Finite Group Databasebeta

← #29675

Subgroups — #29675

Order 256.

601 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 127 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×#540C211yesmaximal, characteristic, complemented
2C2×#463C212yesmaximal, complemented
3#2620C211yesmaximal, characteristic, complemented
4#2026C211yesmaximal, characteristic, complemented
5#2025C211yesmaximal, characteristic, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2011C211yesmaximal, characteristic, complemented
8#2000C212yesmaximal, complemented
9#1999C212yesmaximal, complemented
10#1802C212yesmaximal, complemented
11#1798C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14#540—28no—
15C2×#127C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#123—22no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121—22no—
21C2×#121C2212yes—
22C2×#121C2211yescharacteristic
23C2×C4×D4C2212yes—
24C2×#117C2211yescharacteristic
25#463—28no—
26C2×C4⋊C8C2212yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×C22⋊C8C2212yes—
32#396—22no—
33#394—22no—
34#391C2211yescharacteristic
35#390C2211yescharacteristic
36#384C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C22×Q8C2311yescharacteristic
42C22×D4—22no—
43C22×D4—21no—
44C22×D4—22no—
45C22×D4C2311yescharacteristic
46C22×D4C2312yes—
47C23×C4—22no—
48C23×C4C2311yescharacteristic
49C2×SD16—24no—
50C2×SD16—42no—
51C2×SD16—42no—
52C2×D8—24no—
53C2×D8—42no—
54C2×D8—42no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57#127—24no—
58#125—24no—
59#124—24no—
60#123—216no—
61C22⋊Q8—42no—
62#121—216no—
63#121—24no—
64#121—24no—
65#121—42no—
66C4×D4—24no—
67C4×D4D418yescomplemented
68#117—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D412yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4D412yes—
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85C4⋊C8—24no—
86#103—24no—
87#102—24no—
88#102—42no—
89#102—42no—
90C22⋊C8—42no—
91#95—22no—
92C24—22no—
93C24—21no—
94C24C2×D411yescharacteristic
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×D4—28no—
100C2×D4—28no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—42no—
111C2×D4—82no—
112C2×D4—82no—
113C2×D4—82no—
114C2×D4—82no—
115C2×D4—82no—
116C2×D4C2×D414yes—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—28no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—82no—
127C22×C4—82no—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4C2411yescharacteristic
132SD16—48no—
133D8—48no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—28no—
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C4⋊C4—82no—
147C4⋊C4C2×D414yes—
148C22⋊C4—28no—
149C22⋊C4—28no—
150C22⋊C4—24no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—44no—
154C22⋊C4—82no—
155C42—22no—
156C42C2×D414yes—
157C23—21no—
158C23—24no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—22no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—42no—
173C23—42no—
174C23C22×D411yescharacteristic
175Q8—42no—
176Q8—42no—
177D4—24no—
178D4—24no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—88no—
184D4—88no—
185D4—84no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—88no—
207C2×C4#12014yes—
208C2×C4C22×D412yes—
209C2×C4H(5,𝔽2)11yescharacteristic
210C2×C4H(5,𝔽2)12yes—
211C2×C42−1+411yescharacteristic
212C8—42no—
213C8—42no—
214C22—41no—
215C22—44no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22—44no—
220C22—42no—
221C22—44no—
222C22—82no—
223C22—82no—
224C22—82no—
225C22—82no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—82no—
230C22—82no—
231C22C2×#12011yescharacteristic
232C22#53511yescharacteristic
233C22#53711yescharacteristic
234C22#57612yes—
235C22#57712yes—
236C4—24no—
237C4—41no—
238C4—42no—
239C4—44no—
240C4—81no—
241C4—81no—
242C4—81no—
243C4#53512yes—
244C4#53712yes—
245C2—81no—
246C2—84no—
247C2—161no—
248C2#261511yescharacteristic
249C2#301212yes—
250C2#322011yescharacteristic
251C2#322111yescharacteristic
252C2#325712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#540C2semidirect
2. C2×#463C2semidirect
3. #2620C2semidirect
4. #2026C2semidirect
5. #2025C2semidirect
6. #2014C2semidirect
7. #2011C2semidirect
8. #2000C2semidirect
9. #1999C2semidirect
10. #1802C2semidirect
11. #1798C2semidirect
67. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5402
C2×#4632
#26202
#20262
#20252
#20142
#20112
#20002
#19992
#18022
#17982