Finite Group Databasebeta

← #24357

Subgroups — #24357

Order 256.

763 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 763 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C23×SD16C211yesmaximal, characteristic
2C2×#464C212yesmaximal, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4#2435C211yesmaximal, characteristic
5Q8×C22⋊C4C211yesmaximal, characteristic, complemented
6#2055C212yesmaximal, complemented
7#2015C212yesmaximal, complemented
8#1892C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1850C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×Q16C2211yescharacteristic, complemented
15C22×Q16C2211yescharacteristic, complemented
16C22×SD16—24no—
17C22×SD16—28no—
18C22×SD16C2212yes—
19C22×SD16C2212yes—
20C23×C8C2211yescharacteristic
21C2×#124C2212yes—
22C2×C22⋊Q8C2211yescharacteristic
23C2×C4×Q8C2212yescomplemented
24#464—28no—
25#451—24no—
26C2×C4⋊C8C2212yescomplemented
27C2×#103C2211yescharacteristic, complemented
28C2×#103C2211yescharacteristic, complemented
29C2×#103C2212yescomplemented
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33#394—22no—
34#391—22no—
35#391C2212yescomplemented
36#390C2212yes—
37#380—21no—
38#380C2211yescharacteristic
39#379—21no—
40#379—21no—
41C4×C22⋊C4—21no—
42C4×C22⋊C4C2211yescharacteristic
43#340C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—24no—
49C22×Q8—42no—
50C22×Q8D414yes—
51C22×Q8C2311yescharacteristic
52C22×Q8C2311yescharacteristic
53C22×Q8C2311yescharacteristic
54C22×D4—24no—
55C22×D4—22no—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—42no—
59C22×D4C2312yes—
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C2×Q16—22no—
63C2×Q16—22no—
64C2×Q16—24no—
65C2×Q16—42no—
66C2×SD16—232no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—24no—
71C2×SD16—24no—
72C2×SD16D418yes—
73C22×C8—22no—
74C22×C8—24no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2312yes—
78#124—24no—
79#124—42no—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C4×Q8—24no—
83C4×Q8—42no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—22no—
97C2×C42C2312yes—
98C4⋊C8—24no—
99#103—22no—
100#103—22no—
101#103—24no—
102#102—24no—
103#102—42no—
104C22⋊C8—22no—
105#95—22no—
106#95—22no—
107#95—21no—
108#95D412yes—
109C24—22no—
110C24—44no—
111C24—42no—
112C24—42no—
113C24—42no—
114C24—82no—
115C24C2×D411yescharacteristic
116C2×Q8—24no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—28no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—48no—
130C2×Q8C2×D412yes—
131C2×Q8C2×D412yes—
132C2×Q8C2×D414yes—
133C2×D4—28no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—48no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—42no—
143C2×D4C2×D414yes—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—44no—
153C22×C4C2×D412yes—
154C22×C4C2×D412yes—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D412yes—
157C22×C4#4112yes—
158C22×C4C2411yescharacteristic
159Q16—44no—
160Q16—44no—
161SD16—216no—
162SD16—216no—
163C2×C8—28no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8C2×D414yes—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C42—22no—
186C42—22no—
187C42—44no—
188C23—22no—
189C23—22no—
190C23—24no—
191C23—48no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—88no—
199C23—88no—
200C23—84no—
201C23—84no—
202C23#12112yes—
203C23#13512yes—
204C23#13712yes—
205C23C22×D411yescharacteristic
206Q8—22no—
207Q8—24no—
208Q8—22no—
209Q8—22no—
210Q8—48no—
211Q8—42no—
212Q8—41no—
213Q8—41no—
214Q8—42no—
215Q8#12014yes—
216D4—24no—
217D4—24no—
218D4—48no—
219D4—42no—
220D4—42no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—48no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4#12112yes—
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D412yes—
246C2×C4C2×#4111yescharacteristic
247C2×C4H(5,𝔽2)12yes—
248C8—24no—
249C8—41no—
250C8—41no—
251C22—22no—
252C22—24no—
253C22—24no—
254C22—22no—
255C22—44no—
256C22—44no—
257C22—44no—
258C22—816no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22#46414yescomplemented
264C22C2×#12111yescharacteristic
265C22D4212yes—
266C22C2×#13511yescharacteristic
267C22C2×#13711yescharacteristic
268C22#57712yes—
269C4—22no—
270C4—24no—
271C4—42no—
272C4—42no—
273C4—42no—
274C4—81no—
275C4—82no—
276C4—81no—
277C4C2×#12011yescharacteristic
278C4#53411yescharacteristic
279C4#54612yes—
280C2—24no—
281C2—84no—
282C2#241011yescharacteristic
283C2C2×#46411yescharacteristic
284C2#304311yescharacteristic
285C2#326612yes—
286C2#328212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#464C2semidirect
3. C2×#451C2semidirect
5. Q8×C22⋊C4C2semidirect
6. #2055C2semidirect
7. #2015C2semidirect
8. #1892C2semidirect
9. #1880C2semidirect
10. #1850C2semidirect
14. C22×Q16C22semidirect
15. C22×Q16C22semidirect
23. C2×C4×Q8C22semidirect
26. C2×C4⋊C8C22semidirect
27. C2×#103C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
35. #391C22semidirect
43. #340C22semidirect
44. #336C22semidirect
263. C22#464semidirect

Maximal subgroups

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

subgroupindex
C23×SD162
C2×#4642
C2×#4512
#24352
Q8×C22⋊C42
#20552
#20152
#18922
#18802
#18502
#17862