Finite Group Databasebeta

← #9293

Subgroups — #9293

Order 256.

279 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 69 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#3205C211yesmaximal, characteristic
2#2932C211yesmaximal, characteristic, complemented
3#1961C211yesmaximal, characteristic, complemented
4#1545C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1286C211yesmaximal, characteristic, complemented
7#1275C211yesmaximal, characteristic, complemented
8C2×#131—21no—
9#532C2211yescharacteristic
10C2×#127C2211yescharacteristic
11#493C412yescomplemented
12#486C412yescomplemented
13#459—22no—
14#447—22no—
15#435—22no—
16#435—22no—
17#431C2211yescharacteristic
18#431C2211yescharacteristic
19#423C2211yescharacteristic
20C2×#102—21no—
21C2×#102—21no—
22#408—21no—
23C2×#97C2211yescharacteristic
24#382C2211yescharacteristic
25#331—22no—
26#327—22no—
27C2×#41—21no—
28C22×D4—21no—
29C22×D4—21no—
30C22×D4—21no—
31C2×D8—24no—
32#131—42no—
33#131—42no—
34C2×M16—22no—
35C2×M16—21no—
36C22×C8—22no—
37C22×C8D411yescharacteristic
38C22×C8D411yescharacteristic
39C4⋊Q8C2×C411yescharacteristic
40#127—21no—
41#127—21no—
42#127—21no—
43#127C2×C412yes—
44#127C2×C411yescharacteristic
45#124—21no—
46#121—21no—
47#121D412yes—
48C4×D4—21no—
49C4×D4D412yes—
50C2×C4⋊C4—21no—
51C2×C42C2311yescharacteristic
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8D412yes—
56#102—24no—
57#102—42no—
58#102—42no—
59C22⋊C8—24no—
60C22⋊C8—21no—
61C22⋊C8—21no—
62#97—21no—
63#97—21no—
64C4×C8—21no—
65C4×C8C2×C412yes—
66#95—21no—
67#95—21no—
68C24—41no—
69#41—41no—
70#41—41no—
71#41—41no—
72#41—41no—
73C2×Q8—21no—
74C2×D4—22no—
75C2×D4—22no—
76C2×D4—21no—
77C2×D4—22no—
78C2×D4—21no—
79C2×D4—22no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—82no—
88C2×D4—82no—
89C2×D4C22⋊C412yes—
90C22×C4—22no—
91C22×C4—41no—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4#4111yescharacteristic
95D8—84no—
96M16—44no—
97M16—41no—
98M16—41no—
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—22no—
102C2×C8—21no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—44no—
107C2×C8—41no—
108C2×C8D812yescomplemented
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4C22⋊C412yes—
113C22⋊C4—22no—
114C22⋊C4—22no—
115C42C22×C411yescharacteristic
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42#4111yescharacteristic
119C23—22no—
120C23—42no—
121C23—82no—
122C23—82no—
123C23—81no—
124C23—81no—
125C23#12111yescharacteristic
126Q8—41no—
127Q8—41no—
128D4—42no—
129D4—44no—
130D4—42no—
131D4—42no—
132D4—41no—
133D4—42no—
134D4—41no—
135D4—42no—
136D4—81no—
137D4—82no—
138D4—82no—
139D4—82no—
140D4—81no—
141D4—81no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4C2×C22⋊C411yescharacteristic
151C2×C4C4×D411yescharacteristic
152C2×C4C4×D411yescharacteristic
153C2×C4#12011yescharacteristic
154C2×C4#12111yescharacteristic
155C2×C4#12311yescharacteristic
156C2×C4C2×D811yescharacteristic
157C2×C4#13511yescharacteristic
158C2×C4AGL(1,ℤ/8)11yescharacteristic
159C2×C4AGL(1,ℤ/8)11yescharacteristic
160C8—22no—
161C8—42no—
162C8—42no—
163C8—41no—
164C8—81no—
165C22—21no—
166C22—21no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22—82no—
171C22—82no—
172C22—82no—
173C22—162no—
174C22—162no—
175C22#38611yescharacteristic
176C22#45911yescharacteristic
177C22#46311yescharacteristic
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—42no—
183C4—161no—
184C4C4×D811yescharacteristic
185C4#44211yescharacteristic
186C4#44711yescharacteristic
187C4#44911yescharacteristic
188C2—21no—
189C2—21no—
190C2—82no—
191C2—162no—
192C2#187911yescharacteristic
193C2#190311yescharacteristic
194C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3205C2
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. #2932C2semidirect
3. #1961C2semidirect
4. #1545C2semidirect
5. #1517C2semidirect
6. #1286C2semidirect
7. #1275C2semidirect
11. #493C4semidirect
12. #486C4semidirect
108. C2×C8D8semidirect

Maximal subgroups

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

subgroupindex
#32052
#29322
#19612
#15452
#15172
#12862
#12752