Finite Group Databasebeta

← #9182

Subgroups — #9182

Order 256.

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

normal subgroups only

This group has 250 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#3037C211yesmaximal, characteristic
2#2356C211yesmaximal, 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×Q8C2211yescharacteristic
11#464—22no—
12#462C412yescomplemented
13#460C412yescomplemented
14#431C2211yescharacteristic
15#431C2211yescharacteristic
16#423C2211yescharacteristic
17#417—21no—
18#391—21no—
19#387—22no—
20#383—21no—
21#382C2211yescharacteristic
22C4×C4⋊C4—21no—
23C4×C4⋊C4C2211yescharacteristic
24#376—21no—
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×Q8C2×C412yes—
44C4×Q8D412yes—
45C4×D4—21no—
46C4×D4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4D411yescharacteristic
51C2×C42—21no—
52C2×C42—22no—
53C2×C42—21no—
54C2×C42C2311yescharacteristic
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8C2×C412yes—
59#103—22no—
60#102—22no—
61C22⋊C8—21no—
62C22⋊C8—21no—
63C4×C8—21no—
64C4×C8—21no—
65#95—22no—
66#95—22no—
67#95—22no—
68#95—21no—
69#95D411yescharacteristic
70#95D411yescharacteristic
71#95D412yes—
72#41—41no—
73#41—41no—
74#41—41no—
75#41—41no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—21no—
82C2×Q8C22⋊C412yes—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—41no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96Q16—44no—
97SD16—44no—
98M16—41no—
99M16—41no—
100C2×C8—22no—
101C2×C8—41no—
102C2×C8—41no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—42no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4C22⋊C412yes—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C42—22no—
115C42—22no—
116C42—44no—
117C42—41no—
118C42—41no—
119C42C22×C411yescharacteristic
120C42C2×D411yescharacteristic
121C42#4111yescharacteristic
122C42#4111yescharacteristic
123C23—41no—
124C23#12011yescharacteristic
125Q8—22no—
126Q8—22no—
127Q8—41no—
128Q8—41no—
129Q8—41no—
130Q8—41no—
131D4—41no—
132D4—41no—
133D4—81no—
134D4—81no—
135D4—81no—
136D4—81no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4C2×C22⋊C411yescharacteristic
157C2×C4C4×D411yescharacteristic
158C2×C4C4×D411yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4#12311yescharacteristic
162C2×C4#13511yescharacteristic
163C2×C4#13511yescharacteristic
164C2×C4AGL(1,ℤ/8)11yescharacteristic
165C2×C4AGL(1,ℤ/8)11yescharacteristic
166C8—42no—
167C8—81no—
168C8—81no—
169C22—21no—
170C22—21no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22#38611yescharacteristic
175C22#44911yescharacteristic
176C22#44911yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—42no—
182C4—42no—
183C4—42no—
184C4—81no—
185C4—82no—
186C4—81no—
187C4#41611yescharacteristic
188C4#41811yescharacteristic
189C4#48211yescharacteristic
190C4#48211yescharacteristic
191C2—21no—
192C2—21no—
193C2—161no—
194C2#187511yescharacteristic
195C2#190511yescharacteristic
196C2#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. #2356C2semidirect
5. #1479C2semidirect
6. #1303C2semidirect
7. #1286C2semidirect
12. #462C4semidirect
13. #460C4semidirect

Maximal subgroups

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

subgroupindex
#30372
#23562
#15172
#15022
#14792
#13032
#12862