Finite Group Databasebeta

← #25898

Subgroups — #25898

Order 256.

731 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 137 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×#546C211yesmaximal, characteristic
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#476C211yesmaximal, characteristic, complemented
4#2547C211yesmaximal, characteristic, complemented
5#2035C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#1983C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1788C212yesmaximal, complemented
10#1786C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×D4C2211yescharacteristic
14#546—24no—
15#546—24no—
16C2×C4⋊Q8C2211yescharacteristic, complemented
17C2×#124—21no—
18C2×#124C2211yescharacteristic
19C2×#123—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#120—21no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C22⋊C4—21no—
27#489—24no—
28#476—24no—
29C2×#106C2212yescomplemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×#103C2212yescomplemented
32C2×#103C2212yescomplemented
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yescomplemented
39C2×#97C2211yescharacteristic, complemented
40#397—21no—
41#397—21no—
42#391—21no—
43#380C2212yes—
44#380C2212yescomplemented
45C4×C4⋊C4C2211yescharacteristic, complemented
46#340C2212yescomplemented
47C25—21no—
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—24no—
51C2×#41—42no—
52C22×Q8—21no—
53C22×Q8C2311yescharacteristic
54C22×D4—21no—
55C22×D4—21no—
56C22×D4—22no—
57C22×D4—22no—
58C22×D4—24no—
59C22×D4—21no—
60C22×D4—42no—
61C22×D4D411yescharacteristic
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C22×C8C2312yes—
69C22×C8C2312yes—
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72#124—22no—
73#124—44no—
74#124D414yes—
75#123—28no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78#121—28no—
79#121—22no—
80#121—22no—
81#120—28no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4D414yes—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2312yes—
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—28no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4D411yescharacteristic
101C2×C22⋊C4D411yescharacteristic
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104#106—24no—
105C4⋊C8—22no—
106#103—24no—
107#103—42no—
108#102—24no—
109#102—24no—
110#102—24no—
111#102—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114#97—22no—
115#95—21no—
116#95—22no—
117#95—22no—
118C24—21no—
119C24—24no—
120C24—21no—
121C24—44no—
122C24—42no—
123C24—42no—
124C24—82no—
125C24C2×D411yescharacteristic
126C24C2×D411yescharacteristic
127#41—48no—
128#41—44no—
129#41—44no—
130C2×Q8—21no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8C2×D412yes—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—22no—
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—41no—
148C2×D4—44no—
149C2×D4—48no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4—44no—
153C2×D4—82no—
154C2×D4C2×D412yes—
155C2×D4C2×D412yes—
156C2×D4#4114yes—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—44no—
172C22×C4—82no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2411yescharacteristic
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C4⋊C4—24no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4C2×D412yes—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—22no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C42—42no—
203C42C2×D412yes—
204C42C2×D412yes—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—48no—
218C23—48no—
219C23—41no—
220C23—41no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23—42no—
226C23—88no—
227C23—84no—
228C23—88no—
229C23AGL(1,ℤ/8)12yes—
230C23C22×D411yescharacteristic
231Q8—24no—
232Q8—41no—
233Q8—41no—
234Q8—84no—
235D4—22no—
236D4—24no—
237D4—24no—
238D4—44no—
239D4—48no—
240D4—48no—
241D4—41no—
242D4—41no—
243D4—84no—
244D4#12414yescomplemented
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—48no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—84no—
270C2×C4#12012yes—
271C2×C4#12012yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×D411yescharacteristic
274C2×C4C2×#4112yes—
275C2×C4H(5,𝔽2)12yes—
276C8—42no—
277C8—42no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—44no—
286C22—42no—
287C22—44no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—816no—
297C22—82no—
298C22—82no—
299C22C2×#12011yescharacteristic
300C22#54612yes—
301C22C2×AGL(1,ℤ/8)11yescharacteristic
302C22#57511yescharacteristic
303C22#57712yes—
304C4—22no—
305C4—22no—
306C4—41no—
307C4—41no—
308C4—42no—
309C4—42no—
310C4—82no—
311C4—82no—
312C4C2×#12411yescharacteristic
313C4#53511yescharacteristic
314C4#54612yes—
315C2—42no—
316C2—44no—
317C2—81no—
318C2—82no—
319C2#243511yescharacteristic
320C2#300611yescharacteristic
321C2#300911yescharacteristic
322C2#330612yes—
323C2#331112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#546C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#489C2semidirect
3. C2×#476C2semidirect
4. #2547C2semidirect
5. #2035C2semidirect
6. #2034C2semidirect
7. #1983C2semidirect
8. #1955C2semidirect
9. #1788C2semidirect
10. #1786C2semidirect
11. #1762C2semidirect
16. C2×C4⋊Q8C22semidirect
29. C2×#106C22semidirect
30. C2×C4⋊C8C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
38. C2×C22⋊C8C22semidirect
39. C2×#97C22semidirect
44. #380C22semidirect
45. C4×C4⋊C4C22semidirect
46. #340C22semidirect
244. D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4892
C2×#4762
#25472
#20352
#20342
#19832
#19552
#17882
#17862
#17622