Finite Group Databasebeta

← #23976

Subgroups — #23976

Order 256.

741 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 741 conjugacy classes of subgroups — too many to draw. Only its 133 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×#555C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic, complemented
5#2023C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1895C211yesmaximal, characteristic, complemented
8#1879C212yesmaximal, complemented
9#1875C212yesmaximal, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1825C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2212yes—
14C22×M16C2211yescharacteristic
15#555—24no—
16#555—24no—
17C2×C4⋊Q8C2211yescharacteristic, complemented
18C2×#124—21no—
19C2×#124—21no—
20C2×#121—21no—
21C2×#121C2212yes—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic, complemented
24C2×#120—21no—
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic, complemented
27C2×#117—21no—
28C2×#117C2211yescharacteristic
29C2×#117C2211yescharacteristic, complemented
30#447—24no—
31#418—24no—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yescomplemented
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2212yescomplemented
38#394—21no—
39#392—21no—
40#390C2212yes—
41#380C2211yescharacteristic
42#379—21no—
43#340C2212yescomplemented
44#340C2212yescomplemented
45#340C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C22×Q8—21no—
49C22×D4—24no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—22no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—21no—
56C22×D4—42no—
57C22×D4D411yescharacteristic
58C22×D4C2311yescharacteristic
59C22×D4C2312yes—
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C2×D8—24no—
64C2×D8—24no—
65C2×D8—42no—
66C2×M16—22no—
67C2×M16—42no—
68C22×C8C2312yes—
69C22×C8C2312yes—
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72#124—28no—
73#124—28no—
74#121—28no—
75#121—22no—
76#121—22no—
77#121—22no—
78#121—24no—
79#121—42no—
80#121D414yescomplemented
81#120—44no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86#117—28no—
87#117—22no—
88#117—22no—
89#117—22no—
90#117D414yescomplemented
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4D411yescharacteristic
107C2×C22⋊C4D411yescharacteristic
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111#102—24no—
112#102—24no—
113#102—24no—
114#102—42no—
115C22⋊C8—22no—
116C22⋊C8—42no—
117#95—21no—
118C24—21no—
119C24—22no—
120C24—44no—
121C24—42no—
122C24—42no—
123C24—42no—
124C24—82no—
125C24C2×D411yescharacteristic
126C24C2×D411yescharacteristic
127C2×Q8—24no—
128C2×Q8—82no—
129C2×D4—28no—
130C2×D4—22no—
131C2×D4—24no—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—48no—
143C2×D4—41no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4—44no—
148C2×D4—42no—
149C2×D4—82no—
150C2×D4—82no—
151C2×D4—82no—
152C2×D4C2×D412yes—
153C2×D4#4114yes—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D412yes—
171C22×C4C2×D412yes—
172C22×C4C2×D411yescharacteristic
173C22×C4C2411yescharacteristic
174D8—48no—
175M16—44no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4C2×D412yes—
190C4⋊C4C2×D412yes—
191C4⋊C4C2×D412yes—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—24no—
196C22⋊C4—22no—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—44no—
200C22⋊C4—42no—
201C22⋊C4—44no—
202C22⋊C4—82no—
203C42—24no—
204C42—24no—
205C42—21no—
206C42—21no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—22no—
213C23—24no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—48no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—84no—
230C23—84no—
231C23—88no—
232C23—88no—
233C23#12012yes—
234C23C22×D411yescharacteristic
235Q8—84no—
236D4—24no—
237D4—24no—
238D4—24no—
239D4—44no—
240D4—48no—
241D4—41no—
242D4—41no—
243D4—42no—
244D4—42no—
245D4—84no—
246D4—84no—
247D4—88no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—48no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#12012yes—
273C2×C4AGL(1,ℤ/8)12yes—
274C2×C4C22×D411yescharacteristic
275C2×C4C22×D411yescharacteristic
276C2×C4C2×#4112yes—
277C2×C4H(5,𝔽2)12yes—
278C8—42no—
279C8—42no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—44no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—42no—
293C22—44no—
294C22—42no—
295C22—44no—
296C22—816no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22C2×#12011yescharacteristic
304C22#54612yes—
305C22C2×AGL(1,ℤ/8)11yescharacteristic
306C22#57511yescharacteristic
307C22#57612yes—
308C4—22no—
309C4—22no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—42no—
314C4—42no—
315C4—82no—
316C4#52511yescharacteristic
317C4#53511yescharacteristic
318C4#54612yes—
319C2—42no—
320C2—81no—
321C2—82no—
322C2—84no—
323C2#240411yescharacteristic
324C2#300611yescharacteristic
325C2#300811yescharacteristic
326C2#331912yes—
327C2#332512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#555C2semidirect
2. C2×#447C2semidirect
3. C2×#418C2semidirect
4. #2408C2semidirect
5. #2023C2semidirect
6. #1999C2semidirect
7. #1895C2semidirect
8. #1879C2semidirect
9. #1875C2semidirect
10. #1870C2semidirect
11. #1825C2semidirect
17. C2×C4⋊Q8C22semidirect
23. C2×#121C22semidirect
26. C2×C4×D4C22semidirect
29. C2×#117C22semidirect
35. C2×#102C22semidirect
37. C2×C22⋊C8C22semidirect
43. #340C22semidirect
44. #340C22semidirect
45. #340C22semidirect
46. #336C22semidirect
80. #121D4semidirect
90. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4472
C2×#4182
#24082
#20232
#19992
#18952
#18792
#18752
#18702
#18252