Finite Group Databasebeta

← #9112

Subgroups — #9112

Order 256.

316 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#431C211yesmaximal, characteristic
2#2855C211yesmaximal, characteristic
3C2×#328C212yesmaximal
4#1434C211yesmaximal, characteristic
5#1294C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C22×C42C2211yescharacteristic
8#431—24no—
9C2×C4⋊C8—21no—
10C2×C22⋊C8C2212yes—
11C2×C4×C8—21no—
12#391C411yescharacteristic, complemented
13#391C411yescharacteristic, complemented
14#389C411yescharacteristic, complemented
15#389C411yescharacteristic, complemented
16#387—21no—
17#387—21no—
18#385—21no—
19#381—21no—
20#380C2211yescharacteristic
21C2×#95—21no—
22C2×#95—21no—
23C2×#95C2212yes—
24#328—24no—
25C22×Q8C2×C411yescharacteristic
26C23×C4—22no—
27C23×C4D411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4C2311yescharacteristic
30C22×C8—22no—
31C22⋊Q8—22no—
32C22⋊Q8—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C42—22no—
42C2×C42—22no—
43C2×C42C2×C411yescharacteristic
44C2×C42C2×C411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C4⋊C8—24no—
48C22⋊C8—22no—
49C22⋊C8D414yes—
50C4×C8—24no—
51#95—24no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—22no—
60#95—22no—
61#95—42no—
62C24C2×D411yescharacteristic
63C2×Q8—22no—
64C2×Q8—41no—
65C2×Q8—41no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—24no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4SD1612yes—
89C22×C4Q1612yes—
90C22×C4C22×C411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D412yes—
93C22×C4#4112yes—
94C2×C8—24no—
95C2×C8—42no—
96C2×C8—42no—
97C4⋊C4—22no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C42—22no—
107C42—22no—
108C42—21no—
109C42—21no—
110C42—21no—
111C42—21no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23C2×C22⋊C411yescharacteristic
117C23#12112yes—
118C23C2×SD1611yescharacteristic
119C23C2×Q1611yescharacteristic
120C23#13512yes—
121Q8—82no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—24no—
133C2×C4—24no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—44no—
140C2×C4—44no—
141C2×C4—44no—
142C2×C4—44no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—82no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4#10312yes—
160C2×C4#10312yes—
161C2×C4#10412yes—
162C2×C4C4×D412yes—
163C2×C4#12011yescharacteristic
164C2×C4#12311yescharacteristic
165C8—44no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22#38611yescharacteristic
175C22#41011yescharacteristic
176C22#41011yescharacteristic
177C22C2×#10311yescharacteristic
178C22#41611yescharacteristic
179C22C2×#10411yescharacteristic
180C22#42111yescharacteristic
181C22#46512yes—
182C22#46712yes—
183C4—22no—
184C4—22no—
185C4—22no—
186C4—21no—
187C4—22no—
188C4—21no—
189C4—41no—
190C4—84no—
191C4—81no—
192C4—81no—
193C4—81no—
194C4—81no—
195C2—22no—
196C2—22no—
197C2#150011yescharacteristic
198C2#150311yescharacteristic
199C2#151811yescharacteristic
200C2#152111yescharacteristic
201C2#187111yescharacteristic
202C2#189211yescharacteristic
203C2#189911yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
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
12. #391C4semidirect
13. #391C4semidirect
14. #389C4semidirect
15. #389C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4312
#28552
C2×#3282
#14342
#12942