Finite Group Databasebeta

← #9156

Subgroups — #9156

Order 256.

346 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 83 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#2854C211yesmaximal, characteristic
2C2×#382C211yesmaximal, characteristic, complemented
3#1817C211yesmaximal, characteristic, complemented
4C2×#327C212yesmaximal
5#1294C212yesmaximal, complemented
6C2×#123—21no—
7C2×#121C2211yescharacteristic
8C22×C4⋊C4—21no—
9C22×C42C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11#390C412yescomplemented
12#389C412yescomplemented
13#388—22no—
14#386—21no—
15#382—24no—
16#380C2211yescharacteristic
17C2×#95—21no—
18C2×#95C2212yes—
19#336—22no—
20#327—24no—
21C22×D4—21no—
22C22×D4C2×C411yescharacteristic
23C23×C4—22no—
24C23×C4D411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4C2311yescharacteristic
27C22×C8—22no—
28#123—42no—
29#123—42no—
30#121—22no—
31#121—22no—
32C2×C4⋊C4—24no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—42no—
35C2×C4⋊C4—42no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C412yes—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—21no—
42C2×C42—22no—
43C2×C42—22no—
44C2×C42C2×C412yes—
45C2×C42D412yes—
46C22⋊C8—22no—
47C22⋊C8—22no—
48#95—24no—
49#95—22no—
50#95—22no—
51#95—21no—
52#95D414yes—
53C24—21no—
54C24C2×D411yescharacteristic
55C2×D4—22no—
56C2×D4—42no—
57C2×D4—41no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4—41no—
62C2×D4—41no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—24no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4—44no—
75C22×C4—44no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4C22⋊C412yes—
79C22×C4C22⋊C412yes—
80C22×C4D812yes—
81C22×C4SD1612yes—
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D412yes—
85C22×C4#4112yes—
86C2×C8—44no—
87C2×C8—42no—
88C4⋊C4—22no—
89C4⋊C4—42no—
90C4⋊C4—48no—
91C4⋊C4—44no—
92C22⋊C4—44no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—44no—
96C22⋊C4—42no—
97C42—24no—
98C42—24no—
99C23—22no—
100C23—22no—
101C23—41no—
102C23—42no—
103C23—41no—
104C23—41no—
105C23—42no—
106C23C2×C22⋊C411yescharacteristic
107C23#12011yescharacteristic
108C23#12311yescharacteristic
109C23C2×D811yescharacteristic
110C23C2×SD1611yescharacteristic
111C23AGL(1,ℤ/8)11yescharacteristic
112C23#13711yescharacteristic
113D4—82no—
114D4—81no—
115D4—81no—
116D4—81no—
117D4—81no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—44no—
134C2×C4—44no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—44no—
142C2×C4—84no—
143C2×C4—84no—
144C2×C4#10214yes—
145C2×C4C4×D412yes—
146C2×C4#12112yes—
147C8—82no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—22no—
153C22—22no—
154C22—22no—
155C22—22no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22—82no—
160C22—82no—
161C22#38611yescharacteristic
162C22#41012yes—
163C22C2×#10211yescharacteristic
164C22#41711yescharacteristic
165C22#42112yes—
166C22#44711yescharacteristic
167C22#44811yescharacteristic
168C22#48011yescharacteristic
169C22#48111yescharacteristic
170C4—24no—
171C4—22no—
172C4—22no—
173C4—42no—
174C4—84no—
175C4—81no—
176C4—82no—
177C2—21no—
178C2—21no—
179C2—21no—
180C2—21no—
181C2—161no—
182C2#151212yes—
183C2#151712yes—
184C2#187411yescharacteristic
185C2#189711yescharacteristic
186C2#189911yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#382C2semidirect
3. #1817C2semidirect
5. #1294C2semidirect
11. #390C4semidirect
12. #389C4semidirect

Maximal subgroups

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

subgroupindex
#28542
C2×#3822
#18172
C2×#3272
#12942