Finite Group Databasebeta

← #32959

Subgroups — #32959

Order 256.

860 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×C4×D4C211yesmaximal, characteristic
2C2×#478C212yesmaximal, complemented
3#2851C211yesmaximal, characteristic, complemented
4#1864C212yesmaximal, complemented
5#1862C212yesmaximal, complemented
6#1853C211yesmaximal, characteristic, complemented
7#1852C211yesmaximal, characteristic, complemented
8#1804C212yesmaximal, complemented
9#1797C211yesmaximal, characteristic, complemented
10#1786C211yesmaximal, characteristic
11#1786C211yesmaximal, characteristic
12C23×D4C2211yescharacteristic
13C24×C4—21no—
14C2×#125C2211yescharacteristic, complemented
15C2×C4×D4—28no—
16C2×C4×D4—21no—
17C2×C4×D4—21no—
18C2×C4×D4C2212yes—
19C2×C4×D4C2212yes—
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4—21no—
22C22×C42C2211yescharacteristic
23#478—28no—
24C2×#107C2212yescomplemented
25C2×#106C2212yescomplemented
26C2×C4⋊C8C2212yescomplemented
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#389C2211yescharacteristic, complemented
34#389C2211yescharacteristic, complemented
35#387—21no—
36#387—21no—
37#385—21no—
38#385—21no—
39#382C2211yescharacteristic, complemented
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#340C2212yescomplemented
43#340C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C22×D4—24no—
47C22×D4—24no—
48C22×D4—21no—
49C22×D4—22no—
50C22×D4Q814yes—
51C22×D4C2312yes—
52C23×C4—28no—
53C23×C4—21no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—44no—
57C23×C4—44no—
58C23×C4D411yescharacteristic
59C23×C4D411yescharacteristic
60C23×C4C2311yescharacteristic
61C22×C8C2312yes—
62C22×C8C2312yes—
63#125—24no—
64C4×D4—28no—
65C4×D4—28no—
66C4×D4—24no—
67C4×D4—24no—
68C4×D4—24no—
69C4×D4D418yes—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C22⋊C4—28no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C42—24no—
87C2×C42D411yescharacteristic
88C2×C42D411yescharacteristic
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91#107—24no—
92#106—24no—
93C4⋊C8—24no—
94#102—24no—
95#102—24no—
96#102—24no—
97#102—24no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106C24—24no—
107C24—24no—
108C24—24no—
109C24—22no—
110C24—42no—
111C24—42no—
112C24—42no—
113C24—42no—
114C24C2×D411yescharacteristic
115C2×D4—24no—
116C2×D4—24no—
117C2×D4—216no—
118C2×D4—216no—
119C2×D4—24no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4C2×D414yes—
124C2×D4C2×Q814yes—
125C2×D4#4114yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—28no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—28no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—432no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×Q812yes—
155C22×C4#4112yes—
156C22×C4C2411yescharacteristic
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4C2×D414yes—
174C22⋊C4—28no—
175C22⋊C4—28no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—22no—
179C42—22no—
180C42—22no—
181C42C2×D414yes—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—216no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—48no—
191C23—48no—
192C23—48no—
193C23—48no—
194C23—48no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—48no—
200C23C22⋊Q812yes—
201C23#13512yes—
202C23AGL(1,ℤ/8)12yes—
203C23C22×D411yescharacteristic
204D4—216no—
205D4—24no—
206D4—24no—
207D4—24no—
208D4C22⋊Q818yes—
209C2×C4—28no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—44no—
225C2×C4—416no—
226C2×C4—416no—
227C2×C4—48no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4#12012yes—
243C2×C4#12012yes—
244C2×C4C22⋊Q812yes—
245C2×C4AGL(1,ℤ/8)12yes—
246C2×C4C22×D412yes—
247C2×C4C22×Q811yescharacteristic
248C2×C4C2×#4112yes—
249C2×C4C2×#4111yescharacteristic
250C8—42no—
251C8—42no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—28no—
259C22—432no—
260C22—44no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—44no—
265C22—44no—
266C22#47814yescomplemented
267C22C2×#12011yescharacteristic
268C22C2×C22⋊Q811yescharacteristic
269C22#52511yescharacteristic
270C22C2×#13511yescharacteristic
271C22C2×AGL(1,ℤ/8)11yescharacteristic
272C22C2×AGL(1,ℤ/8)11yescharacteristic
273C22#57511yescharacteristic
274C4—22no—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—41no—
279C4—44no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4C2×C22⋊Q812yes—
285C4#52512yes—
286C2—24no—
287C2—48no—
288C2#284811yescharacteristic
289C2#300211yescharacteristic
290C2#300311yescharacteristic
291C2C2×#47811yescharacteristic
292C2#307711yescharacteristic
293C2#310511yescharacteristic
294C2#311411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#478C2semidirect
3. #2851C2semidirect
4. #1864C2semidirect
5. #1862C2semidirect
6. #1853C2semidirect
7. #1852C2semidirect
8. #1804C2semidirect
9. #1797C2semidirect
14. C2×#125C22semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
26. C2×C4⋊C8C22semidirect
33. #389C22semidirect
34. #389C22semidirect
39. #382C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #336C22semidirect
266. C22#478semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#4782
#28512
#18642
#18622
#18532
#18522
#18042
#17972
#17862