Finite Group Databasebeta

← #18320

Subgroups — #18320

Order 256.

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

normal subgroups only

This group has 504 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×#539C211yesmaximal, characteristic, complemented
2#3193C211yesmaximal, characteristic, complemented
3#3192C211yesmaximal, characteristic
4#2591C211yesmaximal, characteristic, complemented
5#1615C214yesmaximal, complemented
6#1597C214yesmaximal, complemented
7#1508C211yesmaximal, characteristic, complemented
8#1507C211yesmaximal, characteristic, complemented
9#1463C211yesmaximal, characteristic, complemented
10#539—22no—
11#539—22no—
12#539—22no—
13#539C2214yes—
14C2×#126—21no—
15C2×#125—21no—
16C2×#123—21no—
17C2×#123—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4—21no—
23C22×C4⋊C4C2211yescharacteristic
24#484C2212yescomplemented
25#481—21no—
26#481C2212yes—
27#480—21no—
28#469—22no—
29#468—22no—
30#450C2212yescomplemented
31#448—21no—
32#447—21no—
33#447C2212yes—
34#407C2211yescharacteristic
35#407C2211yescharacteristic
36#400C2212yescomplemented
37#394C2212yescomplemented
38#388—22no—
39#386—21no—
40#386—21no—
41C4×C22⋊C4C2211yescharacteristic
42#328C2214yescomplemented
43#327C2214yes—
44#324C2214yescomplemented
45C22×Q8D411yescharacteristic
46C22×D4—21no—
47C22×D4—21no—
48C22×D4D411yescharacteristic
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C23×C4D411yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C2×SD16—22no—
55C2×D8—22no—
56C2×M16—21no—
57C2×M16—21no—
58#126—24no—
59#126—24no—
60#125—24no—
61#125—24no—
62#123—24no—
63#123—24no—
64#123—24no—
65#123—24no—
66C22⋊Q8—24no—
67C22⋊Q8—24no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C22⋊Q8D412yes—
71C22⋊Q8C2312yes—
72#121—21no—
73#121—21no—
74#121—21no—
75#121—21no—
76#121—21no—
77#121—21no—
78#121D412yes—
79#121C2312yes—
80C4×D4—24no—
81C4×D4—24no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D412yes—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2312yes—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C42—21no—
105C2×C42—21no—
106#107—22no—
107#106—22no—
108#103—22no—
109#102—22no—
110#102—22no—
111#102—22no—
112C22⋊C8—21no—
113C22⋊C8—21no—
114C22⋊C8C2312yes—
115C22⋊C8C2312yes—
116#95—22no—
117#95—22no—
118#95C2312yes—
119C24—21no—
120C24—21no—
121C24C2×D411yescharacteristic
122C2×Q8—21no—
123C2×Q8—81no—
124C2×Q8C2×D412yes—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—44no—
131C2×D4—44no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—81no—
135C2×D4—81no—
136C2×D4C2×D412yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4—82no—
151C22×C4C2×D412yes—
152C22×C4C2×D412yes—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159SD16—82no—
160D8—82no—
161M16—81no—
162M16—81no—
163C2×C8—22no—
164C2×C8—22no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4C2×D412yes—
179C4⋊C4C2×D412yes—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—44no—
188C22⋊C4—42no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—82no—
192C42—24no—
193C42—81no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—41no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—42no—
205C23#12011yescharacteristic
206C23#12011yescharacteristic
207C23C22×D411yescharacteristic
208Q8—42no—
209Q8—81no—
210D4—42no—
211D4—42no—
212D4—42no—
213D4—44no—
214D4—44no—
215D4—81no—
216D4—82no—
217D4—82no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—84no—
245C2×C4#12011yescharacteristic
246C2×C4#12011yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C8—42no—
250C8—42no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—44no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—84no—
263C22—84no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22#45412yes—
268C22C2×#12011yescharacteristic
269C22#57512yes—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—42no—
276C4—42no—
277C4—42no—
278C4—42no—
279C4—82no—
280C4—81no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2—82no—
285C2—161no—
286C2#300511yescharacteristic
287C2C2×#45411yescharacteristic
288C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3192C2
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
1. C2×#539C2semidirect
2. #3193C2semidirect
4. #2591C2semidirect
5. #1615C2semidirect
6. #1597C2semidirect
7. #1508C2semidirect
8. #1507C2semidirect
9. #1463C2semidirect
24. #484C22semidirect
30. #450C22semidirect
36. #400C22semidirect
37. #394C22semidirect
42. #328C22semidirect
44. #324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5392
#31932
#31922
#25912
#16152
#15972
#15082
#15072
#14632