Finite Group Databasebeta

← #7800

Subgroups — #7800

Order 256.

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

normal subgroups only

This group has 258 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#2093C211yesmaximal, characteristic, complemented
3#2084C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic, complemented
5#1271C211yesmaximal, characteristic, complemented
6#1270C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic, complemented
8C2×#123—21no—
9C2×#121—21no—
10C2×#107—21no—
11C2×#106—21no—
12C2×#102—21no—
13C2×#102—21no—
14#396—21no—
15#394—21no—
16#392—21no—
17#390C2211yescharacteristic
18#390C2211yescharacteristic
19#389C2211yescharacteristic
20#385—21no—
21#384C2211yescharacteristic
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×D4—21no—
28C22×D4—21no—
29C22×D4D411yescharacteristic
30C22×D4D411yescharacteristic
31C23×C4—21no—
32C22×C8—21no—
33C22×C8—21no—
34C22×C8—21no—
35#123—42no—
36#123—42no—
37#121—42no—
38#121—42no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C4⋊C4D411yescharacteristic
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C42C2311yescharacteristic
56#107—42no—
57#106—42no—
58#102—42no—
59#102—42no—
60#95—21no—
61#95—21no—
62C24—21no—
63C24—21no—
64C2×D4—21no—
65C2×D4—21no—
66C2×D4—21no—
67C2×D4—41no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—81no—
75C2×D4—81no—
76C2×D4—81no—
77C2×D4SD1612yes—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—41no—
85C22×C4—81no—
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C2×C8—22no—
90C2×C8—22no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—41no—
98C4⋊C4—21no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—42no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4D812yes—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—81no—
123C42—41no—
124C23—22no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23#12011yescharacteristic
139D4—42no—
140D4—42no—
141D4—81no—
142D4—81no—
143D4—84no—
144D4—84no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—84no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4C2×D811yescharacteristic
168C2×C4C2×SD1611yescharacteristic
169C2×C4#13511yescharacteristic
170C2×C4AGL(1,ℤ/8)11yescharacteristic
171C2×C4AGL(1,ℤ/8)11yescharacteristic
172C2×C4AGL(1,ℤ/8)11yescharacteristic
173C8—42no—
174C8—42no—
175C8—81no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22#44711yescharacteristic
191C22#44911yescharacteristic
192C22#45011yescharacteristic
193C22#45311yescharacteristic
194C22#45511yescharacteristic
195C22#45611yescharacteristic
196C22#45811yescharacteristic
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—42no—
204C4—81no—
205C4—81no—
206C4—81no—
207C4—161no—
208C2—82no—
209C2—161no—
210C2#161011yescharacteristic
211C2#161211yescharacteristic
212C2#161311yescharacteristic
213C2#162111yescharacteristic
214C2#163911yescharacteristic
215C2#165411yescharacteristic
216C2#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. #2093C2semidirect
3. #2084C2semidirect
4. #2071C2semidirect
5. #1271C2semidirect
6. #1270C2semidirect
7. #1270C2semidirect

Maximal subgroups

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

subgroupindex
#26822
#20932
#20842
#20712
#12712
#12702