Finite Group Databasebeta

← #12227

Subgroups — #12227

Order 256.

260 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 35 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#2463C211yesmaximal, characteristic
2#2062C211yesmaximal, characteristic, complemented
3#1602C211yesmaximal, characteristic, complemented
4#1596C211yesmaximal, characteristic
5#1351C211yesmaximal, characteristic, complemented
6#1343C211yesmaximal, characteristic
7#1326C211yesmaximal, characteristic, complemented
8#543C2211yescharacteristic
9C2×C22⋊Q8—21no—
10C2×C4×D4—21no—
11#451—21no—
12#450—21no—
13#394—21no—
14#393C2211yescharacteristic
15#392C2211yescharacteristic
16#388—21no—
17#387—21no—
18#386—21no—
19#385—21no—
20#385C2211yescharacteristic
21#342—21no—
22#342C2211yescharacteristic
23#328—21no—
24#328—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×D4—21no—
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C23×C4D411yescharacteristic
32C2×Q16—41no—
33C2×SD16—41no—
34C22⋊Q8—21no—
35C22⋊Q8—21no—
36C22⋊Q8—41no—
37C22⋊Q8—41no—
38C22⋊Q8—44no—
39C4×D4—44no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—41no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C22⋊C4D411yescharacteristic
52C2×C22⋊C4C2311yescharacteristic
53C2×C42—21no—
54#103—41no—
55#102—41no—
56C22⋊C8—21no—
57C22⋊C8—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—41no—
62#95D411yescharacteristic
63#95D411yescharacteristic
64C24—21no—
65C24C2×D411yescharacteristic
66C2×Q8—21no—
67C2×Q8—21no—
68C2×Q8—42no—
69C2×Q8—41no—
70C2×D4—42no—
71C2×D4—41no—
72C2×D4—42no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—42no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—81no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91Q16—82no—
92SD16—82no—
93C2×C8—41no—
94C2×C8—41no—
95C4⋊C4—24no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—42no—
99C4⋊C4—42no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—82no—
103C22⋊C4—22no—
104C22⋊C4—21no—
105C22⋊C4—21no—
106C22⋊C4—41no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42—42no—
116C23—21no—
117C23—21no—
118C23—22no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23#12011yescharacteristic
128C23#13511yescharacteristic
129C23#13711yescharacteristic
130Q8—42no—
131Q8—81no—
132Q8—81no—
133Q8—81no—
134D4—42no—
135D4—44no—
136D4—44no—
137D4—81no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
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—42no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4—84no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C8—81no—
168C8—81no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22#45211yescharacteristic
186C22#45611yescharacteristic
187C22B(4,𝔽2)11yescharacteristic
188C4—42no—
189C4—41no—
190C4—42no—
191C4—42no—
192C4—41no—
193C4—81no—
194C4—81no—
195C4—82no—
196C4—81no—
197C4—81no—
198C2—21no—
199C2—21no—
200C2—41no—
201C2—41no—
202C2—82no—
203C2#160311yescharacteristic
204C2#219611yescharacteristic
205C2#219711yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#392C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2062C2semidirect
3. #1602C2semidirect
5. #1351C2semidirect
7. #1326C2semidirect

Maximal subgroups

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

subgroupindex
#24632
#20622
#16022
#15962
#13512
#13432
#13262