Finite Group Databasebeta

← #28331

Subgroups — #28331

Order 256.

620 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 139 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×#484C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4C22×#102C211yesmaximal, characteristic, complemented
5#2504C211yesmaximal, characteristic, complemented
6#2435C211yesmaximal, characteristic, complemented
7#2092C212yesmaximal, complemented
8#2012C212yesmaximal, complemented
9#1970C212yesmaximal, complemented
10#1958C212yesmaximal, complemented
11C8×C22⋊C4C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2211yescharacteristic, complemented
14C22×D8C2212yes—
15C23×C8C2211yescharacteristic
16C2×#124C2212yes—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20#484—24no—
21#465—24no—
22#447—24no—
23C2×#107C2212yescomplemented
24C2×#106C2211yescharacteristic, complemented
25C2×#103C2212yescomplemented
26C2×#103C2211yescharacteristic, complemented
27C2×#102—24no—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2211yescharacteristic, complemented
32C2×C22⋊C8C2211yescharacteristic
33C2×C4×C8C2212yescomplemented
34#397—22no—
35#394—22no—
36#390C2212yes—
37#390C2212yescomplemented
38#386—22no—
39#382C2212yescomplemented
40#379—22no—
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yescomplemented
43C25—21no—
44C22×Q8C2311yescharacteristic
45C22×D4—24no—
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4—42no—
51C22×D4C2311yescharacteristic
52C22×D4C2312yes—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C2×SD16—22no—
56C2×SD16—42no—
57C2×D8—24no—
58C2×D8—24no—
59C2×D8—42no—
60C22×C8—42no—
61C22×C8D414yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2312yes—
65#124—24no—
66#124—42no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69#121—22no—
70#121—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4D412yes—
84C2×C42C2312yes—
85#107—24no—
86#106—22no—
87#103—22no—
88#103—24no—
89#102—24no—
90#102—24no—
91#102—24no—
92#102—22no—
93C22⋊C8D414yes—
94C4×C8—42no—
95#95—24no—
96#95D412yes—
97C24—22no—
98C24—21no—
99C24—44no—
100C24—42no—
101C24—44no—
102C24—82no—
103C24C2×D411yescharacteristic
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—48no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—42no—
122C2×D4—44no—
123C2×D4—42no—
124C2×D4#4114yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—82no—
133C22×C4C2×D414yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138SD16—44no—
139D8—48no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—44no—
145C2×C8C2×D414yes—
146C2×C8C2×D412yes—
147C2×C8C2×D412yes—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C22⋊C4—24no—
157C22⋊C4—48no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—84no—
161C22⋊C4—84no—
162C42—22no—
163C42—22no—
164C23—22no—
165C23—22no—
166C23—24no—
167C23—48no—
168C23—42no—
169C23—42no—
170C23—44no—
171C23—44no—
172C23—41no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—816no—
177C23—84no—
178C23—84no—
179C23#12712yes—
180C23#13514yes—
181C23C22×D411yescharacteristic
182Q8—41no—
183Q8—41no—
184Q8—42no—
185D4—24no—
186D4—24no—
187D4—48no—
188D4—41no—
189D4—42no—
190D4—41no—
191D4—42no—
192D4—42no—
193D4—84no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—84no—
210C2×C4#12712yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4112yes—
214C2×C4H(5,𝔽2)12yes—
215C8—22no—
216C8—22no—
217C8—24no—
218C22—22no—
219C22—28no—
220C22—22no—
221C22—48no—
222C22—44no—
223C22—816no—
224C22—82no—
225C22—82no—
226C22—84no—
227C22—81no—
228C22—81no—
229C22—84no—
230C22—81no—
231C22#49514yes—
232C22C2×#12711yescharacteristic
233C22#54612yes—
234C22C2×#13512yes—
235C22#57612yes—
236C4—22no—
237C4—44no—
238C4—81no—
239C4—81no—
240C4—82no—
241C4—81no—
242C4#52511yescharacteristic
243C4#53411yescharacteristic
244C4#54612yes—
245C2—24no—
246C2—84no—
247C2—161no—
248C2#253911yescharacteristic
249C2C2×#49511yescharacteristic
250C2#315511yescharacteristic
251C2#329714yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
2. C2×#465C2semidirect
4. C22×#102C2semidirect
5. #2504C2semidirect
6. #2435C2semidirect
7. #2092C2semidirect
8. #2012C2semidirect
9. #1970C2semidirect
10. #1958C2semidirect
11. C8×C22⋊C4C2semidirect
13. C22×SD16C22semidirect
23. C2×#107C22semidirect
24. C2×#106C22semidirect
25. C2×#103C22semidirect
26. C2×#103C22semidirect
31. C2×#102C22semidirect
33. C2×C4×C8C22semidirect
37. #390C22semidirect
39. #382C22semidirect
42. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4652
C2×#4472
C22×#1022
#25042
#24352
#20922
#20122
#19702
#19582
C8×C22⋊C42