Finite Group Databasebeta

← #11726

Subgroups — #11726

Order 256.

303 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 53 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#2397C211yesmaximal, characteristic
2C4×#99C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1352C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C2×C4×D4—21no—
7C2×C4×D4—21no—
8C2×C4×D4C411yescharacteristic, complemented
9C2×C4×D4C2211yescharacteristic
10C2×#99—21no—
11C2×C22⋊C8—21no—
12#399—21no—
13#399C411yescharacteristic, complemented
14#399C2211yescharacteristic
15#392—21no—
16#392C411yescharacteristic, complemented
17#392C2211yescharacteristic
18C4×C22⋊C4—21no—
19#376C411yescharacteristic, complemented
20#342C2212yes—
21#324—21no—
22#323C2212yes—
23C22×D4—21no—
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C22×D4D411yescharacteristic
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C22×C8—41no—
31C4×D4—41no—
32C4×D4—41no—
33C4×D4—44no—
34C4×D4—44no—
35C4×D4—41no—
36C4×D4—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4D411yescharacteristic
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C22⋊C4C2311yescharacteristic
49C2×C42—21no—
50C2×C42—41no—
51C2×C42C2×C411yescharacteristic
52#99—24no—
53C22⋊C8—22no—
54C22⋊C8—41no—
55#95—21no—
56#95—41no—
57#95C2×C411yescharacteristic
58#95C2×C411yescharacteristic
59C24—21no—
60C24C22⋊C411yescharacteristic
61C24C2×D411yescharacteristic
62C2×D4—22no—
63C2×D4—22no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—42no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4—42no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22×C411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C2×C8—42no—
93C2×C8—82no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C22⋊C4—24no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—42no—
102C22⋊C4—42no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C42—24no—
108C42—21no—
109C42—21no—
110C42—82no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23—22no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23#10412yes—
127C23C2×C22⋊C411yescharacteristic
128C23#13512yes—
129D4—44no—
130D4—42no—
131D4—48no—
132D4—48no—
133D4—42no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—44no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4#9911yescharacteristic
163C2×C4#9911yescharacteristic
164C8—82no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—22no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22C2×#9911yescharacteristic
185C22#41611yescharacteristic
186C22C2×#10411yescharacteristic
187C4—21no—
188C4—42no—
189C4—42no—
190C4—41no—
191C4—41no—
192C4—41no—
193C4—41no—
194C4—84no—
195C4—81no—
196C4#35312yes—
197C2—21no—
198C2—21no—
199C2—42no—
200C2—42no—
201C2—82no—
202C2#150111yescharacteristic
203C2#212011yescharacteristic
204C2C2×#35311yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
socleC22C2×#99
prime radical#2397C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #1459C2semidirect
4. #1352C2semidirect
5. #1343C2semidirect
8. C2×C4×D4C4semidirect
13. #399C4semidirect
16. #392C4semidirect
19. #376C4semidirect

Maximal subgroups

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

subgroupindex
#23972
C4×#992
#14592
#13522
#13432