Finite Group Databasebeta

← #17631

Subgroups — #17631

Order 256.

569 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 183 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#3497C211yesmaximal, characteristic
2#2983C211yesmaximal, characteristic, complemented
3#2356C211yesmaximal, characteristic, complemented
4#1517C212yesmaximal, complemented
5#1514C214yesmaximal, complemented
6#1505C214yesmaximal, complemented
7#1502C212yesmaximal, complemented
8C2×2−1+4—21no—
9#550C414yescomplemented
10D4×Q8C414yescomplemented
11#548—24no—
12#537—21no—
13#537—22no—
14#532C2212yes—
15#532C2214yes—
16C2×C4×Q8C2211yescharacteristic
17#446—24no—
18#432C2214yes—
19#431C2212yes—
20#391—21no—
21#387—22no—
22#383C2212yes—
23#382C2211yescharacteristic
24C4×C4⋊C4—21no—
25C4×C4⋊C4C2211yescharacteristic
26#376C2212yes—
27#327C2214yes—
28#327C2214yes—
29#327C2218yes—
302−1+4—44no—
312−1+4—44no—
32C2×#41—24no—
33C2×#41—24no—
34C2×#41—22no—
35C22×Q8—24no—
36C22×Q8D411yescharacteristic
37C4⋊Q8C2×C412yes—
38C4⋊Q8C2×C414yes—
39#127C2×C412yes—
40#127C2×C414yes—
41#124—22no—
42#124—24no—
43C22⋊Q8—28no—
44C22⋊Q8—24no—
45#121—24no—
46#121—22no—
47#121—28no—
48#121—24no—
49#121C2314yes—
50C4×Q8C2×C414yes—
51C4×Q8D414yes—
52C4×D4—22no—
53C4×D4—24no—
54C4×D4C2×C414yes—
55C4×D4C2×C418yes—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D412yes—
60C2×C42—22no—
61C2×C42—21no—
62C2×C42C2312yes—
63C2×C42C2311yescharacteristic
64C4⋊C8—22no—
65C4⋊C8—24no—
66C22⋊C8C2314yes—
67#97—24no—
68C4×C8—22no—
69#95—22no—
70#95—22no—
71#95—21no—
72#95—22no—
73#95C2311yescharacteristic
74#95C2312yes—
75#95C2311yescharacteristic
76#41—48no—
77#41—48no—
78#41—416no—
79#41—44no—
80#41—44no—
81C2×Q8—24no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—416no—
86C2×Q8C22⋊C414yes—
87C2×D4—24no—
88C2×D4—24no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—24no—
92C2×D4C22×C414yes—
93C22×C4—21no—
94C22×C4—28no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4C2×D412yes—
99C22×C4C2×D412yes—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2411yescharacteristic
103C2×C8—24no—
104C4⋊C4—22no—
105C4⋊C4—24no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4C22⋊C418yes—
109C4⋊C4C22⋊C414yes—
110C4⋊C4C22×C414yes—
111C22⋊C4—24no—
112C22⋊C4—28no—
113C22⋊C4—24no—
114C42—44no—
115C42—42no—
116C42C22×C414yes—
117C42C22×C412yes—
118C42C2×D412yes—
119C42C2×D414yes—
120C23—24no—
121C23C22×D411yescharacteristic
122Q8—28no—
123Q8—48no—
124Q8—44no—
125D4—44no—
126D4—48no—
127D4—44no—
128D4—44no—
129D4—44no—
130D4—84no—
131D4—82no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—24no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—416no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—48no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4C2×C22⋊C412yes—
150C2×C4C2×C22⋊C414yes—
151C2×C4C2×C22⋊C414yes—
152C2×C4C2×C22⋊C412yes—
153C2×C4AGL(1,ℤ/8)12yes—
154C2×C4AGL(1,ℤ/8)12yes—
155C2×C4C23×C411yescharacteristic
156C2×C4C22×D411yescharacteristic
157C8—44no—
158C22—21no—
159C22—21no—
160C22—44no—
161C22—44no—
162C22—44no—
163C22C22×C22⋊C411yescharacteristic
164C22C2×AGL(1,ℤ/8)11yescharacteristic
165C22C2×AGL(1,ℤ/8)11yescharacteristic
166C4—21no—
167C4—21no—
168C4—24no—
169C4—24no—
170C4—22no—
171C4—42no—
172C4—44no—
173C4—81no—
174C4—82no—
175C4#41814yes—
176C2—21no—
177C2—21no—
178C2—84no—
179C2#288411yescharacteristic
180C2C2×#41811yescharacteristic
181C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3497C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2983C2semidirect
3. #2356C2semidirect
4. #1517C2semidirect
5. #1514C2semidirect
6. #1505C2semidirect
7. #1502C2semidirect
9. #550C4semidirect
10. D4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#34972
#29832
#23562
#15172
#15142
#15052
#15022