Finite Group Databasebeta

← #9409

Subgroups — #9409

Order 256.

420 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 65 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#2539C211yesmaximal, characteristic
2#2273C211yesmaximal, characteristic, complemented
3#1507C212yesmaximal, complemented
4#1462C211yesmaximal, characteristic, complemented
5#1297C212yesmaximal, complemented
6C2×#123—21no—
7C2×#121C2211yescharacteristic
8C2×#120—21no—
9C22×C22⋊C4—21no—
10C22×C22⋊C4C2211yescharacteristic
11#407C2212yes—
12#399C412yescomplemented
13#392C412yescomplemented
14#390—22no—
15C4×C22⋊C4—21no—
16C4×C22⋊C4—24no—
17C4×C22⋊C4C2211yescharacteristic
18C4×C22⋊C4C2212yes—
19#327—24no—
20#323—22no—
21C25D411yescharacteristic
22C22×D4—21no—
23C22×D4—22no—
24C22×D4C2×C411yescharacteristic, complemented
25C23×C4—22no—
26C23×C4D411yescharacteristic
27C23×C4C2311yescharacteristic
28C2×M16—22no—
29#123—44no—
30#121—22no—
31#121—22no—
32#120—44no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4C2×C411yescharacteristic, complemented
35C2×C22⋊C4—28no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4C2×C412yes—
45C2×C22⋊C4D412yes—
46C2×C42—22no—
47C2×C42—22no—
48C2×C42—21no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51#95—22no—
52#95C2×C412yescomplemented
53#95D414yescomplemented
54C24—21no—
55C24—24no—
56C24—21no—
57C24—21no—
58C24—42no—
59C24—81no—
60C24C22⋊C412yes—
61C24C2×D411yescharacteristic
62C2×D4—22no—
63C2×D4—44no—
64C2×D4—42no—
65C2×D4—42no—
66C2×D4—44no—
67C2×D4—41no—
68C2×D4—41no—
69C2×D4—82no—
70C2×D4—81no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—24no—
74C22×C4—28no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—42no—
78C22×C4—81no—
79C22×C4—82no—
80C22×C4—82no—
81C22×C4C22⋊C412yes—
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D412yes—
84C22×C4C2×D411yescharacteristic
85C22×C4#4112yes—
86M16—82no—
87C2×C8—42no—
88C4⋊C4—22no—
89C4⋊C4—44no—
90C22⋊C4—28no—
91C22⋊C4—28no—
92C22⋊C4—24no—
93C22⋊C4—24no—
94C22⋊C4—24no—
95C22⋊C4—44no—
96C22⋊C4—44no—
97C22⋊C4—42no—
98C42—82no—
99C42—81no—
100C42—82no—
101C23—22no—
102C23—22no—
103C23—24no—
104C23—22no—
105C23—22no—
106C23—22no—
107C23—22no—
108C23—41no—
109C23—42no—
110C23—42no—
111C23—42no—
112C23—41no—
113C23—41no—
114C23—42no—
115C23—42no—
116C23—41no—
117C23—42no—
118C23—81no—
119C23—81no—
120C23—88no—
121C23#10414yes—
122C23C2×C22⋊C411yescharacteristic
123C23#12011yescharacteristic
124C23#12311yescharacteristic
125D4—82no—
126D4—88no—
127D4—82no—
128D4—82no—
129C2×C4—28no—
130C2×C4—24no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—44no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—44no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—81no—
146C2×C4—88no—
147C2×C4C4×D412yes—
148C2×C4#12112yes—
149C8—82no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—24no—
156C22—21no—
157C22—21no—
158C22—44no—
159C22—44no—
160C22—44no—
161C22—44no—
162C22—42no—
163C22—42no—
164C22—82no—
165C22—82no—
166C22—82no—
167C22—84no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22#38611yescharacteristic
172C22C2×#10412yes—
173C22#45312yes—
174C4—22no—
175C4—22no—
176C4—48no—
177C4—44no—
178C4—42no—
179C4—81no—
180C2—22no—
181C2—22no—
182C2—41no—
183C2—44no—
184C2—161no—
185C2#188711yescharacteristic
186C2#189611yescharacteristic
187C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2539C2
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. #2273C2semidirect
3. #1507C2semidirect
4. #1462C2semidirect
5. #1297C2semidirect
12. #399C4semidirect
13. #392C4semidirect
24. C22×D4C2×C4semidirect
34. C2×C4⋊C4C2×C4semidirect
52. #95C2×C4semidirect
53. #95D4semidirect

Maximal subgroups

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

subgroupindex
#25392
#22732
#15072
#14622
#12972