Finite Group Databasebeta

← #32908

Subgroups — #32908

Order 256.

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

normal subgroups only

This group has 947 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, complemented
2C2×#463C212yesmaximal, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic
5#2850C211yesmaximal, characteristic, complemented
6#1879C212yesmaximal, complemented
7#1877C212yesmaximal, complemented
8#1874C211yesmaximal, characteristic, complemented
9#1872C211yesmaximal, characteristic, complemented
10#1804C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4—21no—
14C22×SD16C2212yes—
15C22×D8C2212yes—
16C2×#124C2211yescharacteristic, complemented
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4—28no—
20C2×C4×D4—21no—
21C2×C4×D4—21no—
22C2×C4×D4C2212yes—
23C2×C4×D4C2212yes—
24C22×C4⋊C4C2211yescharacteristic
25C22×C22⋊C4—21no—
26C22×C42C2211yescharacteristic
27#463—28no—
28#450—24no—
29#447—24no—
30C2×C4⋊C8C2212yescomplemented
31C2×#103C2212yescomplemented
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yescomplemented
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#391C2211yescharacteristic, complemented
38#390C2211yescharacteristic, complemented
39#388—21no—
40#388—21no—
41#386—21no—
42#386—21no—
43#382C2211yescharacteristic, complemented
44#340C2212yescomplemented
45#340C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—24no—
50C22×D4—24no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4—22no—
54C22×D4D414yes—
55C22×D4C2311yescharacteristic
56C22×D4C2312yes—
57C23×C4—28no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—44no—
62C23×C4—44no—
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4C2311yescharacteristic
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×D8—24no—
70C2×D8—24no—
71C2×D8—24no—
72C22×C8C2312yes—
73C22×C8C2312yes—
74#124—24no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77#121—22no—
78#121—22no—
79C4×D4—28no—
80C4×D4—28no—
81C4×D4—24no—
82C4×D4—24no—
83C4×D4—24no—
84C4×D4D418yescomplemented
85C2×C4⋊C4—21no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2312yes—
91C2×C22⋊C4—28no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—24no—
101C2×C42D411yescharacteristic
102C2×C42D411yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C4⋊C8—24no—
106#103—24no—
107#102—24no—
108#102—24no—
109#102—24no—
110C22⋊C8—22no—
111C22⋊C8—22no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116C24—24no—
117C24—24no—
118C24—24no—
119C24—22no—
120C24—21no—
121C24—42no—
122C24—42no—
123C24—42no—
124C24—42no—
125C24C2×D411yescharacteristic
126C2×Q8—22no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×D4—24no—
131C2×D4—24no—
132C2×D4—216no—
133C2×D4—216no—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—22no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4C2×D414yes—
146C2×D4C2×D414yes—
147C2×D4#4114yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—28no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—28no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—432no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D412yes—
175C22×C4C2×D411yescharacteristic
176C22×C4#4112yes—
177C22×C4C2411yescharacteristic
178SD16—48no—
179D8—48no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C4⋊C4—24no—
187C4⋊C4—22no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C4⋊C4C2×D414yes—
194C22⋊C4—28no—
195C22⋊C4—28no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—22no—
203C42—22no—
204C42—22no—
205C42C2×D414yes—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—216no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—48no—
215C23—48no—
216C23—48no—
217C23—48no—
218C23—48no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—48no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23#12112yes—
232C23#13512yes—
233C23AGL(1,ℤ/8)12yes—
234C23C22×D411yescharacteristic
235Q8—42no—
236Q8—42no—
237D4—216no—
238D4—24no—
239D4—24no—
240D4—24no—
241D4—42no—
242D4—42no—
243D4—84no—
244D4#12118yes—
245C2×C4—28no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—44no—
261C2×C4—416no—
262C2×C4—416no—
263C2×C4—48no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4#12012yes—
276C2×C4#12012yes—
277C2×C4#12112yes—
278C2×C4AGL(1,ℤ/8)12yes—
279C2×C4C22×D411yescharacteristic
280C2×C4C22×D412yes—
281C2×C4C2×#4112yes—
282C2×C4C2×#4111yescharacteristic
283C8—42no—
284C8—42no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—28no—
292C22—432no—
293C22—44no—
294C22—44no—
295C22—44no—
296C22—44no—
297C22—44no—
298C22—44no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22#46314yescomplemented
307C22C2×#12011yescharacteristic
308C22C2×#12111yescharacteristic
309C22#52511yescharacteristic
310C22C2×#13511yescharacteristic
311C22C2×AGL(1,ℤ/8)11yescharacteristic
312C22C2×AGL(1,ℤ/8)11yescharacteristic
313C22#57511yescharacteristic
314C4—22no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—41no—
319C4—44no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4C2×#12112yes—
324C4#52512yes—
325C2—24no—
326C2—48no—
327C2—161no—
328C2#284711yescharacteristic
329C2#300211yescharacteristic
330C2#300311yescharacteristic
331C2C2×#46311yescharacteristic
332C2#303611yescharacteristic
333C2#310511yescharacteristic
334C2#311411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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
1. C22×C4×D4C2semidirect
2. C2×#463C2semidirect
3. C2×#450C2semidirect
5. #2850C2semidirect
6. #1879C2semidirect
7. #1877C2semidirect
8. #1874C2semidirect
9. #1872C2semidirect
10. #1804C2semidirect
11. #1797C2semidirect
16. C2×#124C22semidirect
30. C2×C4⋊C8C22semidirect
31. C2×#103C22semidirect
34. C2×#102C22semidirect
37. #391C22semidirect
38. #390C22semidirect
43. #382C22semidirect
44. #340C22semidirect
45. #340C22semidirect
46. #336C22semidirect
84. C4×D4D4semidirect
306. C22#463semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#4632
C2×#4502
C2×#4472
#28502
#18792
#18772
#18742
#18722
#18042
#17972