Finite Group Databasebeta

← #29952

Subgroups — #29952

Order 256.

562 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 137 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#2094C212yesmaximal, complemented
4#2065C214yesmaximal, complemented
5#2023C212yesmaximal, complemented
6#1853C212yesmaximal, complemented
7#1797C211yesmaximal, characteristic, complemented
8#1786C212yesmaximal
9C23×D4C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic, complemented
11C2×C22⋊Q8—21no—
12C22×C4⋊C4C2211yescharacteristic
13C22×C22⋊C4—21no—
14C22×C22⋊C4—21no—
15C22×C42C2211yescharacteristic
16C2×#106C2214yescomplemented
17C2×C4⋊C8C2212yescomplemented
18C2×#102C2214yes—
19C2×#102C2214yes—
20C2×C22⋊C8C2212yes—
21#391—21no—
22#390—21no—
23#390—21no—
24#390C2212yes—
25#390C2212yes—
26#389—21no—
27#389C2212yescomplemented
28#384C2211yescharacteristic, complemented
29#380—21no—
30#380C2212yes—
31#340C2214yescomplemented
32#336C2212yescomplemented
33C25—21no—
34C22×Q8—21no—
35C22×D4—21no—
36C22×D4—22no—
37C22×D4—42no—
38C22×D4—42no—
39C22×D4Q814yescomplemented
40C22×D4C2312yes—
41C23×C4—21no—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8C2314yes—
46C4⋊Q8—42no—
47C22⋊Q8—44no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—44no—
63C2×C22⋊C4—44no—
64C2×C42—42no—
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#106—44no—
70C4⋊C8—42no—
71#102—44no—
72#102—44no—
73C22⋊C8—24no—
74#95—22no—
75#95—22no—
76C24—24no—
77C24—22no—
78C24—42no—
79C24—42no—
80C24—42no—
81C24—42no—
82C24—42no—
83C24—42no—
84C24C2×D411yescharacteristic
85C2×Q8—42no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×D4—24no—
89C2×D4—22no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—48no—
93C2×D4—42no—
94C2×D4—42no—
95C2×D4—48no—
96C2×D4C2×Q814yes—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4SD1614yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×Q812yes—
117C22×C4#4112yes—
118C22×C4C2411yescharacteristic
119C2×C8—24no—
120C2×C8—24no—
121C2×C8—24no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—42no—
130C22⋊C4—44no—
131C22⋊C4—44no—
132C22⋊C4—44no—
133C22⋊C4—44no—
134C22⋊C4—44no—
135C42—22no—
136C42—22no—
137C42—41no—
138C42—41no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—24no—
144C23—48no—
145C23—48no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—48no—
153C23—84no—
154C23—84no—
155C23—84no—
156C23—84no—
157C23C22⋊Q812yes—
158C23C2×SD1612yes—
159C23AGL(1,ℤ/8)12yes—
160C23C22×D411yescharacteristic
161Q8—84no—
162D4—28no—
163D4—48no—
164D4—44no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4C22⋊Q812yes—
191C2×C4C2×SD1612yes—
192C2×C4C2×SD1612yes—
193C2×C4AGL(1,ℤ/8)12yes—
194C2×C4C22×Q811yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4H(5,𝔽2)12yes—
197C2×C4H(5,𝔽2)12yes—
198C8—44no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—44no—
206C22—44no—
207C22—44no—
208C22—816no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22—82no—
213C22#47614yes—
214C22C2×C22⋊Q811yescharacteristic
215C22#53411yescharacteristic
216C22#53511yescharacteristic
217C22C22×SD1611yescharacteristic
218C22C2×AGL(1,ℤ/8)11yescharacteristic
219C22C2×AGL(1,ℤ/8)11yescharacteristic
220C22#57511yescharacteristic
221C4—22no—
222C4—22no—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—82no—
227C4—82no—
228C4—82no—
229C4#53812yes—
230C4#54312yes—
231C2—24no—
232C2—84no—
233C2#262611yescharacteristic
234C2C2×#47611yescharacteristic
235C2#307711yescharacteristic
236C2#318711yescharacteristic
237C2#319211yescharacteristic
238C2#320811yescharacteristic
239C2#321311yescharacteristic
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. #2094C2semidirect
4. #2065C2semidirect
5. #2023C2semidirect
6. #1853C2semidirect
7. #1797C2semidirect
10. C2×C4⋊Q8C22semidirect
16. C2×#106C22semidirect
17. C2×C4⋊C8C22semidirect
27. #389C22semidirect
28. #384C22semidirect
31. #340C22semidirect
32. #336C22semidirect
39. C22×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
C2×#3902
#20942
#20652
#20232
#18532
#17972
#17862