Finite Group Databasebeta

← #25579

Subgroups — #25579

Order 256.

603 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#554C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#464C211yesmaximal, characteristic, complemented
4#2540C211yesmaximal, characteristic
5#2001C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1988C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1874C212yesmaximal, complemented
10#1872C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14#554—24no—
15#554—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—21no—
18C2×#124C2211yescharacteristic
19C2×C22⋊Q8—22no—
20C2×C22⋊Q8C2212yes—
21C2×C22⋊Q8C2212yes—
22C2×#121C2212yes—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2212yes—
27#489—24no—
28#464—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2212yes—
37C2×#97C2211yescharacteristic
38#399—21no—
39#392—21no—
40#390—22no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4—22no—
49C22×D4D411yescharacteristic
50C22×D4C2311yescharacteristic
51C23×C4—22no—
52C23×C4C2312yes—
53C2×Q16—22no—
54C2×Q16—42no—
55C2×SD16—22no—
56C2×SD16—42no—
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C22×C8C2312yes—
60C4⋊Q8—22no—
61C4⋊Q8D414yescomplemented
62#125—28no—
63#124—22no—
64#124—22no—
65C22⋊Q8—216no—
66C22⋊Q8—24no—
67C22⋊Q8—24no—
68C22⋊Q8—24no—
69C22⋊Q8—42no—
70#121—24no—
71#121—42no—
72C4×Q8—22no—
73C4×Q8D414yescomplemented
74C4×D4—28no—
75C4×D4—22no—
76C4×D4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2312yes—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D412yes—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C4⋊C8—22no—
98#103—22no—
99#103—24no—
100#103—24no—
101#102—22no—
102#102—24no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105#97—22no—
106#95—21no—
107C24—21no—
108C24C2×D412yes—
109C2×Q8—24no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—81no—
118C2×Q8—81no—
119C2×Q8C2×D412yes—
120C2×D4—24no—
121C2×D4—22no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—82no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—28no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4C2×D412yes—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147Q16—44no—
148SD16—44no—
149C2×C8—22no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C4⋊C4—28no—
157C4⋊C4—28no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—44no—
165C4⋊C4C2×D412yes—
166C4⋊C4C2×D414yes—
167C4⋊C4#4114yes—
168C22⋊C4—28no—
169C22⋊C4—24no—
170C22⋊C4—44no—
171C22⋊C4—44no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C42—24no—
175C42—81no—
176C42C2×D412yes—
177C42C2×D412yes—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—41no—
185C23—42no—
186C23—42no—
187C23—41no—
188C23—41no—
189C23#13714yes—
190C23C22×D411yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—42no—
194Q8—41no—
195Q8—41no—
196Q8—42no—
197Q8—84no—
198D4—28no—
199D4—41no—
200D4—41no—
201D4—42no—
202D4—88no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—88no—
227C2×C4#12012yes—
228C2×C4#12012yes—
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C2×#4112yes—
232C2×C4H(5,𝔽2)12yes—
233C8—41no—
234C8—41no—
235C8—42no—
236C22—24no—
237C22—24no—
238C22—24no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—82no—
247C22—82no—
248C22C2×#12011yescharacteristic
249C22#54612yes—
250C22C2×#13712yes—
251C22#57611yescharacteristic
252C22#57711yescharacteristic
253C4—22no—
254C4—22no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—44no—
259C4—44no—
260C4—81no—
261C4#53211yescharacteristic
262C4#53511yescharacteristic
263C4#55012yes—
264C2—44no—
265C2—161no—
266C2#243111yescharacteristic
267C2#300411yescharacteristic
268C2#301211yescharacteristic
269C2#331412yes—
270C2#331612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#554C2semidirect
2. C2×#489C2semidirect
3. C2×#464C2semidirect
5. #2001C2semidirect
6. #2000C2semidirect
7. #1988C2semidirect
8. #1955C2semidirect
9. #1874C2semidirect
10. #1872C2semidirect
11. #1763C2semidirect
61. C4⋊Q8D4semidirect
73. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4892
C2×#4642
#25402
#20012
#20002
#19882
#19552
#18742
#18722
#17632