Finite Group Databasebeta

← #33350

Subgroups — #33350

Order 256.

620 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3485C211yesmaximal, characteristic
2#2888C211yesmaximal, characteristic, complemented
3#2883C211yesmaximal, characteristic, complemented
4#2123C212yesmaximal, complemented
5#2121C214yesmaximal, complemented
6#2120C212yesmaximal, complemented
7C2×#352C214yesmaximal, complemented
8C2×H(5,𝔽2)C2211yescharacteristic
9C22×#41C411yescharacteristic, complemented
10#568—21no—
11#547—21no—
12#547—22no—
13#547—21no—
14#546—21no—
15#546—22no—
16#546—21no—
17#536C411yescharacteristic, complemented
18#536C2211yescharacteristic
19#534C412yescomplemented
20#534C2212yes—
21#525C411yescharacteristic, complemented
22#525C2211yescharacteristic
23C2×#123C412yescomplemented
24C2×#123C2212yes—
25#518—21no—
26#518C411yescharacteristic, complemented
27#413—22no—
28#413C2212yes—
29C2×#100—21no—
30C2×#100C2214yes—
31#410—21no—
32#410C2212yes—
33C2×#99—22no—
34C2×#99C2214yes—
35#352C2214yescomplemented
36#352C2218yescomplemented
37AΣL(1,𝔽16)C2214yescomplemented
38H(5,𝔽2)—44no—
39C2×#41—21no—
40C2×#41—22no—
41C2×#41—21no—
42C2×#41—44no—
43C2×#41C2×C411yescharacteristic
44C2×#41D412yes—
45C2×#41D411yescharacteristic
46C2×#41D411yescharacteristic
47C2×#41C2311yescharacteristic
48C22×Q8C2×C411yescharacteristic, complemented
49C22×D4C2×C412yes—
50C22×D4C2×C411yescharacteristic
51C22×D4C2×C412yescomplemented
52C22×D4D412yes—
53C22×D4D411yescharacteristic
54C22×D4D411yescharacteristic
55C22×D4C2312yes—
56C23×C4C2×C412yescomplemented
57C23×C4C2×C411yescharacteristic, complemented
58#131—44no—
59C2×M16—22no—
60C2×M16—24no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63#124—21no—
64#124—21no—
65#123—22no—
66#123—22no—
67#123—22no—
68#123—22no—
69#123—22no—
70#123—22no—
71#123C2×C411yescharacteristic, complemented
72#123C2×C412yescomplemented
73#123C2×C412yescomplemented
74#123C2×C412yescomplemented
75#123C2×C412yescomplemented
76#123C2311yescharacteristic
77#123C2312yes—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—22no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83#121—21no—
84#121—22no—
85#121—22no—
86#121—21no—
87#121—22no—
88#121—22no—
89#120—22no—
90#120—22no—
91#120C2×C412yescomplemented
92#120C2×C411yescharacteristic, complemented
93#120C2311yescharacteristic
94C4×D4—22no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—44no—
101#117—22no—
102#117C2×C411yescharacteristic, complemented
103#117C2×C411yescharacteristic, complemented
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
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—
117C24C22⋊C412yes—
118C24C22×C412yes—
119C24C22×C411yescharacteristic
120C24C2×D412yes—
121#41—24no—
122#41—42no—
123#41—44no—
124#41—42no—
125#41—42no—
126#41—48no—
127#41—44no—
128#41—44no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—42no—
132C2×Q8C22⋊C411yescharacteristic
133C2×Q8C2×D411yescharacteristic
134C2×D4—24no—
135C2×D4—24no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4C22⋊C412yes—
147C2×D4C22⋊C411yescharacteristic
148C2×D4C22⋊C412yes—
149C2×D4C22⋊C412yes—
150C2×D4C22⋊C411yescharacteristic
151C2×D4C22⋊C411yescharacteristic
152C2×D4C22×C411yescharacteristic
153C2×D4C22×C412yes—
154C2×D4C2×D412yes—
155C2×D4C2×D411yescharacteristic
156C2×D4C2×D412yes—
157C2×D4C2411yescharacteristic
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4C22⋊C412yes—
168C22×C4C22⋊C412yes—
169C22×C4C22×C411yescharacteristic
170C22×C4C22×C411yescharacteristic
171C22×C4C22×C412yes—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D412yes—
174C22×C4C2×D411yescharacteristic
175M16—24no—
176M16—44no—
177M16—42no—
178C2×C8—44no—
179C2×C8—42no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C22⋊C4—22no—
191C22⋊C4—21no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—21no—
198C22⋊C4—22no—
199C22⋊C4—21no—
200C22⋊C4—22no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C22⋊C4—44no—
204C22⋊C4C22×C412yes—
205C22⋊C4C22×C411yescharacteristic
206C22⋊C4C22×C411yescharacteristic
207C42—21no—
208C42—21no—
209C42—42no—
210C23—24no—
211C23—24no—
212C23—21no—
213C23—42no—
214C23—41no—
215C23—82no—
216C23—81no—
217C23—82no—
218C23—82no—
219C23C2×C22⋊C412yes—
220C23C2×C22⋊C411yescharacteristic
221C23C2×C22⋊C412yes—
222C23C2×C22⋊C412yes—
223C23C23×C411yescharacteristic
224C23C22×D411yescharacteristic
225Q8—22no—
226Q8—41no—
227Q8—42no—
228Q8—42no—
229D4—24no—
230D4—24no—
231D4—24no—
232D4—22no—
233D4—22no—
234D4—41no—
235D4—42no—
236D4—42no—
237D4—41no—
238D4—42no—
239D4—41no—
240D4—44no—
241D4—44no—
242D4—82no—
243D4—81no—
244D4—81no—
245D4#9914yes—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—164no—
267C2×C4C2×C22⋊C411yescharacteristic
268C2×C4C2×C22⋊C411yescharacteristic
269C2×C4C2×C22⋊C412yes—
270C2×C4C2×C22⋊C411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C8—44no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—84no—
288C22—84no—
289C22—82no—
290C22—162no—
291C22C2×#9914yes—
292C22C22×C22⋊C411yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—22no—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—41no—
301C4—42no—
302C4—84no—
303C4C2×#9912yes—
304C2—21no—
305C2—24no—
306C2—41no—
307C2—41no—
308C2—42no—
309C2—42no—
310C2—41no—
311C2—81no—
312C2C22×#9911yescharacteristic
subgroupG/H
center, socleC2C22×#99
commutatorC23C23×C4
Frattini, cosocleC2×D4C24
prime radical#3485C2
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. #2123C2semidirect
5. #2121C2semidirect
6. #2120C2semidirect
7. C2×#352C2semidirect
9. C22×#41C4semidirect
17. #536C4semidirect
19. #534C4semidirect
21. #525C4semidirect
23. C2×#123C4semidirect
26. #518C4semidirect
35. #352C22semidirect
36. #352C22semidirect
37. AΣL(1,𝔽16)C22semidirect
48. C22×Q8C2×C4semidirect
51. C22×D4C2×C4semidirect
56. C23×C4C2×C4semidirect
57. C23×C4C2×C4semidirect
71. #123C2×C4semidirect
72. #123C2×C4semidirect
73. #123C2×C4semidirect
74. #123C2×C4semidirect
75. #123C2×C4semidirect
91. #120C2×C4semidirect
92. #120C2×C4semidirect
102. #117C2×C4semidirect
103. #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
#34852
#28882
#28832
#21232
#21212
#21202
C2×#3522