Finite Group Databasebeta

← #33348

Subgroups — #33348

Order 256.

705 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3484C211yesmaximal, characteristic
2#2888C211yesmaximal, characteristic, complemented
3#2883C211yesmaximal, characteristic, complemented
4#2122C212yesmaximal, complemented
5#2121C214yesmaximal, complemented
6#2120C212yesmaximal, complemented
7C2×AΣL(1,𝔽16)C214yesmaximal, complemented
8C2×H(5,𝔽2)C2211yescharacteristic
9C23×D4C411yescharacteristic, complemented
10#568—21no—
11#546—21no—
12#546—22no—
13#546—21no—
14D42—22no—
15D42—21no—
16D42—21no—
17#535C411yescharacteristic, complemented
18#535C2211yescharacteristic
19#534C412yescomplemented
20#534C2212yes—
21#525C411yescharacteristic, complemented
22#525C2211yescharacteristic
23C2×#120C412yescomplemented
24C2×#120C2212yes—
25#518—21no—
26#518C411yescharacteristic, complemented
27#413—22no—
28#413C2212yescomplemented
29C2×#100—21no—
30C2×#100C2214yescomplemented
31#410—21no—
32#410C2212yescomplemented
33C2×#99—22no—
34C2×#99C2214yescomplemented
35#352C2214yescomplemented
36AΣL(1,𝔽16)C2214yescomplemented
37AΣL(1,𝔽16)C2218yescomplemented
38C25C2×C412yescomplemented
39H(5,𝔽2)—44no—
40C2×#41C2×C411yescharacteristic
41C2×#41D412yes—
42C2×#41D411yescharacteristic
43C2×#41D411yescharacteristic
44C2×#41C2311yescharacteristic
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—44no—
51C22×D4C2×C412yes—
52C22×D4C2×C411yescharacteristic
53C22×D4C2×C412yescomplemented
54C22×D4C2×C411yescharacteristic, complemented
55C22×D4D412yes—
56C22×D4D411yescharacteristic
57C22×D4D411yescharacteristic
58C22×D4C2312yes—
59C23×C4C2×C411yescharacteristic, complemented
60#131—44no—
61C2×M16—22no—
62C2×M16—24no—
63#127—21no—
64#127—21no—
65#124—21no—
66#124—21no—
67#123—22no—
68#123—22no—
69#123C2×C412yescomplemented
70#123C2×C411yescharacteristic, complemented
71#123C2311yescharacteristic
72C22⋊Q8—21no—
73C22⋊Q8—21no—
74#121—21no—
75#121—22no—
76#121—22no—
77#121—21no—
78#121—22no—
79#121—22no—
80#121—22no—
81#121—21no—
82#121—21no—
83#120—22no—
84#120—22no—
85#120—22no—
86#120—22no—
87#120—22no—
88#120—22no—
89#120C2×C411yescharacteristic, complemented
90#120C2×C412yescomplemented
91#120C2×C412yescomplemented
92#120C2×C412yescomplemented
93#120C2×C412yescomplemented
94#120C2311yescharacteristic
95#120C2312yes—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—22no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—44no—
103#117—22no—
104#117C2×C411yescharacteristic, complemented
105#117C2×C411yescharacteristic, complemented
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4C2×C412yescomplemented
110C2×C22⋊C4C2×C412yescomplemented
111#101—22no—
112#100—24no—
113#100C2314yes—
114#99—22no—
115#99—24no—
116#99C2314yes—
117C24—22no—
118C24—22no—
119C24—44no—
120C24—84no—
121C24C22⋊C412yes—
122C24C22×C412yes—
123C24C22×C411yescharacteristic
124C24C22×C412yes—
125C24C2×D412yes—
126#41—24no—
127#41—42no—
128#41—44no—
129#41—42no—
130#41—42no—
131C2×Q8C22⋊C411yescharacteristic
132C2×Q8C2×D411yescharacteristic
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—42no—
147C2×D4—44no—
148C2×D4—42no—
149C2×D4—44no—
150C2×D4—44no—
151C2×D4—48no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—84no—
156C2×D4—84no—
157C2×D4C22⋊C412yes—
158C2×D4C22⋊C411yescharacteristic
159C2×D4C22⋊C412yes—
