Finite Group Databasebeta

← #28329

Subgroups — #28329

Order 256.

713 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 155 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×#480C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic, complemented
4C22×#102C211yesmaximal, characteristic, complemented
5#2501C211yesmaximal, characteristic, complemented
6#2433C211yesmaximal, characteristic, complemented
7#2093C212yesmaximal, complemented
8#2011C212yesmaximal, complemented
9#1970C212yesmaximal, complemented
10#1957C212yesmaximal, complemented
11C8×C22⋊C4C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2212yes—
14C22×D8C2211yescharacteristic, complemented
15C23×C8C2211yescharacteristic
16C2×#127C2212yes—
17C2×#123—22no—
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#120—22no—
21C22×C4⋊C4C2211yescharacteristic
22#480—24no—
23#466—24no—
24#447—24no—
25C2×#107C2212yescomplemented
26C2×#107C2211yescharacteristic, complemented
27C2×#102—24no—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yescomplemented
32C2×#102C2211yescharacteristic, complemented
33C2×#102C2211yescharacteristic, complemented
34C2×C22⋊C8C2211yescharacteristic
35C2×C4×C8C2212yescomplemented
36#399—22no—
37#392—22no—
38#390C2212yes—
39#390C2212yescomplemented
40#384C2212yescomplemented
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#123—48no—
70#121—22no—
71#121—22no—
72#121—22no—
73#121—22no—
74#120—48no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—24no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4D412yes—
87C2×C42C2312yes—
88#107—22no—
89#107—24no—
90#102—24no—
91#102—24no—
92#102—24no—
93#102—22no—
94#102—22no—
95#102—24no—
96C22⋊C8D414yescomplemented
97C4×C8—42no—
98#95D412yes—
99C24—22no—
100C24—21no—
101C24—21no—
102C24—44no—
103C24—42no—
104C24—44no—
105C24—82no—
106C24C2×D411yescharacteristic
107C2×D4—28no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—48no—
122C2×D4—44no—
123C2×D4—44no—
124C2×D4—48no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—42no—
130C2×D4—48no—
131C2×D4—84no—
132C2×D4#4114yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—44no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2×D414yes—
144C22×C4C2411yescharacteristic
145D8—48no—
146D8—44no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—44no—
152C2×C8C2×D414yes—
153C2×C8C2×D412yes—
154C2×C8C2×D412yes—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—42no—
162C4⋊C4—48no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—48no—
166C22⋊C4—48no—
167C22⋊C4D818yescomplemented
168C42—22no—
169C42—22no—
170C23—22no—
171C23—22no—
172C23—24no—
173C23—48no—
174C23—42no—
175C23—42no—
176C23—44no—
177C23—44no—
178C23—41no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—816no—
187C23—84no—
188C23—84no—
189C23#12712yes—
190C23C2×D814yes—
191C23C22×D411yescharacteristic
192D4—24no—
193D4—24no—
194D4—48no—
195D4—41no—
196D4—41no—
197D4—42no—
198D4—41no—
199D4—41no—
200D4—42no—
201D4—42no—
202D4—42no—
203D4—816no—
204D4—84no—
205D4—84no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—84no—
219C2×C4#12712yes—
220C2×C4C2×D818yes—
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C2×#4112yes—
224C2×C4H(5,𝔽2)12yes—
225C8—22no—
226C8—22no—
227C8—24no—
228C22—22no—
229C22—28no—
230C22—22no—
231C22—48no—
232C22—44no—
233C22—816no—
234C22—82no—
235C22—82no—
236C22—84no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—84no—
242C22—84no—
243C22—81no—
244C22—81no—
245C22#49314yes—
246C22C2×#12711yescharacteristic
247C22#54612yes—
248C22C22×D812yes—
249C22#57712yes—
250C4—28no—
251C4—22no—
252C4—82no—
253C4—81no—
254C4—81no—
255C4#52511yescharacteristic
256C4#53411yescharacteristic
257C4#54612yes—
258C2—24no—
259C2—84no—
260C2—161no—
261C2—161no—
262C2#253911yescharacteristic
263C2C2×#49311yescharacteristic
264C2#315411yescharacteristic
265C2#329414yes—
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×#480C2semidirect
2. C2×#466C2semidirect
3. C2×#447C2semidirect
4. C22×#102C2semidirect
5. #2501C2semidirect
6. #2433C2semidirect
7. #2093C2semidirect
8. #2011C2semidirect
9. #1970C2semidirect
10. #1957C2semidirect
11. C8×C22⋊C4C2semidirect
14. C22×D8C22semidirect
25. C2×#107C22semidirect
26. C2×#107C22semidirect
31. C2×#102C22semidirect
32. C2×#102C22semidirect
33. C2×#102C22semidirect
35. C2×C4×C8C22semidirect
39. #390C22semidirect
40. #384C22semidirect
42. #336C22semidirect
63. C22×C8D4semidirect
96. C22⋊C8D4semidirect
167. C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C2×#4662
C2×#4472
C22×#1022
#25012
#24332
#20932
#20112
#19702
#19572
C8×C22⋊C42