Finite Group Databasebeta

← #27978

Subgroups — #27978

Order 256.

456 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 131 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#2850C211yesmaximal, characteristic
2C2×#384C211yesmaximal, characteristic, complemented
3#2072C212yesmaximal, complemented
4#2071C212yesmaximal, complemented
5#1959C212yesmaximal, complemented
6#1874C212yesmaximal
7#1872C212yesmaximal, complemented
8#1836C212yesmaximal, complemented
9#1749C211yesmaximal, characteristic, complemented
10C2×#124C2211yescharacteristic
11C2×C22⋊Q8C2211yescharacteristic
12C2×#121C2211yescharacteristic
13C22×C4⋊C4—21no—
14C22×C4⋊C4—21no—
15C22×C4⋊C4C2212yes—
16C22×C42C2211yescharacteristic
17C2×#103C2214yes—
18C2×#102C2214yes—
19C2×C22⋊C8C2212yes—
20C2×#97C2212yes—
21#391C2211yescharacteristic
22#390C2211yescharacteristic
23#388—21no—
24#388—21no—
25#386—21no—
26#386—21no—
27#384—21no—
28#384—21no—
29#384C2212yes—
30#384C2212yes—
31#382C2211yescharacteristic
32#340C2214yes—
33#340C2214yes—
34#336C2212yes—
35C22×Q8C2311yescharacteristic
36C22×D4—21no—
37C22×D4C2311yescharacteristic
38C23×C4—22no—
39C23×C4D411yescharacteristic
40C23×C4D411yescharacteristic
41C23×C4C2311yescharacteristic
42C22×C8C2314yes—
43#124—24no—
44C22⋊Q8—24no—
45#121—24no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—44no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4—44no—
55C2×C4⋊C4C2314yes—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C42—24no—
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#103—44no—
70#102—44no—
71C22⋊C8—24no—
72#97—24no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—21no—
77C24—21no—
78C24C2×D411yescharacteristic
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—22no—
82C2×D4—22no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—42no—
86C2×D4—42no—
87C2×D4—42no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—24no—
100C22×C4—44no—
101C22×C4—44no—
102C22×C4—44no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4#4114yes—
109C22×C4C2411yescharacteristic
110C2×C8—24no—
111C2×C8—24no—
112C2×C8—24no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—24no—
116C4⋊C4—24no—
117C4⋊C4—44no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—44no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C22⋊C4—42no—
124C22⋊C4—42no—
125C22⋊C4—42no—
126C22⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C42—22no—
130C42—22no—
131C42—22no—
132C42#4114yes—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23#12412yes—
145C23AGL(1,ℤ/8)12yes—
146C23#13712yes—
147C23C22×D411yescharacteristic
148Q8—44no—
149D4—44no—
150D4—84no—
151C2×C4—28no—
152C2×C4—28no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—28no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—88no—
177C2×C4#12312yes—
178C2×C4#12312yes—
179C2×C4#12412yes—
180C2×C4AGL(1,ℤ/8)12yes—
181C2×C4#13712yes—
182C2×C4C2×#4112yes—
183C2×C4C2×#4112yes—
184C2×C4H(5,𝔽2)12yes—
185C8—44no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22#48914yes—
200C22C2×#12311yescharacteristic
201C22C2×#12411yescharacteristic
202C22#53511yescharacteristic
203C22C2×AGL(1,ℤ/8)11yescharacteristic
204C22C2×AGL(1,ℤ/8)11yescharacteristic
205C22C2×#13711yescharacteristic
206C22C2×#13711yescharacteristic
207C4—22no—
208C4—22no—
209C4—22no—
210C4—22no—
211C4—22no—
212C4—81no—
213C4—81no—
214C4—84no—
215C4—81no—
216C4#55314yes—
217C2—24no—
218C2—161no—
219C2#250411yescharacteristic
220C2#309211yescharacteristic
221C2#309211yescharacteristic
222C2C2×#48911yescharacteristic
223C2#314111yescharacteristic
224C2#321711yescharacteristic
225C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#384C2semidirect
3. #2072C2semidirect
4. #2071C2semidirect
5. #1959C2semidirect
7. #1872C2semidirect
8. #1836C2semidirect
9. #1749C2semidirect

Maximal subgroups

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

subgroupindex
#28502
C2×#3842
#20722
#20712
#19592
#18742
#18722
#18362
#17492