Finite Group Databasebeta

← #9199

Subgroups — #9199

Order 256.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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#2626C211yesmaximal, characteristic
2#2315C211yesmaximal, characteristic
3#1520C212yesmaximal, complemented
4C2×#323C211yesmaximal, characteristic
5#1289C212yesmaximal, complemented
6C2×C22⋊Q8C2211yescharacteristic
7C22×C22⋊C4—21no—
8C22×C22⋊C4C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10#399C412yescomplemented
11#396—22no—
12#393C412yescomplemented
13#385—21no—
14#385—24no—
15#380—21no—
16#380C2211yescharacteristic
17#380C2212yes—
18#379—21no—
19#328—24no—
20#323—22no—
21C25D411yescharacteristic
22C22×Q8C2×C411yescharacteristic, complemented
23C23×C4—22no—
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C22⋊Q8—22no—
28C22⋊Q8—22no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4C2×C411yescharacteristic, complemented
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—42no—
44C2×C22⋊C4—42no—
45C2×C22⋊C4C2×C412yes—
46C2×C22⋊C4D412yes—
47C22⋊C8—22no—
48C22⋊C8—22no—
49#95—22no—
50#95—22no—
51#95C2×C412yescomplemented
52#95D414yes—
53C24—21no—
54C24—21no—
55C24—21no—
56C24—21no—
57C24—81no—
58C24—81no—
59C24C22⋊C412yes—
60C24SD1612yes—
61C24Q1612yes—
62C24C2×D411yescharacteristic
63C2×Q8—22no—
64C2×Q8—81no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—21no—
70C22×C4—24no—
71C22×C4—44no—
72C22×C4—81no—
73C22×C4—81no—
74C22×C4—82no—
75C22×C4—82no—
76C22×C4C22⋊C412yes—
77C22×C4C22×C411yescharacteristic
78C22×C4C2×D412yes—
79C22×C4C2×D411yescharacteristic
80C22×C4#4112yes—
81C2×C8—42no—
82C2×C8—82no—
83C4⋊C4—22no—
84C4⋊C4—42no—
85C4⋊C4—81no—
86C4⋊C4—82no—
87C4⋊C4—82no—
88C4⋊C4—82no—
89C22⋊C4—24no—
90C22⋊C4—24no—
91C22⋊C4—44no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—42no—
95C22⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—41no—
98C22⋊C4—82no—
99C22⋊C4—82no—
100C23—22no—
101C23—22no—
102C23—22no—
103C23—22no—
104C23—22no—
105C23—22no—
106C23—41no—
107C23—41no—
108C23—42no—
109C23—42no—
110C23—42no—
111C23—42no—
112C23—81no—
113C23—81no—
114C23—82no—
115C23—82no—
116C23—82no—
117C23—82no—
118C23#10314yes—
119C23C2×C22⋊C411yescharacteristic
120C23#12011yescharacteristic
121C23#12311yescharacteristic
122C23C2×SD1611yescharacteristic
123C23C2×Q1611yescharacteristic
124C23AGL(1,ℤ/8)11yescharacteristic
125C23#13711yescharacteristic
126Q8—82no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—44no—
139C2×C4—44no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—44no—
144C2×C4—81no—
145C2×C4—88no—
146C2×C4C4×D412yes—
147C2×C4#12112yes—
148C8—82no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—24no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22—41no—
160C22—44no—
161C22—44no—
162C22—44no—
163C22—42no—
164C22—42no—
165C22—82no—
166C22—82no—
167C22—82no—
168C22#38611yescharacteristic
169C22C2×#10311yescharacteristic
170C22#41711yescharacteristic
171C22#45011yescharacteristic
172C22#45111yescharacteristic
173C22#48311yescharacteristic
174C22#48411yescharacteristic
175C4—22no—
176C4—22no—
177C4—42no—
178C4—82no—
179C4—84no—
180C4—81no—
181C4—81no—
182C2—21no—
183C2—21no—
184C2—21no—
185C2—21no—
186C2—21no—
187C2—21no—
188C2—44no—
189C2#187211yescharacteristic
190C2#191111yescharacteristic
191C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
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
3. #1520C2semidirect
5. #1289C2semidirect
10. #399C4semidirect
12. #393C4semidirect
22. C22×Q8C2×C4semidirect
34. C2×C4⋊C4C2×C4semidirect
51. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#26262
#23152
#15202
C2×#3232
#12892