Finite Group Databasebeta

← #9367

Subgroups — #9367

Order 256.

271 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 61 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#3094C211yesmaximal, characteristic
2#2346C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic
4#1826C211yesmaximal, characteristic, complemented
5#1507C211yesmaximal, characteristic
6#1305C211yesmaximal, characteristic, complemented
7#1297C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×#117C2211yescharacteristic
10#482—22no—
11#481—21no—
12#480—21no—
13#428C412yescomplemented
14#418C412yescomplemented
15C2×#99—22no—
16#407C2211yescharacteristic
17#407C2211yescharacteristic
18#389—21no—
19#388—22no—
20#382—21no—
21#380C2211yescharacteristic
22C4×C4⋊C4—21no—
23C4×C22⋊C4C2211yescharacteristic
24C4×C22⋊C4C2211yescharacteristic
25#376—21no—
26#339—22no—
27#327—21no—
28#327—21no—
29C22×D4—21no—
30C22×D4C2×C411yescharacteristic
31C23×C4C2311yescharacteristic
32C2×M16—21no—
33C2×M16C2×C412yes—
34#121—21no—
35#121—21no—
36#121—21no—
37#121—21no—
38#117—21no—
39#117—21no—
40#117C2×C412yes—
41#117D412yes—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4C2×C411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C22⋊C4D412yes—
52C2×C42—21no—
53C2×C42—22no—
54C2×C42—21no—
55C2×C42—21no—
56#107—22no—
57#106—22no—
58#102—22no—
59#102—22no—
60#99—44no—
61C22⋊C8—21no—
62C22⋊C8—21no—
63C22⋊C8—21no—
64C22⋊C8—21no—
65#95—22no—
66#95—22no—
67#95—21no—
68#95—21no—
69#95D411yescharacteristic
70#95D411yescharacteristic
71C24—21no—
72C24C2×D411yescharacteristic
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—42no—
76C2×D4—42no—
77C2×D4—42no—
78C2×D4—41no—
79C2×D4—41no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—42no—
87C22×C4—41no—
88C22×C4C22×C411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4#4111yescharacteristic
93C22×C4#4111yescharacteristic
94M16—42no—
95M16—42no—
96C2×C8—22no—
97C2×C8—41no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—42no—
103C4⋊C4—82no—
104C4⋊C4C22⋊C412yes—
105C4⋊C4C22⋊C412yes—
106C22⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—42no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—42no—
114C42—22no—
115C42—22no—
116C42—42no—
117C42—44no—
118C42—42no—
119C23—21no—
120C23—21no—
121C23—41no—
122C23—41no—
123C23—42no—
124C23—41no—
125C23—81no—
126C23—81no—
127C23C4×D411yescharacteristic
128C23#12011yescharacteristic
129C23#12111yescharacteristic
130D4—42no—
131D4—81no—
132D4—82no—
133D4—81no—
134D4—81no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4C2×C22⋊C411yescharacteristic
161C2×C4C4×D411yescharacteristic
162C2×C4#12111yescharacteristic
163C2×C4#12311yescharacteristic
164C8—42no—
165C8—81no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—162no—
180C22#38611yescharacteristic
181C22#45311yescharacteristic
182C22#45411yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—42no—
187C4—42no—
188C4—42no—
189C4—81no—
190C4—82no—
191C4—82no—
192C4—81no—
193C4#41012yes—
194C2—21no—
195C2—21no—
196C2—41no—
197C2—41no—
198C2—161no—
199C2#188311yescharacteristic
200C2#189911yescharacteristic
201C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2346C2semidirect
4. #1826C2semidirect
6. #1305C2semidirect
7. #1297C2semidirect
13. #428C4semidirect
14. #418C4semidirect

Maximal subgroups

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

subgroupindex
#30942
#23462
#19642
#18262
#15072
#13052
#12972