Finite Group Databasebeta

← #9180

Subgroups — #9180

Order 256.

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

normal subgroups only

This group has 272 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#3037C211yesmaximal, characteristic
2#2351C211yesmaximal, characteristic, complemented
3#1517C211yesmaximal, characteristic
4#1502C211yesmaximal, characteristic
5#1479C211yesmaximal, characteristic, complemented
6#1303C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C4×Q8—21no—
11C2×C4×Q8C2211yescharacteristic
12#464—22no—
13#462C412yescomplemented
14#460C412yescomplemented
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18#417—21no—
19#389—21no—
20#389—22no—
21#382—21no—
22#382C2211yescharacteristic
23C4×C4⋊C4—21no—
24C4×C4⋊C4C2211yescharacteristic
25#327—21no—
26#327—21no—
27#327—22no—
28Q8⋊C8—22no—
29C2×#41—21no—
30C22×Q8D411yescharacteristic
31#137—42no—
32#137—42no—
33C2×Q16—22no—
34C2×SD16—22no—
35C2×M16—21no—
36C4⋊Q8C2×C411yescharacteristic
37#127C2×C411yescharacteristic
38#124—21no—
39#121—21no—
40#121—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×Q8—42no—
44C4×Q8—42no—
45C4×Q8C2×C412yes—
46C4×Q8D412yes—
47C4×D4—21no—
48C4×D4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4D411yescharacteristic
55C2×C42—21no—
56C2×C42—22no—
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8C2×C412yes—
63#103—22no—
64#102—22no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67C4×C8—21no—
68C4×C8—21no—
69#95—22no—
70#95—21no—
71#95D411yescharacteristic
72#95D411yescharacteristic
73#95D412yes—
74#41—41no—
75#41—41no—
76#41—41no—
77#41—41no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—21no—
82C2×Q8C22⋊C412yes—
83C2×Q8D812yes—
84C2×Q8SD1612yes—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—41no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—41no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98Q16—44no—
99SD16—44no—
100M16—41no—
101M16—41no—
102C2×C8—22no—
103C2×C8—41no—
104C2×C8—41no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—42no—
111C4⋊C4—44no—
112C4⋊C4—44no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4C22⋊C412yes—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C42—22no—
119C42—22no—
120C42—44no—
121C42—41no—
122C42—41no—
123C42C22×C411yescharacteristic
124C42C2×D411yescharacteristic
125C42#4111yescharacteristic
126C42#4111yescharacteristic
127C23—41no—
128C23#12011yescharacteristic
129Q8—22no—
130Q8—22no—
131Q8—22no—
132Q8—41no—
133Q8—41no—
134Q8#10214yes—
135D4—41no—
136D4—41no—
137D4—81no—
138D4—81no—
139D4—81no—
140D4—81no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4C2×C22⋊C411yescharacteristic
161C2×C4C4×D411yescharacteristic
162C2×C4C4×D411yescharacteristic
163C2×C4#12111yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12311yescharacteristic
166C2×C4C2×D811yescharacteristic
167C2×C4C2×SD1611yescharacteristic
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C2×C4#13711yescharacteristic
170C8—42no—
171C8—81no—
172C8—81no—
173C22—21no—
174C22—21no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#38611yescharacteristic
179C22#44711yescharacteristic
180C22#44811yescharacteristic
181C4—24no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—42no—
187C4—42no—
188C4—82no—
189C4—81no—
190C4—81no—
191C4C2×#10211yescharacteristic
192C4#41711yescharacteristic
193C4#48011yescharacteristic
194C4#48111yescharacteristic
195C2—21no—
196C2—21no—
197C2—161no—
198C2#187411yescharacteristic
199C2#190511yescharacteristic
200C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
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. #2351C2semidirect
5. #1479C2semidirect
6. #1303C2semidirect
7. #1286C2semidirect
13. #462C4semidirect
14. #460C4semidirect

Maximal subgroups

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

subgroupindex
#30372
#23512
#15172
#15022
#14792
#13032
#12862