Finite Group Databasebeta

← #28341

Subgroups — #28341

Order 256.

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

normal subgroups only

This group has 679 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×#481C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic, complemented
4C22×#102C211yesmaximal, characteristic, complemented
5#2504C211yesmaximal, characteristic, complemented
6#2433C211yesmaximal, characteristic, complemented
7#2092C212yesmaximal, complemented
8#2011C212yesmaximal, complemented
9#1972C212yesmaximal, complemented
10#1961C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2212yes—
14C22×D8C2211yescharacteristic, complemented
15C23×C8C2211yescharacteristic
16C2×#127C2212yes—
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#120—22no—
20C22×C4⋊C4C2211yescharacteristic
21#481—24no—
22#466—24no—
23#447—24no—
24C2×#107C2211yescharacteristic, complemented
25C2×#106C2212yescomplemented
26C2×#102—24no—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yescomplemented
31C2×#102C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic, complemented
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2212yescomplemented
35#397—22no—
36#392—22no—
37#390C2212yes—
38#390C2212yescomplemented
39#386—22no—
40#382C2212yescomplemented
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yescomplemented
43C25—21no—
44C22×D4—24no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4—24no—
51C22×D4—42no—
52C22×D4C2311yescharacteristic
53C22×D4C2311yescharacteristic
54C22×D4C2312yes—
55C23×C4—21no—
56C23×C4C2311yescharacteristic
57C2×D8—24no—
58C2×D8—24no—
59C2×D8—22no—
60C2×D8—42no—
61C2×D8—42no—
62C22×C8—42no—
63C22×C8D414yescomplemented
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2312yes—
67#127—24no—
68#127—42no—
69#121—22no—
70#121—22no—
71#121—22no—
72#121—22no—
73#120—48no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—24no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4D412yes—
85C2×C42C2312yes—
86#107—22no—
87#106—24no—
88#102—24no—
89#102—24no—
90#102—24no—
91#102—22no—
92#102—22no—
93#102—24no—
94C22⋊C8D414yescomplemented
95#97—42no—
96#95—24no—
97#95D412yes—
98C24—22no—
99C24—21no—
100C24—21no—
101C24—44no—
102C24—42no—
103C24—44no—
104C24—82no—
105C24C2×D411yescharacteristic
106C2×D4—28no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—48no—
121C2×D4—44no—
122C2×D4—44no—
123C2×D4—48no—
124C2×D4—48no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—42no—
130C2×D4—84no—
131C2×D4#4114yes—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—42no—
138C22×C4—82no—
139C22×C4C2×D414yes—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2411yescharacteristic
144D8—48no—
145D8—44no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—44no—
151C2×C8C2×D414yes—
152C2×C8C2×D412yes—
153C2×C8C2×D412yes—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—42no—
161C22⋊C4—24no—
162C22⋊C4—48no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—84no—
166C42—22no—
167C42—22no—
168C23—22no—
169C23—22no—
170C23—24no—
171C23—48no—
172C23—42no—
173C23—42no—
174C23—44no—
175C23—44no—
176C23—41no—
177C23—41no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—816no—
185C23—84no—
186C23—84no—
187C23#12712yes—
188C23AGL(1,ℤ/8)14yes—
189C23C22×D411yescharacteristic
190D4—24no—
191D4—24no—
192D4—48no—
193D4—41no—
194D4—41no—
195D4—42no—
196D4—41no—
197D4—41no—
198D4—42no—
199D4—42no—
200D4—42no—
201D4—816no—
202D4—84no—
203D4—84no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—84no—
218C2×C4#12712yes—
219C2×C4C22×D411yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C2×#4112yes—
222C2×C4H(5,𝔽2)12yes—
223C8—22no—
224C8—22no—
225C8—24no—
226C22—22no—
227C22—22no—
228C22—28no—
229C22—44no—
230C22—48no—
231C22—816no—
232C22—82no—
233C22—82no—
234C22—84no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—84no—
242C22—84no—
243C22#49614yes—
244C22C2×#12711yescharacteristic
245C22#54612yes—
246C22C2×AGL(1,ℤ/8)12yes—
247C22#57612yes—
248C4—22no—
249C4—44no—
250C4—81no—
251C4—81no—
252C4—82no—
253C4#52511yescharacteristic
254C4#53411yescharacteristic
255C4#54612yes—
256C2—24no—
257C2—84no—
258C2—161no—
259C2—161no—
260C2#253911yescharacteristic
261C2C2×#49611yescharacteristic
262C2#315511yescharacteristic
263C2#331814yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#481C2semidirect
2. C2×#466C2semidirect
3. C2×#447C2semidirect
4. C22×#102C2semidirect
5. #2504C2semidirect
6. #2433C2semidirect
7. #2092C2semidirect
8. #2011C2semidirect
9. #1972C2semidirect
10. #1961C2semidirect
11. #1752C2semidirect
14. C22×D8C22semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
30. C2×#102C22semidirect
31. C2×#102C22semidirect
32. C2×#102C22semidirect
34. C2×#97C22semidirect
38. #390C22semidirect
40. #382C22semidirect
42. #336C22semidirect
63. C22×C8D4semidirect
94. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4662
C2×#4472
C22×#1022
#25042
#24332
#20922
#20112
#19722
#19612
#17522