Finite Group Databasebeta

← #11729

Subgroups — #11729

Order 256.

243 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 51 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#2635C211yesmaximal, characteristic
2C4×#99C211yesmaximal, characteristic, complemented
3#1460C211yesmaximal, characteristic, complemented
4#1352C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C2×#121C411yescharacteristic, complemented
7C2×C4×D4C2211yescharacteristic
8#431—21no—
9C2×#99—21no—
10#400C411yescharacteristic, complemented
11#399C2211yescharacteristic
12#394—21no—
13#394C411yescharacteristic, complemented
14#392—21no—
15#392C2211yescharacteristic
16#388—21no—
17#386—21no—
18#381C411yescharacteristic, complemented
19C4×C22⋊C4—21no—
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C22×D4—21no—
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C22×D4D411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29#121—41no—
30#121—41no—
31C4×D4—41no—
32C4×D4—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4D411yescharacteristic
44C2×C22⋊C4C2311yescharacteristic
45C2×C42—41no—
46C2×C42D411yescharacteristic
47C4⋊C8—41no—
48#99—24no—
49C22⋊C8—22no—
50C4×C8—41no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—41no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24—21no—
58C24C22⋊C411yescharacteristic
59C24C2×D411yescharacteristic
60C2×D4—22no—
61C2×D4—22no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—42no—
65C2×D4—42no—
66C2×D4—81no—
67C2×D4—81no—
68C2×D4—81no—
69C2×D4—81no—
70C2×D4—81no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—42no—
79C22×C4—81no—
80C22×C4—81no—
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22×C411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C2×C8—42no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—81no—
90C22⋊C4—24no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—42no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—81no—
98C22⋊C4—81no—
99C22⋊C4—81no—
100C22⋊C4—81no—
101C42—21no—
102C42—21no—
103C42—82no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23C2×C22⋊C411yescharacteristic
121C23#13512yes—
122D4—44no—
123D4—82no—
124D4—84no—
125D4—84no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—44no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4#9911yescharacteristic
152C2×C4#9911yescharacteristic
153C2×C4#10412yes—
154C8—82no—
155C22—21no—
156C22—21no—
157C22—22no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22C2×#9911yescharacteristic
175C22#41611yescharacteristic
176C22C2×#10411yescharacteristic
177C4—21no—
178C4—21no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—81no—
184C4—81no—
185C4—84no—
186C4—81no—
187C2—21no—
188C2—21no—
189C2—42no—
190C2—81no—
191C2—161no—
192C2#150111yescharacteristic
193C2#212111yescharacteristic
194C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2635C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#99C2semidirect
3. #1460C2semidirect
4. #1352C2semidirect
5. #1343C2semidirect
6. C2×#121C4semidirect
10. #400C4semidirect
13. #394C4semidirect
18. #381C4semidirect

Maximal subgroups

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

subgroupindex
#26352
C4×#992
#14602
#13522
#13432