Finite Group Databasebeta

← #29109

Subgroups — #29109

Order 256.

561 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 127 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#2857C211yesmaximal, characteristic, complemented
2C2×#390C211yesmaximal, characteristic
3#2071C212yesmaximal, complemented
4#2023C212yesmaximal, complemented
5#1936C214yesmaximal, complemented
6#1824C212yesmaximal, complemented
7#1786C212yesmaximal
8#1749C211yesmaximal, characteristic, complemented
9C23×D4C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic, complemented
11C2×C22⋊Q8—21no—
12C22×C4⋊C4C2211yescharacteristic
13C22×C22⋊C4—21no—
14C22×C22⋊C4—21no—
15C22×C42C2211yescharacteristic
16C2×#102C2214yes—
17C2×#102C2214yes—
18C2×C22⋊C8C2212yes—
19C2×#97C2212yescomplemented
20#391—21no—
21#390—21no—
22#390—21no—
23#390C2212yes—
24#390C2212yes—
25#389—21no—
26#389C2212yescomplemented
27#384C2211yescharacteristic, complemented
28#380—21no—
29#380C2212yes—
30#340C2214yescomplemented
31#340C2214yescomplemented
32#336C2212yescomplemented
33C25—21no—
34C22×Q8—21no—
35C22×D4—24no—
36C22×D4—21no—
37C22×D4—24no—
38C22×D4—42no—
39C22×D4C2312yes—
40C23×C4—21no—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C22×C8C2314yes—
45C4⋊Q8—42no—
46C22⋊Q8—44no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—44no—
62C2×C22⋊C4—44no—
63C2×C42—42no—
64C2×C42D411yescharacteristic
65C2×C42D411yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#102—28no—
69#102—44no—
70C22⋊C8—24no—
71#97—42no—
72#95—22no—
73#95—22no—
74C24—24no—
75C24—22no—
76C24—42no—
77C24—42no—
78C24—42no—
79C24—42no—
80C24—42no—
81C24—42no—
82C24C2×D411yescharacteristic
83C2×Q8—42no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×D4—24no—
87C2×D4—24no—
88C2×D4—22no—
89C2×D4—24no—
90C2×D4—416no—
91C2×D4—42no—
92C2×D4—42no—
93C2×D4#4114yes—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4#4114yes—
114C22×C4C2411yescharacteristic
115C2×C8—24no—
116C2×C8—24no—
117C2×C8—24no—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C22⋊C4—44no—
127C22⋊C4—44no—
128C22⋊C4—44no—
129C22⋊C4—44no—
130C22⋊C4—44no—
131C42—22no—
132C42—22no—
133C42—41no—
134C42—41no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—24no—
140C23—48no—
141C23—48no—
142C23—48no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—84no—
150C23—84no—
151C23—84no—
152C23—84no—
153C23#12412yes—
154C23AGL(1,ℤ/8)12yes—
155C23#13712yes—
156C23C22×D411yescharacteristic
157Q8—84no—
158D4—28no—
159D4—48no—
160D4—44no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—28no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—84no—
186C2×C4#12412yes—
187C2×C4AGL(1,ℤ/8)12yes—
188C2×C4AGL(1,ℤ/8)12yes—
189C2×C4C2×#4112yes—
190C2×C4C2×#4112yes—
191C2×C4H(5,𝔽2)12yes—
192C8—44no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—44no—
200C22—44no—
201C22—44no—
202C22—816no—
203C22—82no—
204C22—82no—
205C22—82no—
206C22—82no—
207C22#48914yes—
208C22#52511yescharacteristic
209C22C2×#12411yescharacteristic
210C22#53411yescharacteristic
211C22C2×AGL(1,ℤ/8)11yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C22C2×AGL(1,ℤ/8)11yescharacteristic
214C22C2×#13711yescharacteristic
215C4—22no—
216C4—22no—
217C4—22no—
218C4—41no—
219C4—41no—
220C4—82no—
221C4—82no—
222C4—82no—
223C4#55114yes—
224C2—24no—
225C2—84no—
226C2#256111yescharacteristic
227C2#310911yescharacteristic
228C2#311011yescharacteristic
229C2C2×#48911yescharacteristic
230C2#313811yescharacteristic
231C2#319211yescharacteristic
232C2#319311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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
1. #2857C2semidirect
3. #2071C2semidirect
4. #2023C2semidirect
5. #1936C2semidirect
6. #1824C2semidirect
8. #1749C2semidirect
10. C2×C4⋊Q8C22semidirect
19. C2×#97C22semidirect
26. #389C22semidirect
27. #384C22semidirect
30. #340C22semidirect
31. #340C22semidirect
32. #336C22semidirect

Maximal subgroups

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

subgroupindex
#28572
C2×#3902
#20712
#20232
#19362
#18242
#17862
#17492