Finite Group Databasebeta

← #33014

Subgroups — #33014

Order 256.

737 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 143 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×#518C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4#2624C211yesmaximal, characteristic, complemented
5#1879C212yesmaximal, complemented
6#1879C212yesmaximal, complemented
7#1864C212yesmaximal, complemented
8#1862C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic, complemented
10#1818C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2212yes—
14C2×#121C2211yescharacteristic
15#518—24no—
16#518—24no—
17C2×C4×D4—21no—
18C2×C4×D4—21no—
19C2×C4×D4C2212yes—
20C2×C4×D4C2212yes—
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C22⋊C4—22no—
24#482—24no—
25#447—24no—
26C2×#107C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic, complemented
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#399—21no—
36#397—21no—
37#394—22no—
38#390C2212yescomplemented
39#389C2212yescomplemented
40#380C2211yescharacteristic
41#340C2212yescomplemented
42#340C2212yescomplemented
43#336C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C22×D4—24no—
47C22×D4—24no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—22no—
51C22×D4D414yes—
52C22×D4C2311yescharacteristic
53C22×D4C2312yes—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4C2311yescharacteristic
57C2×D8—24no—
58C2×D8—24no—
59C2×D8—24no—
60C22×C8C2312yes—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63#121—22no—
64#121—22no—
65C4×D4—28no—
66C4×D4—28no—
67C4×D4—24no—
68C4×D4—24no—
69C4×D4—24no—
70C4×D4—24no—
71#117—22no—
72#117—22no—
73#117—22no—
74#117D414yes—
75C2×C4⋊C4—22no—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—216no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4D412yes—
91C2×C42—21no—
92C2×C42C2312yes—
93#107—22no—
94#106—22no—
95#102—24no—
96#102—24no—
97#102—24no—
98#102—22no—
99#102—22no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102#95—21no—
103#95—22no—
104C24—28no—
105C24—22no—
106C24—24no—
107C24—21no—
108C24—81no—
109C24—82no—
110C24—81no—
111C24C2×D411yescharacteristic
112C2×D4—28no—
113C2×D4—28no—
114C2×D4—28no—
115C2×D4—216no—
116C2×D4—24no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—42no—
125C2×D4—82no—
126C2×D4C2×D414yes—
127C2×D4#4114yes—
128C2×D4#4114yes—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—28no—
134C22×C4—28no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×D412yes—
146C22×C4C2411yescharacteristic
147D8—48no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—82no—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D412yes—
166C22⋊C4—28no—
167C22⋊C4—28no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—42no—
171C22⋊C4—82no—
172C22⋊C4—82no—
173C22⋊C4—81no—
174C42—24no—
175C42—22no—
176C42—22no—
177C23—216no—
178C23—24no—
179C23—22no—
180C23—24no—
181C23—24no—
182C23—21no—
183C23—22no—
184C23—21no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—41no—
190C23—42no—
191C23—41no—
192C23—42no—
193C23—41no—
194C23—84no—
195C23—84no—
196C23—88no—
197C23—88no—
198C23#12012yes—
199C23C22×D411yescharacteristic
200D4—216no—
201D4—24no—
202D4—24no—
203D4—24no—
204D4—42no—
205D4—41no—
206D4—41no—
207D4—84no—
208D4#12318yes—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—88no—
230C2×C4—88no—
231C2×C4#12012yes—
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C2×C4C2×#4112yes—
235C2×C4C2×#4112yes—
236C8—42no—
237C8—41no—
238C8—41no—
239C22—22no—
240C22—28no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—44no—
246C22—42no—
247C22—44no—
248C22—44no—
249C22—44no—
250C22—44no—
251C22—816no—
252C22—81no—
253C22—82no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C2×#12011yescharacteristic
258C22#52512yes—
259C22#57612yes—
260C22#57611yescharacteristic
261C22#57711yescharacteristic
262C4—22no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—42no—
267C4—82no—
268C4—81no—
269C4C2×#12312yes—
270C4#52511yescharacteristic
271C4#52511yescharacteristic
272C2—42no—
273C2—48no—
274C2—161no—
275C2#284911yescharacteristic
276C2#300811yescharacteristic
277C2#301211yescharacteristic
278C2#311812yes—
279C2#312212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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×#518C2semidirect
2. C2×#482C2semidirect
4. #2624C2semidirect
5. #1879C2semidirect
6. #1879C2semidirect
7. #1864C2semidirect
8. #1862C2semidirect
9. #1825C2semidirect
10. #1818C2semidirect
11. #1786C2semidirect
26. C2×#107C22semidirect
27. C2×#106C22semidirect
31. C2×#102C22semidirect
32. C2×#102C22semidirect
38. #390C22semidirect
39. #389C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #336C22semidirect
44. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5182
C2×#4822
C2×#4472
#26242
#18792
#18642
#18622
#18252
#18182
#17862