Finite Group Databasebeta

← #12274

Subgroups — #12274

Order 256.

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

normal subgroups only

This group has 206 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#2828C211yesmaximal, characteristic
2#2027C211yesmaximal, characteristic, complemented
3#1617C211yesmaximal, characteristic, complemented
4#1599C211yesmaximal, characteristic, complemented
5#1353C211yesmaximal, characteristic, complemented
6#1345C211yesmaximal, characteristic, complemented
7#1318C211yesmaximal, characteristic, complemented
8#544C2211yescharacteristic
9#484—21no—
10#450—21no—
11#418—21no—
12#400C2211yescharacteristic
13#397—21no—
14#394—21no—
15#394—21no—
16#394C2211yescharacteristic
17#392—21no—
18#392—21no—
19#390C2211yescharacteristic
20#388—21no—
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×D4—21no—
28C22×D4—21no—
29C22×D4D411yescharacteristic
30C22×D4D411yescharacteristic
31C2×SD16—41no—
32C2×M16—21no—
33C4⋊Q8—41no—
34#125—41no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37#117—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4C2311yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C42D411yescharacteristic
50#107—41no—
51#103—41no—
52#102—41no—
53#102—41no—
54#102—41no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95D411yescharacteristic
61#95D411yescharacteristic
62C24—21no—
63C24—21no—
64C2×Q8—21no—
65C2×Q8—21no—
66C2×D4—21no—
67C2×D4—21no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—81no—
72C2×D4—81no—
73C2×D4—81no—
74C2×D4—81no—
75C2×D4—81no—
76C2×D4—81no—
77C2×D4—81no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85SD16—82no—
86M16—41no—
87M16—41no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—81no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—81no—
104C22⋊C4—81no—
105C22⋊C4—81no—
106C22⋊C4—81no—
107C22⋊C4—81no—
108C22⋊C4—81no—
109C22⋊C4—81no—
110C42—41no—
111C42—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23#12011yescharacteristic
127Q8—81no—
128Q8—81no—
129D4—42no—
130D4—42no—
131D4—81no—
132D4—81no—
133D4—84no—
134D4—84no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
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#13511yescharacteristic
156C2×C4#13711yescharacteristic
157C8—81no—
158C8—81no—
159C8—81no—
160C22—21no—
161C22—21no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#45211yescharacteristic
177C22#45511yescharacteristic
178C22B(4,𝔽2)11yescharacteristic
179C4—21no—
180C4—21no—
181C4—41no—
182C4—41no—
183C4—81no—
184C4—81no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4—81no—
189C2—21no—
190C2—21no—
191C2—161no—
192C2—161no—
193C2#160611yescharacteristic
194C2#220011yescharacteristic
195C2#220211yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#2828C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2027C2semidirect
3. #1617C2semidirect
4. #1599C2semidirect
5. #1353C2semidirect
6. #1345C2semidirect
7. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#28282
#20272
#16172
#15992
#13532
#13452
#13182