Finite Group Databasebeta

← #18196

Subgroups — #18196

Order 256.

685 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 129 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
2C2×#450C212yesmaximal, complemented
3#2408C211yesmaximal, characteristic
4#1599C214yesmaximal, complemented
5#1596C214yesmaximal, complemented
6#1520C212yesmaximal, complemented
7#1459C211yesmaximal, characteristic, complemented
8C23×D4C2211yescharacteristic
9C22×SD16—22no—
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544C2212yescomplemented
14#543—21no—
15#543C2212yescomplemented
16#533—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×#121—21no—
20C2×#120—21no—
21#518—21no—
22C2×C4×D4C2211yescharacteristic
23#450—24no—
24#450C2218yes—
25C2×#102—22no—
26C2×C22⋊C8C2212yes—
27#394C2212yes—
28#392C2212yes—
29#390—21no—
30#380C2211yescharacteristic
31#379—21no—
32#328C2218yescomplemented
33#324C2212yescomplemented
34#323C2212yescomplemented
35C25—21no—
36C22×Q8—21no—
37C22×Q8D412yescomplemented
38C22×D4—22no—
39C22×D4—22no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4—44no—
43C22×D4D411yescharacteristic
44C22×D4C2312yes—
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C2×SD16—28no—
48C2×SD16—48no—
49C22×C8—22no—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8—24no—
53#125—22no—
54#125—24no—
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—24no—
65C22⋊Q8D414yescomplemented
66C22⋊Q8C2314yes—
67#121—44no—
68#120—44no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×D4—24no—
72C4×D4—24no—
73C4×D4—22no—
74C4×D4—22no—
75#117—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D412yescomplemented
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4D411yescharacteristic
89C2×C22⋊C4C2311yescharacteristic
90C2×C42D411yescharacteristic
91#102—28no—
92C22⋊C8—22no—
93C22⋊C8C2314yes—
94#95C2312yes—
95C24—22no—
96C24—44no—
97C24—42no—
98C24—42no—
99C24—42no—
100C24—82no—
101C24C2×D411yescharacteristic
102C24C2×D411yescharacteristic
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—44no—
107C2×Q8—82no—
108C2×Q8C2×D414yes—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—28no—
112C2×D4—24no—
113C2×D4—24no—
114C2×D4—42no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—48no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—82no—
123C2×D4SD1614yescomplemented
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4C2411yescharacteristic
140SD16—416no—
141C2×C8—24no—
142C2×C8—44no—
143C4⋊C4—22no—
144C4⋊C4—24no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—42no—
154C4⋊C4C2×D414yes—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—22no—
158C22⋊C4—24no—
159C22⋊C4—22no—
160C22⋊C4—44no—
161C22⋊C4—42no—
162C22⋊C4—44no—
163C22⋊C4—82no—
164C42—24no—
165C42—21no—
166C42—21no—
167C23—21no—
168C23—21no—
169C23—24no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—42no—
175C23—48no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—84no—
182C23—88no—
183C23—88no—
184C23—84no—
185C23#12012yes—
186C23C2×SD1612yes—
187C23C2×SD1612yescomplemented
188C23#13712yes—
189C23C22×D411yescharacteristic
190Q8—44no—
191Q8—44no—
192Q8—82no—
193Q8—82no—
194D4—28no—
195D4—28no—
196D4—44no—
197D4—44no—
198D4—44no—
199D4—88no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—81no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4C2×SD1612yescomplemented
224C2×C4C22×D412yes—
225C8—44no—
226C22—21no—
227C22—22no—
228C22—21no—
229C22—22no—
230C22—22no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—44no—
235C22—44no—
236C22—41no—
237C22—41no—
238C22—44no—
239C22—81no—
240C22—82no—
241C22—82no—
242C22—816no—
243C22—82no—
244C22—82no—
245C22#44814yes—
246C22C2×#12011yescharacteristic
247C22C22×SD1611yescharacteristic
248C22C2×#13711yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—41no—
253C4—44no—
254C4—41no—
255C4—44no—
256C4—82no—
257C4B(4,𝔽2)12yes—
258C2—21no—
259C2—21no—
260C2—24no—
261C2—42no—
262C2—84no—
263C2C2×#44811yescharacteristic
264C2#302111yescharacteristic
265C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
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. C2×#450C2semidirect
4. #1599C2semidirect
5. #1596C2semidirect
6. #1520C2semidirect
7. #1459C2semidirect
13. #544C22semidirect
15. #543C22semidirect
32. #328C22semidirect
33. #324C22semidirect
34. #323C22semidirect
37. C22×Q8D4semidirect
65. C22⋊Q8D4semidirect
80. C2×C4⋊C4D4semidirect
123. C2×D4SD16semidirect
187. C23C2×SD16semidirect
223. C2×C4C2×SD16semidirect

Maximal subgroups

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

subgroupindex
#35012
C2×#4502
#24082
#15992
#15962
#15202
#14592