Finite Group Databasebeta

← #28114

Subgroups — #28114

Order 256.

647 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#480C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic
3C4×#121C211yesmaximal, characteristic, complemented
4C2×#390C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2054C212yesmaximal, complemented
7#1925C212yesmaximal, complemented
8#1879C212yesmaximal, complemented
9#1823C211yesmaximal, characteristic, complemented
10#1797C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×D8C2212yes—
14C2×#121C2211yescharacteristic
15C2×C4×D4—21no—
16C2×C4×D4C2211yescharacteristic, complemented
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4—21no—
19C22×C22⋊C4—21no—
20C22×C42C2211yescharacteristic
21#480—24no—
22#447—24no—
23C2×#107C2212yescomplemented
24C2×C4⋊C8C2212yescomplemented
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yescomplemented
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31#390—21no—
32#390—21no—
33#390C2212yes—
34#390C2212yes—
35#390C2211yescharacteristic, complemented
36#389C2211yescharacteristic, complemented
37#388—21no—
38#386—21no—
39#380C2211yescharacteristic
40C4×C4⋊C4C2211yescharacteristic, complemented
41C4×C22⋊C4—21no—
42#340C2212yescomplemented
43#340C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C22×D4—21no—
47C22×D4—24no—
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—42no—
52C22×D4C2311yescharacteristic
53C22×D4C2312yes—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4C2311yescharacteristic
59C2×D8—24no—
60C2×D8—24no—
61C2×D8—42no—
62C22×C8C2312yes—
63C22×C8C2312yes—
64#121—22no—
65#121D414yes—
66C4×D4—42no—
67C4×D4—44no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—44no—
87C2×C42—21no—
88C2×C42—42no—
89C2×C42D411yescharacteristic
90C2×C42D411yescharacteristic
91C2×C42C2311yescharacteristic
92C2×C42C2311yescharacteristic
93#107—24no—
94C4⋊C8—42no—
95#102—24no—
96#102—24no—
97#102—24no—
98#102—42no—
99C22⋊C8—22no—
100C22⋊C8—22no—
101#95—21no—
102#95—21no—
103#95—21no—
104C24—24no—
105C24—21no—
106C24—22no—
107C24—42no—
108C24—42no—
109C24—42no—
110C24—42no—
111C24—42no—
112C24—42no—
113C24C2×D411yescharacteristic
114C2×D4—24no—
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—48no—
125C2×D4—48no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4C2×D412yes—
131C2×D4#4114yes—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4D814yescomplemented
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D412yes—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4#4112yes—
157C22×C4C2411yescharacteristic
158D8—48no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4C2×D412yes—
173C22⋊C4—24no—
174C22⋊C4—42no—
175C22⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C22⋊C4—42no—
182C42—22no—
183C42—22no—
184C42—41no—
185C42—41no—
186C42—42no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—24no—
193C23—48no—
194C23—48no—
195C23—48no—
196C23—42no—
197C23—41no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—42no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—42no—
207C23—42no—
208C23—84no—
209C23—84no—
210C23—84no—
211C23—84no—
212C23#12112yes—
213C23C2×D812yes—
214C23AGL(1,ℤ/8)12yes—
215C23C22×D411yescharacteristic
216D4—24no—
217D4—24no—
218D4—24no—
219D4—48no—
220D4—48no—
221D4—42no—
222D4—42no—
223D4—42no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4#12112yes—
255C2×C4C2×D812yes—
256C2×C4C2×D812yes—
257C2×C4C22×D411yescharacteristic
258C2×C4C2×#4112yes—
259C2×C4C2×#4111yescharacteristic
260C2×C4H(5,𝔽2)12yes—
261C8—42no—
262C8—42no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—44no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—44no—
274C22—42no—
275C22—816no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—82no—
280C22—81no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22#45914yes—
285C22C2×#12111yescharacteristic
286C22#52511yescharacteristic
287C22#53411yescharacteristic
288C22C22×D811yescharacteristic
289C22C2×AGL(1,ℤ/8)11yescharacteristic
290C22#57511yescharacteristic
291C22#57511yescharacteristic
292C4—22no—
293C4—22no—
294C4—22no—
295C4—41no—
296C4—41no—
297C4—42no—
298C4—81no—
299C4—81no—
300C4—82no—
301C4#54612yes—
302C4#55112yes—
303C2—24no—
304C2—82no—
305C2—84no—
306C2#253411yescharacteristic
307C2C2×#45911yescharacteristic
308C2#304011yescharacteristic
309C2#310011yescharacteristic
310C2#311411yescharacteristic
311C2#318611yescharacteristic
312C2#319411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#480C2semidirect
3. C4×#121C2semidirect
4. C2×#390C2semidirect
5. #2067C2semidirect
6. #2054C2semidirect
7. #1925C2semidirect
8. #1879C2semidirect
9. #1823C2semidirect
10. #1797C2semidirect
11. #1786C2semidirect
16. C2×C4×D4C22semidirect
23. C2×#107C22semidirect
24. C2×C4⋊C8C22semidirect
28. C2×#102C22semidirect
35. #390C22semidirect
36. #389C22semidirect
40. C4×C4⋊C4C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #336C22semidirect
149. C22×C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C2×#4472
C4×#1212
C2×#3902
#20672
#20542
#19252
#18792
#18232
#17972
#17862