Finite Group Databasebeta

← #24370

Subgroups — #24370

Order 256.

835 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 835 conjugacy classes of subgroups — too many to draw. Only its 147 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, complemented
2C2×#460C212yesmaximal, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic, complemented
5#2341C211yesmaximal, characteristic, complemented
6#2054C212yesmaximal, complemented
7#2011C212yesmaximal, complemented
8#1895C211yesmaximal, 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×SD16C2212yes—
17C22×SD16C2211yescharacteristic, complemented
18C22×D8C2212yes—
19C22×D8C2211yescharacteristic, complemented
20C22×M16C2211yescharacteristic
21C2×#127C2212yes—
22C2×#121C2211yescharacteristic
23C2×#120—22no—
24C2×C4×Q8C2212yescomplemented
25#460—28no—
26#449—24no—
27C2×C4⋊C8C2212yescomplemented
28C2×#103C2211yescharacteristic, complemented
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2211yescharacteristic, complemented
32C2×#102C2212yescomplemented
33C2×C22⋊C8C2211yescharacteristic
34#392—21no—
35#392—21no—
36#390—22no—
37#390C2212yes—
38#390C2212yescomplemented
39#380—21no—
40#380C2211yescharacteristic
41C4×C22⋊C4—21no—
42C4×C22⋊C4C2211yescharacteristic
43#340C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C2×#41—22no—
47C2×#41—24no—
48C2×#41—42no—
49C2×#41D414yescomplemented
50C22×Q8C2311yescharacteristic
51C22×D4—24no—
52C22×D4—21no—
53C22×D4—21no—
54C22×D4—22no—
55C22×D4—21no—
56C22×D4—22no—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4—42no—
60C22×D4C2311yescharacteristic
61C22×D4C2312yes—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65AGL(1,ℤ/8)—232no—
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—22no—
69C2×SD16—24no—
70C2×SD16D418yescomplemented
71C2×D8—24no—
72C2×D8—24no—
73C2×D8—22no—
74C2×D8—24no—
75C2×D8—42no—
76C2×M16—22no—
77C2×M16—24no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2312yes—
81#127—24no—
82#127—42no—
83#121—22no—
84#121—22no—
85#120—48no—
86C4×Q8—24no—
87C4×Q8—42no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4D412yes—
98C2×C42—22no—
99C2×C42C2312yes—
100C4⋊C8—24no—
101#103—22no—
102#102—24no—
103#102—22no—
104#102—24no—
105#102—42no—
106C22⋊C8—22no—
107#95—21no—
108#95D411yescharacteristic
109#95D411yescharacteristic
110C24—21no—
111C24—22no—
112C24—21no—
113C24—44no—
114C24—44no—
115C24—42no—
116C24—81no—
117C24—81no—
118C24C2×D411yescharacteristic
119#41—28no—
120#41—44no—
121#41—48no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8C2×D412yes—
125C2×Q8C2×D414yes—
126C2×D4—28no—
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—48no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4—44no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—44no—
151C2×D4—82no—
152C2×D4—82no—
153C2×D4C2×D414yes—
154C2×D4C2×D412yes—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—44no—
162C22×C4C2×D412yes—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D412yes—
165C22×C4C2×D412yes—
166C22×C4#4112yes—
167C22×C4C2411yescharacteristic
168SD16—216no—
169SD16—44no—
170D8—216no—
171D8—44no—
172M16—28no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8C2×D414yes—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C22⋊C4—24no—
186C22⋊C4—44no—
187C22⋊C4—42no—
188C22⋊C4—44no—
189C22⋊C4—44no—
190C22⋊C4—82no—
191C22⋊C4—81no—
192C42—22no—
193C42—22no—
194C42—44no—
195C23—22no—
196C23—22no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—24no—
201C23—48no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23—84no—
218C23—84no—
219C23—88no—
220C23—88no—
221C23#12112yes—
222C23C22×D411yescharacteristic
223Q8—22no—
224Q8—24no—
225Q8#12014yescomplemented
226D4—22no—
227D4—24no—
228D4—22no—
229D4—24no—
230D4—48no—
231D4—48no—
232D4—41no—
233D4—42no—
234D4—41no—
235D4—42no—
236D4—42no—
237D4—42no—
238D4—88no—
239D4—88no—
240D4—84no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—48no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4#12112yes—
259C2×C4AGL(1,ℤ/8)14yescomplemented
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D412yes—
262C2×C4C2×#4111yescharacteristic
263C2×C4H(5,𝔽2)12yes—
264C8—24no—
265C8—41no—
266C8—41no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—44no—
274C22—44no—
275C22—44no—
276C22—42no—
277C22—42no—
278C22—816no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—81no—
286C22—82no—
287C22—82no—
288C22—81no—
289C22—81no—
290C22C2×#12111yescharacteristic
291C22D4212yes—
292C22C2×AGL(1,ℤ/8)12yes—
293C22#57611yescharacteristic
294C22#57711yescharacteristic
295C4—22no—
296C4—24no—
297C4—24no—
298C4—42no—
299C4—82no—
300C4—81no—
301C4C2×#12011yescharacteristic
302C4#53411yescharacteristic
303C4#54612yes—
304C2—42no—
305C2—82no—
306C2—84no—
307C2—161no—
308C2#241011yescharacteristic
309C2#303811yescharacteristic
310C2#304511yescharacteristic
311C2#326512yes—
312C2#327412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C22×AGL(1,ℤ/8)C2semidirect
2. C2×#460C2semidirect
3. C2×#449C2semidirect
4. #2433C2semidirect
5. #2341C2semidirect
6. #2054C2semidirect
7. #2011C2semidirect
8. #1895C2semidirect
9. #1880C2semidirect
10. #1851C2semidirect
11. #1786C2semidirect
17. C22×SD16C22semidirect
19. C22×D8C22semidirect
24. C2×C4×Q8C22semidirect
27. C2×C4⋊C8C22semidirect
28. C2×#103C22semidirect
31. C2×#102C22semidirect
32. C2×#102C22semidirect
38. #390C22semidirect
43. #340C22semidirect
44. #336C22semidirect
49. C2×#41D4semidirect
70. C2×SD16D4semidirect
225. Q8#120semidirect
259. C2×C4AGL(1,ℤ/8)semidirect

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×#4602
C2×#4492
#24332
#23412
#20542
#20112
#18952
#18802
#18512
#17862