Finite Group Databasebeta

← #26236

Subgroups — #26236

Order 256.

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

normal subgroups only

This group has 467 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×#484C212yesmaximal, complemented
2#2542C211yesmaximal, characteristic
3D4×C4⋊C4C211yesmaximal, characteristic
4#2036C212yesmaximal, complemented
5#2036C212yesmaximal, complemented
6#1981C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1954C211yesmaximal, characteristic
9#1947C211yesmaximal, characteristic, complemented
10#1872C212yesmaximal, complemented
11C4×#107C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2212yes—
14C2×C22⋊Q8C2212yes—
15C2×C4×D4—21no—
16C2×C4×D4C2211yescharacteristic
17C22×C4⋊C4C2212yes—
18#484—28no—
19C2×#107C2212yes—
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×C22⋊C8C2212yes—
28C2×C22⋊C8C2212yes—
29C2×C4×C8C2211yescharacteristic
30#399—22no—
31#390—21no—
32#389—21no—
33#385—22no—
34#384—21no—
35#384C2211yescharacteristic
36#380C2212yes—
37C4×C4⋊C4C2211yescharacteristic
38C4×C4⋊C4C2211yescharacteristic
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42C22×Q8C2312yes—
43C22×D4D411yescharacteristic
44C23×C4—22no—
45C23×C4C2312yes—
46C22×C8C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2311yescharacteristic
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C22⋊Q8—24no—
52C22⋊Q8—24no—
53C22⋊Q8—42no—
54C22⋊Q8—42no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4—44no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4D412yes—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80#107—24no—
81#107—22no—
82#107D414yescomplemented
83C4⋊C8—22no—
84#103—24no—
85#103—24no—
86#103—42no—
87#103—42no—
88C22⋊C8—24no—
89C22⋊C8—42no—
90C4×C8—22no—
91#95—21no—
92#95—21no—
93#95—22no—
94C24C2×D412yes—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4Q1614yes—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—82no—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2411yescharacteristic
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8C2×D412yes—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—81no—
147C4⋊C4C2×D412yes—
148C4⋊C4C2×D412yes—
149C4⋊C4#4112yes—
150C22⋊C4—24no—
151C22⋊C4—44no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C42—42no—
155C42—42no—
156C42#4112yes—
157C42#4112yes—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23C2×Q1614yes—
165C23C22×D411yescharacteristic
166Q8—42no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170D4—28no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—88no—
199C2×C4#12312yes—
200C2×C4#12312yes—
201C2×C4C2×Q1612yes—
202C2×C4AGL(1,ℤ/8)12yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4H(5,𝔽2)11yescharacteristic
208C2×C4H(5,𝔽2)11yescharacteristic
209C8—22no—
210C8—41no—
211C8—42no—
212C22—24no—
213C22—24no—
214C22—24no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22C2×#12311yescharacteristic
220C22D4211yescharacteristic
221C22#54611yescharacteristic
222C22C22×Q1611yescharacteristic
223C22C2×AGL(1,ℤ/8)11yescharacteristic
224C22#57711yescharacteristic
225C22#57811yescharacteristic
226C4—22no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—81no—
232C4—82no—
233C4—82no—
234C4#48414yescomplemented
235C4#53211yescharacteristic
236C4#54011yescharacteristic
237C4#55011yescharacteristic
238C4#55911yescharacteristic
239C2—44no—
240C2#244511yescharacteristic
241C2C2×#48411yescharacteristic
242C2#309711yescharacteristic
243C2#327211yescharacteristic
244C2D4×Q1611yescharacteristic
245C2#329911yescharacteristic
246C2#333011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
4. #2036C2semidirect
5. #2036C2semidirect
6. #1981C2semidirect
7. #1956C2semidirect
9. #1947C2semidirect
10. #1872C2semidirect
11. C4×#107C2semidirect
82. #107D4semidirect
234. C4#484semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
#25422
D4×C4⋊C42
#20362
#19812
#19562
#19542
#19472
#18722
C4×#1072