Finite Group Databasebeta

← #18167

Subgroups — #18167

Order 256.

609 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 117 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#3007C212yesmaximal, complemented
3#2388C211yesmaximal, characteristic
4#1605C212yesmaximal, complemented
5#1602C212yesmaximal, complemented
6#1599C212yesmaximal, complemented
7#1596C212yesmaximal, complemented
8#1508C212yesmaximal, complemented
9#1465C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C2×#137—22no—
12#554—24no—
13#554—22no—
14D4×Q8—22no—
15#544C2212yes—
16#543—21no—
17#543C2212yes—
18#533—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2212yes—
21#518—21no—
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24#452—24no—
25#451C2214yes—
26#450C2214yes—
27#417—22no—
28#407C2212yes—
29#395C2211yescharacteristic
30#394—21no—
31#394C2211yescharacteristic
32#393C2211yescharacteristic
33#392—21no—
34#392C2211yescharacteristic
35C4×C22⋊C4—21no—
36C4×C22⋊C4C2211yescharacteristic
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—44no—
44C22×Q8—21no—
45C22×Q8D412yescomplemented
46C22×Q8C2311yescharacteristic
47C22×D4—21no—
48C22×D4—21no—
49C22×D4D411yescharacteristic
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54#137—48no—
55C2×Q16—24no—
56C2×SD16—24no—
57C2×M16—22no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—24no—
61#125—22no—
62#125—24no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67C22⋊Q8—24no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—24no—
73C22⋊Q8D414yescomplemented
74C22⋊Q8C2314yes—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×D4—24no—
78C4×D4—24no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—44no—
82#117—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D412yescomplemented
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4C2311yescharacteristic
98C2×C42—21no—
99C2×C42D411yescharacteristic
100#103—24no—
101#102—24no—
102C22⋊C8—22no—
103C22⋊C8C2314yes—
104#95—21no—
105#95—21no—
106#95C2311yescharacteristic
107#95C2311yescharacteristic
108C24—21no—
109C24C2×D411yescharacteristic
110C24C2×D411yescharacteristic
111#41—28no—
112#41—44no—
113#41—44no—
114#41—44no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—42no—
120C2×Q8—44no—
121C2×Q8—82no—
122C2×Q8C2×D414yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—42no—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D412yes—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D412yes—
155C22×C4C2411yescharacteristic
156Q16—48no—
157SD16—48no—
158M16—44no—
159C2×C8—24no—
160C4⋊C4—22no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4C2×D414yes—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—24no—
176C22⋊C4—22no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C42—24no—
183C42—21no—
184C42—21no—
185C42—42no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23#12012yes—
202C23#13712yescomplemented
203C23C22×D411yescharacteristic
204Q8—24no—
205Q8—44no—
206Q8—42no—
207Q8—44no—
208Q8—82no—
209Q8—82no—
210D4—28no—
211D4—24no—
212D4—42no—
213D4—48no—
214D4—48no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—81no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4#12011yescharacteristic
245C2×C4#12011yescharacteristic
246C2×C4#13712yescomplemented
247C2×C4C22×D412yes—
248C8—44no—
249C22—21no—
250C22—21no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22C2×#12011yescharacteristic
269C22C2×#13711yescharacteristic
270C22C2×#13711yescharacteristic
271C4—22no—
272C4—22no—
273C4—21no—
274C4—42no—
275C4—41no—
276C4—44no—
277C4—42no—
278C4—41no—
279C4—44no—
280C4—81no—
281C4B(4,𝔽2)12yes—
282C2—21no—
283C2—21no—
284C2—42no—
285C2—42no—
286C2—82no—
287C2#300411yescharacteristic
288C2#301811yescharacteristic
289C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2388C2
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. #3007C2semidirect
4. #1605C2semidirect
5. #1602C2semidirect
6. #1599C2semidirect
7. #1596C2semidirect
8. #1508C2semidirect
9. #1465C2semidirect
45. C22×Q8D4semidirect
73. C22⋊Q8D4semidirect
87. C2×C4⋊C4D4semidirect
202. C23#137semidirect
246. C2×C4#137semidirect

Maximal subgroups

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

subgroupindex
#35012
#30072
#23882
#16052
#16022
#15992
#15962
#15082
#14652