Finite Group Databasebeta

← #32893

Subgroups — #32893

Order 256.

1815 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1815 conjugacy classes of subgroups — too many to draw. Only its 183 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×#585C211yesmaximal, characteristic
2C2×#456C212yesmaximal, complemented
3C2×#455C212yesmaximal, complemented
4#3007C212yesmaximal, complemented
5#2432C211yesmaximal, characteristic, complemented
6#1793C212yesmaximal, complemented
7#1792C212yesmaximal, complemented
8#1789C212yesmaximal, complemented
9#1785C211yesmaximal, characteristic, complemented
10#585—28no—
11#585—28no—
12C2×2−1+4—22no—
13C2×2−1+4C2212yes—
14C2×H(5,𝔽2)—22no—
15C2×H(5,𝔽2)—22no—
16C2×H(5,𝔽2)C2212yes—
17C22×#41—24no—
18C22×#41—22no—
19C22×#41C2212yes—
20C22×#41C2211yescharacteristic
21C2×#137C2214yescomplemented
22C2×C4⋊Q8C2211yescharacteristic, complemented
23C2×#124C2211yescharacteristic, complemented
24C2×C22⋊Q8C2211yescharacteristic, complemented
25#456—28no—
26#455—28no—
27#452—24no—
28#451—22no—
29#450—22no—
30C2×#104C2214yescomplemented
31C2×#104C2214yescomplemented
32#417C2214yescomplemented
33C2×#101C2212yescomplemented
34C2×#100C2212yescomplemented
35#407C2212yescomplemented
36#394—22no—
37#391C2211yescharacteristic, complemented
38#390C2211yescharacteristic, complemented
39#385—22no—
40C4×C22⋊C4C2211yescharacteristic, complemented
412−1+4—232no—
422−1+4—28no—
432−1+4—24no—
442−1+4D418yes—
45H(5,𝔽2)—216no—
46H(5,𝔽2)—232no—
47H(5,𝔽2)—216no—
48H(5,𝔽2)—28no—
49H(5,𝔽2)—24no—
50H(5,𝔽2)D418yes—
51C2×#41—24no—
52C2×#41—28no—
53C2×#41—24no—
54C2×#41—232no—
55C2×#41—232no—
56C2×#41—216no—
57C2×#41—216no—
58C2×#41—28no—
59C2×#41—28no—
60C2×#41—24no—
61C2×#41—24no—
62C2×#41—24no—
63C2×#41—24no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—24no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41D414yes—
70C2×#41C2314yes—
71C22×Q8—22no—
72C22×Q8—24no—
73C22×Q8D412yes—
74C22×Q8C2311yescharacteristic
75C22×Q8C2311yescharacteristic
76C22×D4—28no—
77C22×D4—22no—
78C22×D4—24no—
79C22×D4—22no—
80C22×D4—24no—
81C22×D4D412yes—
82C22×D4D412yes—
83C22×D4C2311yescharacteristic
84C23×C4—24no—
85C23×C4—22no—
86C23×C4D412yes—
87C23×C4C2311yescharacteristic
88#137—28no—
89#137—28no—
90C2×Q16—24no—
91C2×SD16—24no—
92C2×M16C2314yes—
93C4⋊Q8—24no—
94#124—24no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#104—28no—
107#104—28no—
108#103—24no—
109#102—24no—
110#101—24no—
111#100—24no—
112C22⋊C8—22no—
113C22⋊C8—22no—
114#95—21no—
115#95—22no—
116C24—24no—
117C24—22no—
118C24C2×D412yes—
119C24C2×D411yescharacteristic
120#41—232no—
121#41—28no—
122#41—28no—
123#41—216no—
124#41—264no—
125#41—232no—
126#41—216no—
127#41—216no—
128#41—216no—
129#41—216no—
130#41—216no—
131#41—28no—
132#41—216no—
133#41—28no—
134#41—216no—
135#41—216no—
136#41—24no—
137#41—24no—
138#41C2×D418yes—
139#41C2×D418yes—
140C2×Q8—24no—
141C2×Q8—216no—
142C2×Q8—216no—
143C2×Q8—216no—
144C2×Q8—28no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8C2×D414yes—
152C2×Q8C2×D414yes—
153C2×D4—28no—
154C2×D4—24no—
155C2×D4—264no—
156C2×D4—216no—
157C2×D4—216no—
158C2×D4—28no—
159C2×D4—28no—
160C2×D4—216no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—216no—
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—216no—
167C2×D4—28no—
168C2×D4—22no—
169C2×D4—24no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4C2×D414yes—
173C2×D4C2×D414yes—
174C22×C4—24no—
175C22×C4—28no—
176C22×C4—232no—
177C22×C4—216no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D412yes—
193C22×C4C2×D412yes—
194C22×C4C2×D412yes—
195C22×C4C2411yescharacteristic
196Q16—48no—
197SD16—48no—
198M16—24no—
199M16—24no—
200C2×C8—24no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C22⋊C4—42no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C42—22no—
210C42—22no—
211C23—28no—
212C23—216no—
213C23—22no—
214C23—24no—
215C23—21no—
216C23—22no—
217C23—21no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—82no—
223C23—82no—
224C23—84no—
225C23—84no—
226C23#12012yes—
227C23#12014yes—
228C23C22×D411yescharacteristic
229Q8—216no—
230Q8—28no—
231Q8—24no—
232Q8—28no—
233Q8—44no—
234Q8#12018yes—
235D4—232no—
236D4—216no—
237D4—216no—
238D4—216no—
239D4—28no—
240D4—28no—
241D4—28no—
242D4—28no—
243D4—24no—
244D4#12018yes—
245C2×C4—216no—
246C2×C4—28no—
247C2×C4—216no—
248C2×C4—28no—
249C2×C4—232no—
250C2×C4—28no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—28no—
258C2×C4—28no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4#12014yes—
272C2×C4#12011yescharacteristic
273C2×C4#12011yescharacteristic
274C2×C4C22×D412yes—
275C2×C4C22×D412yes—
276C2×C4C22×D412yes—
277C8—44no—
278C22—21no—
279C22—21no—
280C22—22no—
281C22—24no—
282C22—28no—
283C22—22no—
284C22—22no—
285C22—44no—
286C22—42no—
287C22—44no—
288C22—42no—
289C22—44no—
290C22—81no—
291C22—82no—
292C22—81no—
293C22—88no—
294C22—816no—
295C22C2×#12011yescharacteristic
296C22C2×#12011yescharacteristic
297C22C2×#12011yescharacteristic
298C4—21no—
299C4—28no—
300C4—24no—
301C4—21no—
302C4—42no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4C2×#12014yes—
307C2—21no—
308C2—21no—
309C2—42no—
310C2—44no—
311C2—48no—
312C2#284611yescharacteristic
313C2#301811yescharacteristic
314C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#585C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#456C2semidirect
3. C2×#455C2semidirect
4. #3007C2semidirect
5. #2432C2semidirect
6. #1793C2semidirect
7. #1792C2semidirect
8. #1789C2semidirect
9. #1785C2semidirect
21. C2×#137C22semidirect
22. C2×C4⋊Q8C22semidirect
23. C2×#124C22semidirect
24. C2×C22⋊Q8C22semidirect
30. C2×#104C22semidirect
31. C2×#104C22semidirect
32. #417C22semidirect
33. C2×#101C22semidirect
34. C2×#100C22semidirect
35. #407C22semidirect
37. #391C22semidirect
38. #390C22semidirect
40. C4×C22⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5852
C2×#4562
C2×#4552
#30072
#24322
#17932
#17922
#17892
#17852