160C2×D4C22⋊C412yes—
161C2×D4C22⋊C411yescharacteristic
162C2×D4C22⋊C411yescharacteristic
163C2×D4C22×C411yescharacteristic
164C2×D4C22×C412yes—
165C2×D4C2×D412yes—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D412yes—
168C2×D4C2411yescharacteristic
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4C22⋊C412yes—
176C22×C4C22⋊C412yes—
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D412yes—
181C22×C4C2×D411yescharacteristic
182M16—24no—
183M16—44no—
184M16—42no—
185C2×C8—44no—
186C2×C8—42no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—21no—
204C22⋊C4—22no—
205C22⋊C4—24no—
206C22⋊C4—44no—
207C22⋊C4C22×C411yescharacteristic
208C22⋊C4C22×C412yes—
209C22⋊C4C22×C411yescharacteristic
210C42—21no—
211C42—21no—
212C42—42no—
213C23—21no—
214C23—24no—
215C23—24no—
216C23—22no—
217C23—42no—
218C23—41no—
219C23—42no—
220C23—44no—
221C23—44no—
222C23—44no—
223C23—42no—
224C23—44no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—82no—
229C23—82no—
230C23—82no—
231C23—88no—
232C23—88no—
233C23C2×C22⋊C412yes—
234C23C2×C22⋊C412yes—
235C23C2×C22⋊C411yescharacteristic
236C23C2×C22⋊C412yes—
237C23C23×C411yescharacteristic
238C23C22×D411yescharacteristic
239Q8—22no—
240Q8—41no—
241D4—24no—
242D4—24no—
243D4—24no—
244D4—22no—
245D4—22no—
246D4—41no—
247D4—42no—
248D4—42no—
249D4—41no—
250D4—42no—
251D4—41no—
252D4—42no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—42no—
257D4—44no—
258D4—42no—
259D4—82no—
260D4—82no—
261D4—82no—
262D4—82no—
263D4#9914yes—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—164no—
281C2×C4C2×C22⋊C411yescharacteristic
282C2×C4C2×C22⋊C411yescharacteristic
283C2×C4C2×C22⋊C412yes—
284C2×C4C2×C22⋊C411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C8—44no—
287C22—22no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—84no—
307C22—84no—
308C22—88no—
309C22—88no—
310C22—84no—
311C22—162no—
312C22C2×#9914yes—
313C22C22×C22⋊C411yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—22no—
318C4—42no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—84no—
323C4C2×#9912yes—
324C2—21no—
325C2—24no—
326C2—41no—
327C2—42no—
328C2—41no—
329C2—42no—
330C2—42no—
331C2—41no—
332C2—82no—
333C2C22×#9911yescharacteristic
subgroupG/H
center, socleC2C22×#99
commutatorC23C23×C4
Frattini, cosocleC2×D4C24
prime radical#3484C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ G
LECSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2888C2semidirect
3. #2883C2semidirect
4. #2122C2semidirect
5. #2121C2semidirect
6. #2120C2semidirect
7. C2×AΣL(1,𝔽16)C2semidirect
9. C23×D4C4semidirect
17. #535C4semidirect
19. #534C4semidirect
21. #525C4semidirect
23. C2×#120C4semidirect
26. #518C4semidirect
28. #413C22semidirect
30. C2×#100C22semidirect
32. #410C22semidirect
34. C2×#99C22semidirect
35. #352C22semidirect
36. AΣL(1,𝔽16)C22semidirect
37. AΣL(1,𝔽16)C22semidirect
38. C25C2×C4semidirect
53. C22×D4C2×C4semidirect
54. C22×D4C2×C4semidirect
59. C23×C4C2×C4semidirect
69. #123C2×C4semidirect
70. #123C2×C4semidirect
89. #120C2×C4semidirect
90. #120C2×C4semidirect
91. #120C2×C4semidirect
92. #120C2×C4semidirect
93. #120C2×C4semidirect
104. #117C2×C4semidirect
105. #117C2×C4semidirect
109. C2×C22⋊C4C2×C4semidirect
110. C2×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34842
#28882
#28832
#21222
#21212
#21202
C2×AΣL(1,𝔽16)2