Finite Group Databasebeta

← #18149

Subgroups — #18149

Order 256.

537 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 113 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×#538C211yesmaximal, characteristic
2#3095C211yesmaximal, characteristic, complemented
3#3094C211yesmaximal, characteristic, complemented
4#2482C211yesmaximal, characteristic, complemented
5#1618C214yesmaximal, complemented
6#1612C214yesmaximal, complemented
7#1511C211yesmaximal, characteristic, complemented
8#1510C211yesmaximal, characteristic, complemented
9#1462C211yesmaximal, characteristic, complemented
10#538—22no—
11#538—22no—
12#538—22no—
13#538C2214yes—
14C2×#126—21no—
15C2×#124—21no—
16C2×#123—21no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121—21no—
19C2×#121C2211yescharacteristic
20C2×#120—21no—
21C2×C4×D4—21no—
22C2×#117C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24#485—22no—
25#484C2212yescomplemented
26#483C2212yescomplemented
27#482—22no—
28#481C2212yescomplemented
29#480C2212yescomplemented
30#428—21no—
31#428—21no—
32#419—21no—
33#418—21no—
34#407C2211yescharacteristic
35#407C2211yescharacteristic
36#399C2212yescomplemented
37#397C2212yescomplemented
38#388—22no—
39#385—21no—
40#381C2211yescharacteristic
41#379—21no—
42#328C2214yescomplemented
43#327C2214yescomplemented
44#323C2214yescomplemented
45C25D411yescharacteristic
46C22×Q8D411yescharacteristic
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4D411yescharacteristic
51C23×C4—21no—
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C2×M16—21no—
55C2×M16—21no—
56#126—24no—
57#126—24no—
58#124—24no—
59#124—24no—
60#123—24no—
61#123—24no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C22⋊Q8D412yes—
65C22⋊Q8C2312yes—
66#121—24no—
67#121—24no—
68#121—21no—
69#121—21no—
70#121D412yes—
71#121C2312yes—
72#120—24no—
73#120—24no—
74C4×D4—24no—
75C4×D4—24no—
76#117—21no—
77#117—21no—
78#117—21no—
79#117—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4D412yes—
101C2×C22⋊C4C2312yes—
102C2×C42—21no—
103C2×C42—21no—
104#107—22no—
105#107—22no—
106#106—22no—
107#106—22no—
108#103—22no—
109#103—22no—
110#102—22no—
111#102—22no—
112C22⋊C8—21no—
113C22⋊C8—21no—
114C22⋊C8C2312yes—
115C22⋊C8C2312yes—
116#95—21no—
117#95—22no—
118#95C2312yes—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—42no—
123C24—42no—
124C24—81no—
125C24C2×D411yescharacteristic
126C24C2×D412yes—
127C2×Q8—21no—
128C2×Q8—42no—
129C2×Q8C2×D412yes—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—24no—
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—42no—
136C2×D4—44no—
137C2×D4—82no—
138C2×D4—82no—
139C2×D4C2×D412yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—81no—
151C22×C4—82no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D412yes—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159M16—42no—
160M16—42no—
161C2×C8—22no—
162C2×C8—22no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—82no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—42no—
187C22⋊C4—44no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—82no—
191C42—24no—
192C42—41no—
193C42—41no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—81no—
211C23—81no—
212C23—88no—
213C23#12011yescharacteristic
214C23#12011yescharacteristic
215C23C22×D411yescharacteristic
216Q8—42no—
217Q8—81no—
218D4—42no—
219D4—44no—
220D4—44no—
221D4—81no—
222D4—84no—
223D4—84no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—84no—
248C2×C4#12011yescharacteristic
249C2×C4#12011yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C8—42no—
253C8—42no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—82no—
270C22—84no—
271C22—84no—
272C22—84no—
273C22#45812yes—
274C22C2×#12011yescharacteristic
275C22#57512yes—
276C4—21no—
277C4—21no—
278C4—21no—
279C4—21no—
280C4—42no—
281C4—42no—
282C4—42no—
283C4—42no—
284C4—82no—
285C4—81no—
286C4—81no—
287C2—22no—
288C2—22no—
289C2—41no—
290C2—82no—
291C2—82no—
292C2#300511yescharacteristic
293C2#301911yescharacteristic
294C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#538C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3095C2semidirect
3. #3094C2semidirect
4. #2482C2semidirect
5. #1618C2semidirect
6. #1612C2semidirect
7. #1511C2semidirect
8. #1510C2semidirect
9. #1462C2semidirect
25. #484C22semidirect
26. #483C22semidirect
28. #481C22semidirect
29. #480C22semidirect
36. #399C22semidirect
37. #397C22semidirect
42. #328C22semidirect
43. #327C22semidirect
44. #323C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5382
#30952
#30942
#24822
#16182
#16122
#15112
#15102
#14622