Finite Group Databasebeta

← #24627

Subgroups — #24627

Order 256.

469 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 131 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×#461C211yesmaximal, characteristic, complemented
2C2×C4×D8C211yesmaximal, characteristic
3#2547C211yesmaximal, characteristic, complemented
4#2354C211yesmaximal, characteristic, complemented
5#2055C212yesmaximal, complemented
6#2012C212yesmaximal
7#1983C212yesmaximal, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1939C211yesmaximal, characteristic, complemented
10#1928C212yesmaximal, complemented
11#1802C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×D4C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×#117—21no—
19#461—24no—
20C4×D8—24no—
21C2×#107C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2212yes—
30C2×C4×C8C2211yescharacteristic
31#397—22no—
32#391C2212yes—
33#390C2212yes—
34#388—22no—
35#384C2211yescharacteristic
36#380—21no—
37#380—21no—
38#380—21no—
39C4×C4⋊C4C2212yes—
40C4×C4⋊C4C2211yescharacteristic
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C2×SD16—24no—
49C2×SD16—42no—
50C2×SD16—42no—
51C2×D8—22no—
52C2×D8—22no—
53C2×D8—22no—
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59#124—24no—
60#124—42no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4D414yes—
65#117—44no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4D411yescharacteristic
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C42—22no—
82C2×C42C2312yes—
83C2×C42C2311yescharacteristic
84#107—22no—
85C4⋊C8—22no—
86C4⋊C8—42no—
87#103—24no—
88#103—24no—
89#103—42no—
90#102—22no—
91#102—22no—
92#102—42no—
93C4×C8—22no—
94#95—22no—
95#95—22no—
96#95—22no—
97#95D412yes—
98C24—21no—
99C24—21no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4C2×D412yes—
117C2×D4#4112yes—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4C2×D412yes—
129C22×C4C2×D412yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133SD16—48no—
134D8—28no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—22no—
143C2×C8#4112yes—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—81no—
155C4⋊C4C2×D412yes—
156C22⋊C4—24no—
157C22⋊C4—44no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C22⋊C4—42no—
161C22⋊C4—82no—
162C42—22no—
163C42—22no—
164C42—44no—
165C42C2×D412yes—
166C42#4112yes—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23C22×D411yescharacteristic
182Q8—42no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186D4—22no—
187D4—22no—
188D4—22no—
189D4—22no—
190D4—41no—
191D4—41no—
192D4—41no—
193D4—41no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—48no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—84no—
216C2×C4#12112yes—
217C2×C4#12112yes—
218C2×C4#13512yes—
219C2×C4AGL(1,ℤ/8)12yes—
220C2×C4C22×D411yescharacteristic
221C2×C4C2×#4111yescharacteristic
222C2×C4C2×#4111yescharacteristic
223C2×C4C2×#4111yescharacteristic
224C2×C4H(5,𝔽2)11yescharacteristic
225C2×C4H(5,𝔽2)11yescharacteristic
226C8—22no—
227C8—41no—
228C8—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22C2×#12111yescharacteristic
244C22#54611yescharacteristic
245C22#55011yescharacteristic
246C22C2×#13511yescharacteristic
247C22C2×AGL(1,ℤ/8)11yescharacteristic
248C22#57711yescharacteristic
249C22#57811yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—82no—
255C4—82no—
256C4—81no—
257C4#46314yes—
258C4#52511yescharacteristic
259C4#53511yescharacteristic
260C4#55311yescharacteristic
261C4#55511yescharacteristic
262C2—82no—
263C2—82no—
264C2#241411yescharacteristic
265C2C2×#46311yescharacteristic
266C2#304611yescharacteristic
267C2#330311yescharacteristic
268C2#333011yescharacteristic
269C2#339311yescharacteristic
270C2#339711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#461C2semidirect
3. #2547C2semidirect
4. #2354C2semidirect
5. #2055C2semidirect
7. #1983C2semidirect
8. #1956C2semidirect
9. #1939C2semidirect
10. #1928C2semidirect
11. #1802C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×C4×D82
#25472
#23542
#20552
#20122
#19832
#19562
#19392
#19282
#18022