Finite Group Databasebeta

← #25612

Subgroups — #25612

Order 256.

547 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 131 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×#541C211yesmaximal, characteristic, complemented
2C2×#497C211yesmaximal, characteristic, complemented
3C2×#488C211yesmaximal, characteristic, complemented
4#2620C211yesmaximal, characteristic
5#2001C212yesmaximal, complemented
6#2000C212yesmaximal
7#1992C211yesmaximal, characteristic, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1878C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14#541—24no—
15#541—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#126—22no—
18C2×#125—22no—
19C2×#125C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2212yes—
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2212yes—
25C2×#117C2211yescharacteristic
26#497—24no—
27#488—24no—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2212yes—
33C2×#97C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#396—22no—
36#394—22no—
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#384C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×Q8C2312yes—
45C22×D4—21no—
46C22×D4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C2×Q16—24no—
49C2×Q16—24no—
50C2×Q16—42no—
51C2×SD16—24no—
52C2×SD16—24no—
53C2×SD16—42no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8D414yescomplemented
58#126—216no—
59#125—216no—
60#125—22no—
61#125D414yescomplemented
62#124—22no—
63#124—22no—
64C22⋊Q8—24no—
65C22⋊Q8—24no—
66C22⋊Q8—42no—
67#121—42no—
68C4×Q8—24no—
69C4×Q8—24no—
70#117—22no—
71#117—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4D412yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4D412yes—
88C2×C42—22no—
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91#103—24no—
92#103—42no—
93#103—42no—
94#102—24no—
95C22⋊C8—24no—
96#97—22no—
97C4×C8—22no—
98#95—22no—
99C24—21no—
100C24C2×D411yescharacteristic
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8C2×D414yes—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—42no—
116C2×D4—82no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D412yes—
129C22×C4C2411yescharacteristic
130Q16—48no—
131SD16—48no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8#4114yes—
138C4⋊C4—28no—
139C4⋊C4—28no—
140C4⋊C4—28no—
141C4⋊C4—28no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—82no—
148C4⋊C4C2×D414yes—
149C22⋊C4—28no—
150C22⋊C4—24no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—44no—
154C22⋊C4—82no—
155C42—28no—
156C42—21no—
157C42—21no—
158C42C2×D412yes—
159C42C2×D412yes—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—22no—
164C23—22no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—42no—
169C23—42no—
170C23C22×D411yescharacteristic
171Q8—24no—
172Q8—24no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176Q8—42no—
177D4—42no—
178D4—42no—
179D4—84no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4#12014yes—
204C2×C4C22×D411yescharacteristic
205C2×C4C22×D411yescharacteristic
206C2×C4C2×#4112yes—
207C2×C4H(5,𝔽2)12yes—
208C8—24no—
209C8—42no—
210C22—42no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—42no—
215C22—82no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22—82no—
220C22C2×#12011yescharacteristic
221C22#53211yescharacteristic
222C22#53511yescharacteristic
223C22#57712yes—
224C22#57812yes—
225C4—24no—
226C4—41no—
227C4—44no—
228C4—42no—
229C4—44no—
230C4—81no—
231C4—81no—
232C4—81no—
233C4#54612yes—
234C4#55012yes—
235C2—81no—
236C2—161no—
237C2#243111yescharacteristic
238C2#301312yes—
239C2#317812yes—
240C2#322211yescharacteristic
241C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#541C2semidirect
2. C2×#497C2semidirect
3. C2×#488C2semidirect
5. #2001C2semidirect
7. #1992C2semidirect
8. #1956C2semidirect
9. #1880C2semidirect
10. #1878C2semidirect
11. #1750C2semidirect
57. C4⋊Q8D4semidirect
61. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4972
C2×#4882
#26202
#20012
#20002
#19922
#19562
#18802
#18782
#17502