Finite Group Databasebeta

← #18124

Subgroups — #18124

Order 256.

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

normal subgroups only

This group has 506 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#3458C211yesmaximal, characteristic, complemented
2#3101C212yesmaximal, complemented
3#2392C211yesmaximal, characteristic
4#1611C212yesmaximal, complemented
5#1608C212yesmaximal, complemented
6#1599C212yesmaximal, complemented
7#1596C212yesmaximal
8#1503C212yesmaximal, complemented
9#1458C211yesmaximal, characteristic, complemented
10#544—22no—
11#544—22no—
12#544—21no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—24no—
17#533—21no—
18#533C2212yes—
19C2×C4×D4—21no—
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22C2×#117C2211yescharacteristic
23#483C2214yes—
24#476—22no—
25#461—22no—
26#450C2214yes—
27C4×SD16—24no—
28#431C2212yes—
29#398C2211yescharacteristic
30#397C2211yescharacteristic
31#396—21no—
32#394—21no—
33#394C2211yescharacteristic
34#392C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—21no—
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C22×D4—21no—
42C22×D4C2311yescharacteristic
43C23×C4—21no—
44C23×C4D411yescharacteristic
45C2×SD16—24no—
46C4⋊Q8—24no—
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49C4⋊Q8—24no—
50C4⋊Q8D412yescomplemented
51C4⋊Q8D412yescomplemented
52#125—24no—
53#125—22no—
54#125—24no—
55#125—22no—
56C22⋊Q8—24no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—24no—
63C22⋊Q8C2314yes—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8D414yescomplemented
68C4×D4—22no—
69C4×D4—22no—
70C4×D4—44no—
71#117—22no—
72#117—22no—
73#117—22no—
74#117—22no—
75#117—22no—
76#117—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4D411yescharacteristic
91C2×C22⋊C4C2311yescharacteristic
92C2×C42—22no—
93C2×C42—21no—
94C2×C42D411yescharacteristic
95C2×C42C2311yescharacteristic
96#106—24no—
97C4⋊C8—22no—
98#103—24no—
99#102—24no—
100C22⋊C8C2314yes—
101C4×C8—22no—
102#95—21no—
103#95—21no—
104#95C2311yescharacteristic
105#95C2311yescharacteristic
106C24—21no—
107C24C2×D411yescharacteristic
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8C2×D414yes—
112C2×D4—22no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—42no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4SD1614yescomplemented
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136SD16—48no—
137C2×C8—24no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—22no—
150C4⋊C4—24no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4C2×D414yes—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—22no—
158C22⋊C4—22no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C42—24no—
165C42—21no—
166C42—24no—
167C42—21no—
168C42—42no—
169C42C2×D412yes—
170C42C2×D412yes—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23C2×SD1612yes—
183C23C22×D411yescharacteristic
184Q8—44no—
185Q8—82no—
186Q8—82no—
187Q8—82no—
188D4—24no—
189D4—42no—
190D4—48no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4#12012yes—
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4C2×SD1612yes—
220C2×C4C2×SD1612yes—
221C2×C4#13712yes—
222C2×C4C22×D412yes—
223C8—44no—
224C22—21no—
225C22—21no—
226C22—22no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22C2×#12011yescharacteristic
239C22C22×SD1611yescharacteristic
240C22C2×#13711yescharacteristic
241C4—22no—
242C4—22no—
243C4—22no—
244C4—42no—
245C4—41no—
246C4—44no—
247C4—41no—
248C4—44no—
249C4—81no—
250C4—81no—
251C4#44812yes—
252C4#44812yes—
253C2—21no—
254C2—21no—
255C2—42no—
256C2—82no—
257C2C2×#44811yescharacteristic
258C2#301811yescharacteristic
259C2#302511yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC22×C4C24
socleC22C2×#120
prime radical#392C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3458C2semidirect
2. #3101C2semidirect
4. #1611C2semidirect
5. #1608C2semidirect
6. #1599C2semidirect
8. #1503C2semidirect
9. #1458C2semidirect
50. C4⋊Q8D4semidirect
51. C4⋊Q8D4semidirect
67. C4×Q8D4semidirect
129. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#34582
#31012
#23922
#16112
#16082
#15992
#15962
#15032
#14582