Finite Group Databasebeta

← #18382

Subgroups — #18382

Order 256.

528 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 127 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
1#3501C211yesmaximal, characteristic, complemented
2#3187C212yesmaximal, complemented
3#2543C211yesmaximal, characteristic
4#1611C212yesmaximal, complemented
5#1608C212yesmaximal, complemented
6#1599C212yesmaximal, complemented
7#1596C212yesmaximal, complemented
8C2×#328C212yesmaximal, complemented
9#1459C211yesmaximal, characteristic, complemented
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21#518—21no—
22C2×C4×D4C2211yescharacteristic
23#483C2214yes—
24#481—22no—
25#465—24no—
26#450C2214yes—
27#448—22no—
28C2×C22⋊C8C2212yes—
29#398C2211yescharacteristic
30#397C2211yescharacteristic
31#394C2211yescharacteristic
32#392C2211yescharacteristic
33#388—21no—
34#386—21no—
35#386—21no—
36C2×#95C2211yescharacteristic
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C22×Q8—21no—
42C22×Q8D412yescomplemented
43C22×D4—21no—
44C22×D4—21no—
45C22×D4D411yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C2×SD16—24no—
51C22×C8—22no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—24no—
55#125—22no—
56#125—24no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C22⋊Q8—24no—
67C22⋊Q8D414yescomplemented
68C22⋊Q8C2314yes—
69#121—22no—
70#121—22no—
71#121—44no—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×D4—24no—
75C4×D4—24no—
76C4×D4—22no—
77C4×D4—22no—
78#117—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D412yescomplemented
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4C2311yescharacteristic
95C2×C42D411yescharacteristic
96#106—24no—
97#103—24no—
98#102—24no—
99C22⋊C8—22no—
100C22⋊C8C2314yes—
101#95—21no—
102#95—21no—
103#95—21no—
104#95C2311yescharacteristic
105#95C2311yescharacteristic
106C24—21no—
107C24C2×D411yescharacteristic
108C24C2×D411yescharacteristic
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—82no—
113C2×Q8—82no—
114C2×Q8C2×D414yes—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—81no—
124C2×D4—81no—
125C2×D4SD1614yescomplemented
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—81no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×D412yes—
146C22×C4C2411yescharacteristic
147SD16—84no—
148C2×C8—24no—
149C2×C8—82no—
150C4⋊C4—22no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4C2×D414yes—
163C22⋊C4—24no—
164C22⋊C4—24no—
165C22⋊C4—22no—
166C22⋊C4—24no—
167C22⋊C4—22no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—41no—
171C22⋊C4—41no—
172C22⋊C4—42no—
173C22⋊C4—81no—
174C42—24no—
175C42—21no—
176C42—21no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23#12012yes—
191C23C2×SD1612yes—
192C23C2×SD1612yes—
193C23#13712yes—
194C23C22×D411yescharacteristic
195Q8—44no—
196Q8—44no—
197Q8—82no—
198Q8—82no—
199D4—28no—
200D4—44no—
201D4—84no—
202D4—84no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—84no—
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4C2×SD1612yes—
234C2×C4C22×D412yes—
235C8—44no—
236C22—21no—
237C22—22no—
238C22—21no—
239C22—22no—
240C22—22no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22#44814yes—
255C22C2×#12011yescharacteristic
256C22C22×SD1611yescharacteristic
257C22C2×#13711yescharacteristic
258C4—22no—
259C4—22no—
260C4—21no—
261C4—21no—
262C4—42no—
263C4—41no—
264C4—44no—
265C4—41no—
266C4—44no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—24no—
272C2—42no—
273C2—161no—
274C2C2×#44811yescharacteristic
275C2#301811yescharacteristic
276C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2543C2
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. #3501C2semidirect
2. #3187C2semidirect
4. #1611C2semidirect
5. #1608C2semidirect
6. #1599C2semidirect
7. #1596C2semidirect
8. C2×#328C2semidirect
9. #1459C2semidirect
42. C22×Q8D4semidirect
67. C22⋊Q8D4semidirect
83. C2×C4⋊C4D4semidirect
125. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#35012
#31872
#25432
#16112
#16082
#15992
#15962
C2×#3282
#14592