Finite Group Databasebeta

← #25640

Subgroups — #25640

Order 256.

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

normal subgroups only

This group has 551 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×#557C211yesmaximal, characteristic, complemented
2C2×#497C211yesmaximal, characteristic
3C2×#464C211yesmaximal, characteristic
4#2592C211yesmaximal, characteristic
5#2016C212yesmaximal, complemented
6#2013C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1981C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1878C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×Q16C2211yescharacteristic
14C22×SD16C2212yes—
15C22×SD16C2211yescharacteristic
16#557—24no—
17#557—24no—
18C2×C4⋊Q8C2212yes—
19C2×C4⋊Q8C2211yescharacteristic
20C2×#125—22no—
21C2×#125—21no—
22C2×#124C2211yescharacteristic
23C2×C4×Q8—21no—
24C2×C4×Q8C2212yes—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27#497—24no—
28#464—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#397—21no—
37#393—21no—
38#391C2212yes—
39#390C2212yes—
40#388—22no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8D411yescharacteristic
46C22×Q8C2311yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×Q8C2312yes—
49C22×D4C2311yescharacteristic
50C2×Q16—22no—
51C2×Q16—24no—
52C2×Q16—24no—
53C2×Q16—42no—
54C2×Q16—42no—
55C2×SD16—22no—
56C2×SD16—24no—
57C2×SD16—24no—
58C2×SD16—42no—
59C2×SD16—42no—
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2312yes—
63C4⋊Q8—24no—
64C4⋊Q8—24no—
65C4⋊Q8—22no—
66C4⋊Q8D414yes—
67#125—216no—
68#125—28no—
69#124—22no—
70#124—22no—
71C4×Q8—28no—
72C4×Q8—24no—
73C4×Q8—24no—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8D414yes—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D412yes—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C2312yes—
96C2×C42C2311yescharacteristic
97C4⋊C8—22no—
98#103—24no—
99#103—22no—
100#103—42no—
101#103—42no—
102#102—22no—
103#97—22no—
104#95—21no—
105#95—22no—
106#95—22no—
107C24—21no—
108C2×Q8—22no—
109C2×Q8—24no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8C2×D414yes—
123C2×Q8C2×D412yes—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138Q16—48no—
139Q16—44no—
140SD16—48no—
141SD16—44no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8#4114yes—
149C4⋊C4—28no—
150C4⋊C4—28no—
151C4⋊C4—28no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4C2×D412yes—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—44no—
164C22⋊C4—82no—
165C42—28no—
166C42—24no—
167C42—22no—
168C42—22no—
169C42—81no—
170C42C2×D412yes—
171C42C2×D412yes—
172C23—41no—
173C23—42no—
174C23—42no—
175C23—41no—
176C23—41no—
177C23C22×D411yescharacteristic
178Q8—28no—
179Q8—22no—
180Q8—22no—
181Q8—24no—
182Q8—24no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186Q8—41no—
187Q8—41no—
188Q8—42no—
189D4—41no—
190D4—41no—
191D4—42no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4#12012yes—
214C2×C4#12012yes—
215C2×C4#13714yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4112yes—
219C2×C4H(5,𝔽2)12yes—
220C8—24no—
221C8—41no—
222C8—41no—
223C22—82no—
224C22—82no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22C2×#12011yescharacteristic
229C22#55012yes—
230C22C2×#13712yes—
231C22#57711yescharacteristic
232C22#57811yescharacteristic
233C4—22no—
234C4—24no—
235C4—22no—
236C4—42no—
237C4—44no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4—81no—
242C4#53211yescharacteristic
243C4#53511yescharacteristic
244C4#54612yes—
245C2—161no—
246C2#243111yescharacteristic
247C2#300411yescharacteristic
248C2#301311yescharacteristic
249C2#339812yes—
250C2#340012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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×#557C2semidirect
5. #2016C2semidirect
7. #1985C2semidirect
8. #1981C2semidirect
10. #1878C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4972
C2×#4642
#25922
#20162
#20132
#19852
#19812
#18802
#18782
#17632