Finite Group Databasebeta

← #25940

Subgroups — #25940

Order 256.

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

normal subgroups only

This group has 549 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×#463C211yesmaximal, characteristic
4#2592C211yesmaximal, characteristic, complemented
5#2026C212yesmaximal, complemented
6#2025C212yesmaximal, complemented
7#1988C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal
10#1877C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×D8C2211yescharacteristic
14#554—24no—
15#554—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2212yes—
18C2×#124C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8—22no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2212yes—
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4—21no—
25#489—24no—
26#463—24no—
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#397—21no—
35#393—21no—
36#391C2212yes—
37#390C2212yes—
38#388—22no—
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8D411yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C23×C4—22no—
52C2×SD16—22no—
53C2×SD16—24no—
54C2×D8—22no—
55C2×D8—24no—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C22×C8C2312yes—
59C4⋊Q8—22no—
60C4⋊Q8D414yes—
61#125—24no—
62#125—24no—
63#124—22no—
64#124—22no—
65C22⋊Q8—28no—
66C22⋊Q8—216no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×D4—24no—
70C4×D4—24no—
71C4×D4—22no—
72C4×D4D414yes—
73C2×C4⋊C4—28no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D412yes—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C42—21no—
89C2×C42C2312yes—
90C2×C42C2311yescharacteristic
91C4⋊C8—22no—
92#103—22no—
93#103—24no—
94#102—24no—
95#102—22no—
96#102—24no—
97#97—22no—
98#95—21no—
99#95—22no—
100#95—22no—
101C24—21no—
102C24—21no—
103C2×Q8—22no—
104C2×Q8—24no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—81no—
111C2×Q8—81no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4C2×D412yes—
119C2×D4#4114yes—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—28no—
123C22×C4—24no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—44no—
127C22×C4—82no—
128C22×C4—82no—
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2411yescharacteristic
134SD16—44no—
135D8—44no—
136C2×C8—22no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C4⋊C4—28no—
144C4⋊C4—28no—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4C2×D414yes—
153C4⋊C4C2×D412yes—
154C22⋊C4—28no—
155C22⋊C4—24no—
156C22⋊C4—44no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—82no—
160C42—22no—
161C42—22no—
162C42—81no—
163C42C2×D412yes—
164C42C2×D412yes—
165C23—22no—
166C23—24no—
167C23—42no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23—41no—
175C23C22×D411yescharacteristic
176Q8—44no—
177Q8—41no—
178Q8—41no—
179Q8—42no—
180Q8—84no—
181D4—22no—
182D4—24no—
183D4—41no—
184D4—41no—
185D4—42no—
186D4#12414yes—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—88no—
208C2×C4#12012yes—
209C2×C4#12012yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C2×C4C2×#4112yes—
213C2×C42−1+412yes—
214C8—41no—
215C8—41no—
216C8—42no—
217C22—24no—
218C22—44no—
219C22—44no—
220C22—42no—
221C22—42no—
222C22—82no—
223C22—82no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22C2×#12011yescharacteristic
228C22#54812yes—
229C22#57512yes—
230C22#57611yescharacteristic
231C22#57711yescharacteristic
232C4—22no—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—42no—
237C4—44no—
238C4—82no—
239C4—81no—
240C4C2×#12411yescharacteristic
241C4#53611yescharacteristic
242C4#54812yes—
243C2—44no—
244C2—161no—
245C2#243611yescharacteristic
246C2#300511yescharacteristic
247C2#301211yescharacteristic
248C2#337012yes—
249C2#337612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ 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
4. #2592C2semidirect
5. #2026C2semidirect
6. #2025C2semidirect
7. #1988C2semidirect
8. #1985C2semidirect
10. #1877C2semidirect
11. #1762C2semidirect

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×#4632
#25922
#20262
#20252
#19882
#19852
#18792
#18772
#17622