Finite Group Databasebeta

← #25638

Subgroups — #25638

Order 256.

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

normal subgroups only

This group has 567 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×#556C211yesmaximal, characteristic, complemented
2C2×#497C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic
4#2592C211yesmaximal, characteristic, complemented
5#2015C212yesmaximal, complemented
6#2012C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1981C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1878C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C22×SD16C2211yescharacteristic
15C22×D8C2211yescharacteristic
16#556—24no—
17#556—24no—
18C2×C4⋊Q8C2211yescharacteristic
19C2×#126—22no—
20C2×#124C2211yescharacteristic
21C2×#124C2212yes—
22C2×C22⋊Q8—21no—
23C2×C4×Q8C2212yes—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×#117—21no—
27#497—24no—
28#463—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
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×Q8C2312yes—
48C22×D4C2311yescharacteristic
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C2×Q16—24no—
52C2×Q16—24no—
53C2×Q16—42no—
54C2×SD16—24no—
55C2×SD16—22no—
56C2×SD16—24no—
57C2×SD16—42no—
58C2×SD16—42no—
59C2×D8—22no—
60C2×D8—42no—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2312yes—
64C4⋊Q8—22no—
65C4⋊Q8D414yes—
66#126—216no—
67#124—24no—
68#124—24no—
69#124—22no—
70#124—22no—
71C22⋊Q8—28no—
72C4×Q8—24no—
73C4×Q8—24no—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×D4—22no—
77C4×D4D414yes—
78#117—28no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D412yes—
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42C2312yes—
96C2×C42C2311yescharacteristic
97C4⋊C8—22no—
98#103—22no—
99#103—42no—
100#103—42no—
101#102—24no—
102#102—22no—
103#97—22no—
104#95—21no—
105#95—22no—
106#95—22no—
107C24—21no—
108C24—21no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—42no—
120C2×Q8C2×D414yes—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4C2×D412yes—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—44no—
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2411yescharacteristic
140Q16—48no—
141SD16—48no—
142SD16—44no—
143D8—44no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8#4114yes—
151C4⋊C4—28no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—24no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4C2×D412yes—
160C22⋊C4—28no—
161C22⋊C4—28no—
162C22⋊C4—24no—
163C22⋊C4—44no—
164C22⋊C4—42no—
165C22⋊C4—42no—
166C22⋊C4—82no—
167C42—28no—
168C42—22no—
169C42—22no—
170C42—81no—
171C42C2×D412yes—
172C42C2×D412yes—
173C23—22no—
174C23—42no—
175C23—42no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—42no—
180C23—42no—
181C23—41no—
182C23—41no—
183C23C22×D411yescharacteristic
184Q8—24no—
185Q8—24no—
186Q8—44no—
187Q8—42no—
188Q8—42no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192D4—22no—
193D4—22no—
194D4—41no—
195D4—41no—
196D4—42no—
197D4—42no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—48no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4#12012yes—
219C2×C4#12012yes—
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C2×#4112yes—
223C2×C4H(5,𝔽2)12yes—
224C8—24no—
225C8—41no—
226C8—41no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—82no—
231C22—82no—
232C22—82no—
233C22—82no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22C2×#12011yescharacteristic
238C22#55012yes—
239C22#57512yes—
240C22#57711yescharacteristic
241C22#57811yescharacteristic
242C4—22no—
243C4—22no—
244C4—44no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—82no—
249C4—81no—
250C4#53211yescharacteristic
251C4#53511yescharacteristic
252C4#54612yes—
253C2—82no—
254C2—161no—
255C2#243111yescharacteristic
256C2#300511yescharacteristic
257C2#301311yescharacteristic
258C2#340112yes—
259C2#340212yes—
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×#556C2semidirect
2. C2×#497C2semidirect
4. #2592C2semidirect
5. #2015C2semidirect
7. #1985C2semidirect
8. #1981C2semidirect
9. #1880C2semidirect
10. #1878C2semidirect
11. #1762C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4972
C2×#4632
#25922
#20152
#20122
#19852
#19812
#18802
#18782
#17622