Finite Group Databasebeta

← #24371

Subgroups — #24371

Order 256.

770 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#464C212yesmaximal, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4#2435C211yesmaximal, characteristic, complemented
5#2341C211yesmaximal, characteristic, complemented
6#2054C212yesmaximal, complemented
7#2012C212yesmaximal, complemented
8#1894C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1851C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C2×AGL(1,ℤ/8)—24no—
15C2×AGL(1,ℤ/8)—28no—
16C22×Q16C2211yescharacteristic, complemented
17C22×SD16C2212yes—
18C22×SD16C2211yescharacteristic, complemented
19C22×D8C2212yes—
20C22×M16C2211yescharacteristic
21C2×#124C2212yes—
22C2×C22⋊Q8C2211yescharacteristic
23C2×C4×Q8C2212yescomplemented
24#464—28no—
25#452—24no—
26C2×C4⋊C8C2212yescomplemented
27C2×#103C2211yescharacteristic, complemented
28C2×#103C2212yescomplemented
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2211yescharacteristic, complemented
32C2×C22⋊C8C2211yescharacteristic
33#394—21no—
34#394—21no—
35#390—22no—
36#390C2212yes—
37#390C2212yescomplemented
38#380—21no—
39#380C2211yescharacteristic
40#379—22no—
41C4×C22⋊C4—21no—
42C4×C22⋊C4C2211yescharacteristic
43#340C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C2×#41—22no—
47C2×#41—24no—
48C2×#41—42no—
49C2×#41D414yes—
50C22×Q8C2311yescharacteristic
51C22×Q8C2311yescharacteristic
52C22×D4—24no—
53C22×D4—21no—
54C22×D4—22no—
55C22×D4—21no—
56C22×D4—22no—
57C22×D4—42no—
58C22×D4C2312yes—
59C22×D4C2311yescharacteristic
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62AGL(1,ℤ/8)—232no—
63C2×Q16—22no—
64C2×Q16—24no—
65C2×SD16—24no—
66C2×SD16—24no—
67C2×SD16—22no—
68C2×SD16—42no—
69C2×SD16D418yes—
70C2×D8—24no—
71C2×D8—24no—
72C2×D8—24no—
73C2×M16—22no—
74C2×M16—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—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4D412yes—
97C2×C42—22no—
98C2×C42C2312yes—
99C4⋊C8—24no—
100#103—22no—
101#103—24no—
102#102—24no—
103#102—22no—
104#102—42no—
105C22⋊C8—22no—
106#95—21no—
107#95D411yescharacteristic
108#95D411yescharacteristic
109C24—21no—
110C24—22no—
111C24—44no—
112C24—44no—
113C24—42no—
114C24—81no—
115C24—81no—
116C24C2×D411yescharacteristic
117#41—28no—
118#41—44no—
119#41—48no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8C2×D412yes—
127C2×Q8C2×D414yes—
128C2×D4—28no—
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—48no—
139C2×D4—44no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—42no—
143C2×D4C2×D414yes—
144C2×D4C2×D412yes—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
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—
160SD16—216no—
161SD16—44no—
162D8—216no—
163M16—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—42no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C22⋊C4—42no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C22⋊C4—81no—
186C42—22no—
187C42—22no—
188C42—44no—
189C23—22no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23—24no—
195C23—48no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23—41no—
206C23—84no—
207C23—84no—
208C23—88no—
209C23—88no—
210C23#12112yes—
211C23C22×D411yescharacteristic
212Q8—22no—
213Q8—24no—
214Q8—41no—
215Q8—41no—
216Q8—42no—
217Q8#12014yes—
218D4—22no—
219D4—24no—
220D4—22no—
221D4—24no—
222D4—48no—
223D4—48no—
224D4—42no—
225D4—42no—
226D4—42no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—48no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4#12112yes—
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D412yes—
251C2×C4C2×#4111yescharacteristic
252C2×C4H(5,𝔽2)12yes—
253C8—24no—
254C8—41no—
255C8—41no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—44no—
263C22—44no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—816no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22C2×#12111yescharacteristic
275C22D4212yes—
276C22#57512yes—
277C22#57611yescharacteristic
278C22#57711yescharacteristic
279C4—22no—
280C4—24no—
281C4—42no—
282C4—42no—
283C4—81no—
284C4—82no—
285C4—81no—
286C4C2×#12011yescharacteristic
287C4#53411yescharacteristic
288C4#54612yes—
289C2—42no—
290C2—82no—
291C2—84no—
292C2#241011yescharacteristic
293C2#304011yescharacteristic
294C2#304511yescharacteristic
295C2#326812yes—
296C2#327512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×AGL(1,ℤ/8)C2
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×#452C2semidirect
4. #2435C2semidirect
5. #2341C2semidirect
6. #2054C2semidirect
7. #2012C2semidirect
8. #1894C2semidirect
9. #1880C2semidirect
10. #1851C2semidirect
11. #1786C2semidirect
16. C22×Q16C22semidirect
18. C22×SD16C22semidirect
23. C2×C4×Q8C22semidirect
26. C2×C4⋊C8C22semidirect
27. C2×#103C22semidirect
28. C2×#103C22semidirect
31. C2×#102C22semidirect
37. #390C22semidirect
43. #340C22semidirect
44. #336C22semidirect

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)2
C2×#4642
C2×#4522
#24352
#23412
#20542
#20122
#18942
#18802
#18512
#17862