Finite Group Databasebeta

← #9305

Subgroups — #9305

Order 256.

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

normal subgroups only

This group has 271 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#3036C211yesmaximal, characteristic
2#2972C211yesmaximal, characteristic, complemented
3#1804C211yesmaximal, characteristic, complemented
4#1517C211yesmaximal, characteristic, complemented
5#1480C211yesmaximal, characteristic, complemented
6#1301C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#532C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#463—22no—
12#461C412yescomplemented
13#459C412yescomplemented
14C8×D4—22no—
15C8×D4—22no—
16#431C2211yescharacteristic
17#431C2211yescharacteristic
18#423C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20#417—21no—
21C2×#102—21no—
22C2×C22⋊C8—22no—
23#382C2211yescharacteristic
24#340—21no—
25#327—22no—
26D4⋊C8—22no—
27C2×#41—21no—
28C22×D4D411yescharacteristic
29C23×C4—21no—
30AGL(1,ℤ/8)—42no—
31AGL(1,ℤ/8)—42no—
32C2×SD16—22no—
33C2×D8—22no—
34C2×M16—21no—
35C22×C8—24no—
36C22×C8—21no—
37C4⋊Q8C2×C411yescharacteristic
38#127C2×C411yescharacteristic
39#124—21no—
40#121—21no—
41#121—21no—
42C4×D4—21no—
43C4×D4—21no—
44C4×D4—21no—
45C4×D4—21no—
46C4×D4C2×C412yes—
47C4×D4D412yes—
48C2×C4⋊C4—21no—
49C2×C4⋊C4D411yescharacteristic
50C2×C22⋊C4—21no—
51C2×C42C2311yescharacteristic
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8C2×C412yes—
58#103—22no—
59#102—22no—
60#102—42no—
61C22⋊C8—24no—
62C22⋊C8—21no—
63C22⋊C8D412yes—
64C4×C8—21no—
65C4×C8D412yes—
66#95—21no—
67#95—21no—
68C24—21no—
69#41—41no—
70#41—41no—
71#41—41no—
72#41—41no—
73C2×Q8—21no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—41no—
81C2×D4C22⋊C412yes—
82C2×D4D812yes—
83C2×D4SD1612yes—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—82no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4#4111yescharacteristic
93SD16—44no—
94D8—44no—
95M16—41no—
96M16—41no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—84no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4C22⋊C412yes—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C42C22×C411yescharacteristic
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C42#4111yescharacteristic
118C23—22no—
119C23—22no—
120C23—22no—
121C23—41no—
122C23—81no—
123C23—81no—
124C23#12111yescharacteristic
125Q8—41no—
126Q8—41no—
127D4—22no—
128D4—22no—
129D4—22no—
130D4—42no—
131D4—41no—
132D4—41no—
133D4—81no—
134D4—81no—
135D4—81no—
136D4#10214yes—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—44no—
141C2×C4—41no—
142C2×C4—82no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4C2×C22⋊C411yescharacteristic
150C2×C4C4×D411yescharacteristic
151C2×C4C4×D411yescharacteristic
152C2×C4#12011yescharacteristic
153C2×C4#12111yescharacteristic
154C2×C4#12311yescharacteristic
155C2×C4C2×D811yescharacteristic
156C2×C4C2×SD1611yescharacteristic
157C2×C4#13511yescharacteristic
158C2×C4#13511yescharacteristic
159C8—42no—
160C8—42no—
161C8—42no—
162C8—81no—
163C22—21no—
164C22—21no—
165C22—24no—
166C22—42no—
167C22—42no—
168C22—82no—
169C22—82no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22#38611yescharacteristic
174C22#46511yescharacteristic
175C22#46611yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—42no—
181C4—81no—
182C4—161no—
183C4C2×#10211yescharacteristic
184C4#41611yescharacteristic
185C4#46511yescharacteristic
186C4#46611yescharacteristic
187C2—21no—
188C2—21no—
189C2—44no—
190C2—161no—
191C2#188111yescharacteristic
192C2#189311yescharacteristic
193C2#190411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3036C2
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. #2972C2semidirect
3. #1804C2semidirect
4. #1517C2semidirect
5. #1480C2semidirect
6. #1301C2semidirect
7. #1286C2semidirect
12. #461C4semidirect
13. #459C4semidirect

Maximal subgroups

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

subgroupindex
#30362
#29722
#18042
#15172
#14802
#13012
#12862