Finite Group Databasebeta

← #9181

Subgroups — #9181

Order 256.

274 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 67 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#3036C211yesmaximal, characteristic
2#2357C211yesmaximal, characteristic, complemented
3#1517C211yesmaximal, characteristic, complemented
4#1502C211yesmaximal, characteristic, complemented
5#1478C211yesmaximal, characteristic, complemented
6#1303C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#532C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#463—22no—
12#461C412yescomplemented
13#459C412yescomplemented
14#431C2211yescharacteristic
15#431C2211yescharacteristic
16#423C2211yescharacteristic
17#417—21no—
18#390—21no—
19#388—22no—
20#382C2211yescharacteristic
21#381—21no—
22C4×C4⋊C4—21no—
23C4×C4⋊C4C2211yescharacteristic
24C4×C22⋊C4—21no—
25#327—21no—
26#327—21no—
27#327—22no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×D4D411yescharacteristic
31C23×C4—21no—
32AGL(1,ℤ/8)—42no—
33AGL(1,ℤ/8)—42no—
34C2×SD16—22no—
35C2×D8—22no—
36C2×M16—21no—
37C4⋊Q8C2×C411yescharacteristic
38#127C2×C411yescharacteristic
39#124—21no—
40#121—21no—
41#121—21no—
42C4×D4—21no—
43C4×D4—21no—
44C4×D4—21no—
45C4×D4—21no—
46C4×D4C2×C412yes—
47C4×D4D412yes—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4D411yescharacteristic
51C2×C22⋊C4—21no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—22no—
54C2×C42—21no—
55C2×C42—22no—
56C2×C42C2311yescharacteristic
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8C2×C412yes—
61#103—22no—
62#102—22no—
63C22⋊C8—21no—
64C22⋊C8—21no—
65C4×C8—21no—
66C4×C8—21no—
67#95—22no—
68#95—21no—
69#95D411yescharacteristic
70#95D411yescharacteristic
71#95D412yes—
72C24—21no—
73#41—41no—
74#41—41no—
75#41—41no—
76#41—41no—
77C2×Q8—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—41no—
87C2×D4C22⋊C412yes—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—41no—
95C22×C4—82no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99SD16—44no—
100D8—44no—
101M16—41no—
102M16—41no—
103C2×C8—22no—
104C2×C8—41no—
105C2×C8—41no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—42no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4C22⋊C412yes—
113C22⋊C4—22no—
114C22⋊C4—22no—
115C22⋊C4—44no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—82no—
119C42—41no—
120C42—41no—
121C42—82no—
122C42C22×C411yescharacteristic
123C42C2×D411yescharacteristic
124C42#4111yescharacteristic
125C42#4111yescharacteristic
126C23—22no—
127C23—42no—
128C23—42no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23#12011yescharacteristic
133Q8—41no—
134Q8—41no—
135D4—22no—
136D4—22no—
137D4—41no—
138D4—41no—
139D4—41no—
140D4—41no—
141D4—81no—
142D4—81no—
143D4—81no—
144D4—81no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4C2×C22⋊C411yescharacteristic
164C2×C4C4×D411yescharacteristic
165C2×C4C4×D411yescharacteristic
166C2×C4#12111yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12311yescharacteristic
169C2×C4#13511yescharacteristic
170C2×C4#13511yescharacteristic
171C2×C4AGL(1,ℤ/8)11yescharacteristic
172C2×C4AGL(1,ℤ/8)11yescharacteristic
173C8—42no—
174C8—81no—
175C8—81no—
176C22—21no—
177C22—21no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—82no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22#38611yescharacteristic
187C22#44911yescharacteristic
188C22#44911yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—42no—
194C4—42no—
195C4—81no—
196C4—82no—
197C4—81no—
198C4#41611yescharacteristic
199C4#41811yescharacteristic
200C4#48211yescharacteristic
201C4#48211yescharacteristic
202C2—21no—
203C2—21no—
204C2—82no—
205C2—161no—
206C2#187511yescharacteristic
207C2#190411yescharacteristic
208C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3036C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2357C2semidirect
3. #1517C2semidirect
4. #1502C2semidirect
5. #1478C2semidirect
6. #1303C2semidirect
7. #1286C2semidirect
12. #461C4semidirect
13. #459C4semidirect

Maximal subgroups

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

subgroupindex
#30362
#23572
#15172
#15022
#14782
#13032
#12862