Finite Group Databasebeta

← #23479

Subgroups — #23479

Order 256.

805 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#427C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic, complemented
4C2×#416C211yesmaximal, characteristic, complemented
5C22×C22⋊C8C211yesmaximal, characteristic, complemented
6#2280C211yesmaximal, characteristic, complemented
7#2036C212yesmaximal, complemented
8#2035C212yesmaximal, complemented
9#2034C212yesmaximal, complemented
10#2033C212yesmaximal, complemented
11#1735C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C23×C8C2211yescharacteristic
16#525—24no—
17#525—24no—
18C2×#123—21no—
19C2×C22⋊Q8C2212yes—
20C2×#121C2212yes—
21C2×#120—21no—
22C2×C4×D4—22no—
23C2×#117C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#427—24no—
27C2×#107C2212yescomplemented
28C2×#106C2212yescomplemented
29#416—24no—
30#416—24no—
31C2×#103C2211yescharacteristic, complemented
32C2×#103C2212yescomplemented
33C2×#103C2211yescharacteristic, complemented
34C2×#102C2211yescharacteristic, complemented
35C2×#102C2212yescomplemented
36C2×#102C2211yescharacteristic, complemented
37C2×C22⋊C8—24no—
38C2×C22⋊C8C2212yes—
39C2×C22⋊C8C2212yes—
40#381—21no—
41#380C2212yes—
42#380C2212yes—
43#379—21no—
44C4×C22⋊C4—22no—
45#340C2212yescomplemented
46#340C2212yescomplemented
47C25D411yescharacteristic
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—22no—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41D414yes—
54C22×Q8C2311yescharacteristic
55C22×D4—22no—
56C22×D4—21no—
57C22×D4C2311yescharacteristic
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4D411yescharacteristic
66C23×C4D412yes—
67C23×C4C2312yes—
68C23×C4C2311yescharacteristic
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8D414yescomplemented
75C22×C8C2312yes—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#123—28no—
79C22⋊Q8—24no—
80C22⋊Q8—24no—
81#121—24no—
82#121—24no—
83#120—28no—
84C4×D4—216no—
85#117—22no—
86#117—22no—
87#117—22no—
88#117—22no—
89#117—22no—
90#117D414yes—
91C2×C4⋊C4—22no—
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2312yes—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106#107—24no—
107#106—24no—
108#103—24no—
109#103—22no—
110#103—22no—
111#102—24no—
112#102—22no—
113#102—22no—
114C22⋊C8—24no—
115C22⋊C8—24no—
116#95—22no—
117#95—22no—
118C24—22no—
119C24—24no—
120C24—22no—
121C24—22no—
122C24—21no—
123C24—42no—
124C24C2×D412yes—
125C24C2×D411yescharacteristic
126#41—28no—
127#41—28no—
128#41—44no—
129#41—44no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8C2×D412yes—
136C2×D4—28no—
137C2×D4—24no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—44no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—44no—
147C2×D4C2×D412yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4C2×D412yes—
176C22×C4C2×D411yescharacteristic
177C22×C4#4112yes—
178C22×C4#4112yes—
179C22×C4#4112yes—
180C22×C4C2411yescharacteristic
181C2×C8—28no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—44no—
190C2×C8C2×D412yes—
191C2×C8C2×D412yes—
192C4⋊C4—28no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—22no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4C2×D414yes—
201C22⋊C4—28no—
202C22⋊C4—28no—
203C22⋊C4—24no—
204C22⋊C4—44no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—44no—
208C42—24no—
209C42—42no—
210C42—42no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—48no—
222C23—48no—
223C23—44no—
224C23—44no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—42no—
229C23#12012yes—
230C23#12112yes—
231C23#12312yes—
232C23#13514yes—
233C23C22×D411yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—41no—
237Q8—41no—
238D4—22no—
239D4—22no—
240D4—416no—
241D4—48no—
242D4—41no—
243D4—41no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4#12012yes—
268C2×C4#12114yes—
269C2×C4#12114yes—
270C2×C4#12112yes—
271C2×C4#12314yes—
272C2×C4#12312yes—
273C2×C4#12414yes—
274C2×C4C22×D411yescharacteristic
275C2×C4C22×D411yescharacteristic
276C2×C4C22×D411yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C8—22no—
281C8—22no—
282C8—42no—
283C22—24no—
284C22—24no—
285C22—22no—
286C22—22no—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—416no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—44no—
297C22—82no—
298C22—81no—
299C22—81no—
300C22#39414yes—
301C22C2×#12011yescharacteristic
302C22C2×#12111yescharacteristic
303C22C2×#12311yescharacteristic
304C22C2×#13512yes—
305C22#57512yes—
306C4—42no—
307C4—42no—
308C4—44no—
309C4—82no—
310C4—81no—
311C4—81no—
312C4#39414yes—
313C4C2×#12111yescharacteristic
314C4C2×#12111yescharacteristic
315C4C2×#12311yescharacteristic
316C4C2×#12411yescharacteristic
317C2—24no—
318C2—44no—
319C2—82no—
320C2C2×#39411yescharacteristic
321C2#300212yes—
322C2#305011yescharacteristic
323C2#305411yescharacteristic
324C2#309112yes—
subgroupG/H
centerC22×C4C2×D4
commutator, Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC2×#525C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#427C2semidirect
3. C2×#416C2semidirect
4. C2×#416C2semidirect
5. C22×C22⋊C8C2semidirect
6. #2280C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2034C2semidirect
10. #2033C2semidirect
11. #1735C2semidirect
27. C2×#107C22semidirect
28. C2×#106C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
33. C2×#103C22semidirect
34. C2×#102C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
45. #340C22semidirect
46. #340C22semidirect
74. C22×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4272
C2×#4162
C22×C22⋊C82
#22802
#20362
#20352
#20342
#20332
#17352