Finite Group Databasebeta

← #24217

Subgroups — #24217

Order 256.

779 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 151 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×#124C211yesmaximal, characteristic
2C2×#464C212yesmaximal, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4C2×#450C211yesmaximal, characteristic
5C4×C22⋊Q8C211yesmaximal, characteristic, complemented
6#2042C212yesmaximal, complemented
7#2015C212yesmaximal, complemented
8#1983C212yesmaximal, complemented
9#1797C211yesmaximal, characteristic, complemented
10#1787C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×Q16C2212yescomplemented
15C22×SD16C2212yes—
16C2×#124—28no—
17C2×#124—21no—
18C2×#124—21no—
19C2×#124C2212yes—
20C2×#124C2212yes—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic, complemented
23C22×C22⋊C4—21no—
24C22×C22⋊C4—21no—
25C22×C42C2211yescharacteristic
26#464—28no—
27#451—24no—
28#450—24no—
29C2×C4⋊C8C2212yescomplemented
30C2×#103C2212yescomplemented
31C2×#103C2212yescomplemented
32C2×#103C2212yescomplemented
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#391C2211yescharacteristic, complemented
39#389C2211yescharacteristic, complemented
40#387—21no—
41#385—21no—
42#380C2211yescharacteristic
43C4×C4⋊C4—21no—
44C4×C4⋊C4C2211yescharacteristic, complemented
45C4×C22⋊C4—21no—
46#336C2212yescomplemented
47C25—21no—
48C22×Q8—24no—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—42no—
53C22×Q8C2311yescharacteristic
54C22×Q8C2312yes—
55C22×D4—24no—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4—42no—
60C22×D4C2312yes—
61C23×C4—21no—
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4C2311yescharacteristic
65C2×Q16—24no—
66C2×Q16—24no—
67C2×Q16—42no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—42no—
71C22×C8C2312yes—
72C22×C8C2312yes—
73#124—28no—
74#124—28no—
75#124—24no—
76#124—24no—
77#124—24no—
78#124D418yes—
79C22⋊Q8—22no—
80C22⋊Q8D414yes—
81C4×Q8—24no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—28no—
89C2×C22⋊C4—28no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C42—24no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42D411yescharacteristic
100C2×C42D411yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C4⋊C8—24no—
104#103—24no—
105#103—24no—
106#103—42no—
107#102—24no—
108#102—24no—
109#102—42no—
110C22⋊C8—22no—
111C22⋊C8—22no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117C24—24no—
118C24—22no—
119C24—42no—
120C24—42no—
121C24—42no—
122C24—42no—
123C24—42no—
124C24—42no—
125C24C2×D411yescharacteristic
126C2×Q8—24no—
127C2×Q8—22no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—48no—
136C2×Q8—48no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8C2×D414yes—
140C2×Q8C2×D412yes—
141C2×D4—24no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—48no—
148C2×D4—48no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4C2×D414yes—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D412yes—
174C22×C4#4112yes—
175C22×C4C2411yescharacteristic
176Q16—48no—
177SD16—48no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C4⋊C4—24no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4C2×D412yes—
191C22⋊C4—28no—
192C22⋊C4—28no—
193C22⋊C4—24no—
194C22⋊C4—28no—
195C22⋊C4—28no—
196C22⋊C4—42no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—42no—
201C42—42no—
202C42C2×D414yes—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—24no—
208C23—48no—
209C23—48no—
210C23—48no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—84no—
218C23—84no—
219C23—84no—
220C23—84no—
221C23#12112yes—
222C23#13512yes—
223C23#13712yes—
224C23C22×D411yescharacteristic
225Q8—24no—
226Q8—24no—
227Q8—24no—
228Q8—48no—
229Q8—42no—
230Q8—42no—
231Q8—42no—
232D4—24no—
233D4—24no—
234D4—48no—
235D4—42no—
236D4—42no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4#12012yes—
270C2×C4#12012yes—
271C2×C4#12112yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×D412yes—
274C2×C4C2×#4111yescharacteristic
275C2×C4H(5,𝔽2)12yes—
276C8—42no—
277C8—42no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22—816no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22#46414yescomplemented
293C22C2×#12011yescharacteristic
294C22C2×#12111yescharacteristic
295C22#53511yescharacteristic
296C22C2×#13511yescharacteristic
297C22C2×#13711yescharacteristic
298C22#57511yescharacteristic
299C22#57511yescharacteristic
300C4—22no—
301C4—22no—
302C4—22no—
303C4—44no—
304C4—42no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4D4212yes—
311C4#54612yes—
312C2—24no—
313C2—84no—
314C2#240811yescharacteristic
315C2#300211yescharacteristic
316C2#300711yescharacteristic
317C2C2×#46411yescharacteristic
318C2#304011yescharacteristic
319C2#321111yescharacteristic
320C2#321511yescharacteristic
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×#464C2semidirect
3. C2×#451C2semidirect
5. C4×C22⋊Q8C2semidirect
6. #2042C2semidirect
7. #2015C2semidirect
8. #1983C2semidirect
9. #1797C2semidirect
10. #1787C2semidirect
14. C22×Q16C22semidirect
22. C2×C4×Q8C22semidirect
29. C2×C4⋊C8C22semidirect
30. C2×#103C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
38. #391C22semidirect
39. #389C22semidirect
44. C4×C4⋊C4C22semidirect
46. #336C22semidirect
292. C22#464semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#4642
C2×#4512
C2×#4502
C4×C22⋊Q82
#20422
#20152
#19832
#17972
#17872
#17862