Finite Group Databasebeta

← #10508

Subgroups — #10508

Order 256.

207 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 47 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#3213C211yesmaximal, characteristic
2#1983C211yesmaximal, characteristic, complemented
3#1573C211yesmaximal, characteristic, complemented
4#1551C211yesmaximal, characteristic, complemented
5#1502C211yesmaximal, characteristic, complemented
6#1469C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#124C2211yescharacteristic
10#496—21no—
11#489—21no—
12#463—21no—
13#463—21no—
14#448—21no—
15#447—21no—
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18#431C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20C2×#103—21no—
21C2×#102—21no—
22C4×C4⋊C4C2211yescharacteristic
23#335—21no—
24#334—21no—
25#333—21no—
26#331—21no—
27#327—22no—
28#323—22no—
29C2×#41—21no—
30C22×Q8D411yescharacteristic
31C22×D4D411yescharacteristic
32C2×SD16—21no—
33C2×SD16—21no—
34C2×D8—21no—
35C2×D8—21no—
36C22×C8—21no—
37C4⋊Q8D411yescharacteristic
38#127D411yescharacteristic
39#124—21no—
40#124—21no—
41#124—21no—
42#124—21no—
43#124—21no—
44#121—21no—
45#121—21no—
46C4×D4—21no—
47C4×D4—21no—
48C2×C4⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42—21no—
52C2×C42C2311yescharacteristic
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8D411yescharacteristic
58C4⋊C8D411yescharacteristic
59#103—21no—
60#103—21no—
61#103—42no—
62#102—21no—
63#102—21no—
64#102—42no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67C22⋊C8D412yes—
68#97—21no—
69#97—21no—
70C4×C8—21no—
71#95—21no—
72C24—21no—
73#41—81no—
74#41—81no—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—41no—
79C2×Q8—41no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C22×C4—21no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4#4111yescharacteristic
94SD16—81no—
95SD16—81no—
96D8—81no—
97D8—81no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8—22no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C4⋊C4—22no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42—81no—
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42#4111yescharacteristic
119C42#4111yescharacteristic
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—81no—
125C23—81no—
126C23#12111yescharacteristic
127Q8—42no—
128Q8—41no—
129Q8—41no—
130Q8—81no—
131D4—42no—
132D4—41no—
133D4—41no—
134D4—41no—
135D4—41no—
136D4—81no—
137D4—81no—
138D4—81no—
139D4—81no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—82no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4#12011yescharacteristic
153C2×C4#12111yescharacteristic
154C2×C4#12111yescharacteristic
155C2×C4#12311yescharacteristic
156C2×C4#12311yescharacteristic
157C2×C4#12411yescharacteristic
158C8—42no—
159C8—41no—
160C8—41no—
161C8—81no—
162C8—81no—
163C22—21no—
164C22—21no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—161no—
172C22—161no—
173C22#39411yescharacteristic
174C22#47111yescharacteristic
175C22#47211yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—82no—
182C4—81no—
183C4—81no—
184C4#45311yescharacteristic
185C4#45511yescharacteristic
186C2—21no—
187C2—21no—
188C2—161no—
189C2—161no—
190C2#203711yescharacteristic
191C2#205111yescharacteristic
192C2#205311yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radical#3213C2
LCSG ▷ C2×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. #1983C2semidirect
3. #1573C2semidirect
4. #1551C2semidirect
5. #1502C2semidirect
6. #1469C2semidirect
7. #1285C2semidirect

Maximal subgroups

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

subgroupindex
#32132
#19832
#15732
#15512
#15022
#14692
#12852