q8—41no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—41no—
80C2×D4—42no—
81C2×D4—42no—
82C2×D4—42no—
83C2×D4—81no—
84C2×D4—81no—
85C2×D4—81no—
86C2×D4SD1612yescomplemented
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—41no—
94C22×C4—81no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98SD16—48no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—41no—
108C4⋊C4—21no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—42no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4SD1612yescomplemented
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—81no—
131C42—41no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23#12011yescharacteristic
147Q8—42no—
148Q8—42no—
149Q8—81no—
150Q8—81no—
151D4—42no—
152D4—42no—
153D4—81no—
154D4—81no—
155D4—84no—
156D4—84no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—84no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4C2×SD1611yescharacteristic
180C2×C4C2×SD1611yescharacteristic
181C2×C4#13511yescharacteristic
182C2×C4AGL(1,ℤ/8)11yescharacteristic
183C2×C4AGL(1,ℤ/8)11yescharacteristic
184C2×C4AGL(1,ℤ/8)11yescharacteristic
185C8—42no—
186C8—42no—
187C8—81no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22#44911yescharacteristic
203C22#45011yescharacteristic
204C22#45011yescharacteristic
205C22#45311yescharacteristic
206C22#45511yescharacteristic
207C22#45611yescharacteristic
208C22B(4,𝔽2)11yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—82no—
217C4—81no—
218C4—81no—
219C4—81no—
220C2—82no—
221C2—161no—
222C2#160011yescharacteristic
223C2#160111yescharacteristic
224C2#160411yescharacteristic
225C2#162911yescharacteristic
226C2#164211yescharacteristic
227C2#165411yescharacteristic
228C2#166111yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radical#2682C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2058C2semidirect
3. #2023C2semidirect
4. #2013C2semidirect
5. #1271C2semidirect
6. #1270C2semidirect
7. #1270C2semidirect
86. C2×D4SD16semidirect
119. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#26822
#20582
#20232
#20132
#12712
#12702