Finite Group Databasebeta

← #9203

Subgroups — #9203

Order 256.

381 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 75 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#2617C211yesmaximal, characteristic
2#2315C211yesmaximal, characteristic, complemented
3#1519C212yesmaximal, complemented
4C2×#323C211yesmaximal, characteristic, complemented
5#1289C212yesmaximal, complemented
6C2×#121C2211yescharacteristic
7C2×#120—21no—
8C22×C22⋊C4—21no—
9C22×C22⋊C4C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11#397C412yescomplemented
12#394—22no—
13#392C412yescomplemented
14#386—21no—
15#385—24no—
16#380—21no—
17#380C2211yescharacteristic
18#380C2212yes—
19#327—24no—
20#323—22no—
21C25D411yescharacteristic
22C22×D4—21no—
23C22×D4—22no—
24C22×D4C2×C411yescharacteristic, complemented
25C23×C4—22no—
26C23×C4D411yescharacteristic
27C23×C4C2311yescharacteristic
28C22×C8—22no—
29#121—22no—
30#121—22no—
31#120—42no—
32#120—42no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic, complemented
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—42no—
46C2×C22⋊C4—42no—
47C2×C22⋊C4C2×C412yes—
48C2×C22⋊C4D412yes—
49C22⋊C8—22no—
50C22⋊C8—22no—
51#95—22no—
52#95—22no—
53#95C2×C412yescomplemented
54#95D414yescomplemented
55C24—21no—
56C24—21no—
57C24—21no—
58C24—21no—
59C24—21no—
60C24—81no—
61C24—81no—
62C24C22⋊C412yes—
63C24D812yes—
64C24SD1612yes—
65C24C2×D411yescharacteristic
66C2×D4—22no—
67C2×D4—44no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—82no—
75C2×D4—82no—
76C2×D4—81no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—24no—
82C22×C4—44no—
83C22×C4—81no—
84C22×C4—81no—
85C22×C4—82no—
86C22×C4—82no—
87C22×C4C22⋊C412yes—
88C22×C4C22×C411yescharacteristic
89C22×C4C2×D412yes—
90C22×C4C2×D411yescharacteristic
91C22×C4#4112yes—
92C2×C8—42no—
93C2×C8—82no—
94C4⋊C4—22no—
95C4⋊C4—82no—
96C4⋊C4—81no—
97C4⋊C4—82no—
98C22⋊C4—24no—
99C22⋊C4—24no—
100C22⋊C4—44no—
101C22⋊C4—44no—
102C22⋊C4—42no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—82no—
108C23—22no—
109C23—22no—
110C23—22no—
111C23—22no—
112C23—22no—
113C23—22no—
114C23—41no—
115C23—41no—
116C23—42no—
117C23—42no—
118C23—42no—
119C23—42no—
120C23—42no—
121C23—42no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—81no—
126C23—81no—
127C23—82no—
128C23—82no—
129C23—82no—
130C23—82no—
131C23#10214yes—
132C23C2×C22⋊C411yescharacteristic
133C23#12011yescharacteristic
134C23#12311yescharacteristic
135C23C2×D811yescharacteristic
136C23C2×SD1611yescharacteristic
137C23AGL(1,ℤ/8)11yescharacteristic
138C23#13711yescharacteristic
139D4—82no—
140D4—88no—
141D4—81no—
142D4—81no—
143D4—81no—
144D4—81no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—44no—
161C2×C4—81no—
162C2×C4—88no—
163C2×C4C4×D412yes—
164C2×C4#12112yes—
165C8—82no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—24no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—44no—
178C22—44no—
179C22—44no—
180C22—42no—
181C22—42no—
182C22—82no—
183C22—82no—
184C22—82no—
185C22—82no—
186C22—82no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22#38611yescharacteristic
191C22C2×#10211yescharacteristic
192C22#41711yescharacteristic
193C22#44711yescharacteristic
194C22#44811yescharacteristic
195C22#48011yescharacteristic
196C22#48111yescharacteristic
197C4—22no—
198C4—22no—
199C4—42no—
200C4—84no—
201C4—82no—
202C4—81no—
203C2—21no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—21no—
208C2—21no—
209C2—44no—
210C2—161no—
211C2#187411yescharacteristic
212C2#191111yescharacteristic
213C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2617C2
LCSG ▷ C22×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
2. #2315C2semidirect
3. #1519C2semidirect
4. C2×#323C2semidirect
5. #1289C2semidirect
11. #397C4semidirect
13. #392C4semidirect
24. C22×D4C2×C4semidirect
36. C2×C4⋊C4C2×C4semidirect
53. #95C2×C4semidirect
54. #95D4semidirect

Maximal subgroups

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

subgroupindex
#26172
#23152
#15192
C2×#3232
#12892