Finite Group Databasebeta

← #25016

Subgroups — #25016

Order 256.

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

normal subgroups only

This group has 449 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#2538C211yesmaximal, characteristic
3D4×C4⋊C4C211yesmaximal, characteristic
4#2076C212yesmaximal, complemented
5#2036C212yesmaximal, complemented
6#1954C211yesmaximal, characteristic
7#1947C211yesmaximal, characteristic, complemented
8#1852C212yesmaximal, complemented
9#1845C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11C4×#103C211yesmaximal, characteristic, complemented
12C2×C22⋊Q8C2212yes—
13C2×C4×Q8C2211yescharacteristic
14C2×C4×D4—21no—
15C2×C4×D4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17#484—28no—
18C2×#107C2212yes—
19C2×#107C2212yes—
20C2×#107C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×C22⋊C8C2212yes—
26C2×C22⋊C8C2212yes—
27C2×C4×C8C2211yescharacteristic
28#396—22no—
29#390—21no—
30#389—21no—
31#389—21no—
32#385—22no—
33#384C2211yescharacteristic
34#384C2211yescharacteristic
35#380C2212yes—
36#380C2212yes—
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42C22×Q8C2311yescharacteristic
43C22×D4D411yescharacteristic
44C23×C4—22no—
45C23×C4C2312yes—
46C22×C8C2312yes—
47C22×C8C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C22⋊Q8—24no—
50C22⋊Q8—42no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—44no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D412yes—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42C2311yescharacteristic
78#107—24no—
79#107—24no—
80#107—42no—
81C4⋊C8—22no—
82#103—24no—
83#103—22no—
84#103D414yescomplemented
85C22⋊C8—24no—
86C22⋊C8—24no—
87C4×C8—22no—
88#95—21no—
89#95—22no—
90#95—22no—
91C24C2×D412yes—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8C2×D412yes—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4Q1614yes—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—82no—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8C2×D412yes—
128C2×C8#4112yes—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—44no—
140C4⋊C4—44no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—81no—
145C4⋊C4—81no—
146C4⋊C4C2×D412yes—
147C22⋊C4—24no—
148C22⋊C4—44no—
149C22⋊C4—44no—
150C22⋊C4—82no—
151C42—42no—
152C42—42no—
153C42#4112yes—
154C42#4112yes—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23C2×Q1614yes—
162C23C22×D411yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167D4—28no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—88no—
196C2×C4#12312yes—
197C2×C4#12312yes—
198C2×C4C2×Q1612yes—
199C2×C4AGL(1,ℤ/8)12yes—
200C2×C4C22×D411yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4H(5,𝔽2)11yescharacteristic
205C2×C42−1+411yescharacteristic
206C8—22no—
207C8—22no—
208C8—42no—
209C22—24no—
210C22—24no—
211C22—24no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22C2×#12311yescharacteristic
217C22D4211yescharacteristic
218C22#54711yescharacteristic
219C22C22×Q1611yescharacteristic
220C22C2×AGL(1,ℤ/8)11yescharacteristic
221C22#57611yescharacteristic
222C22#57811yescharacteristic
223C4—22no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—81no—
228C4—81no—
229C4—82no—
230C4—82no—
231C4#48414yescomplemented
232C4#53211yescharacteristic
233C4#54211yescharacteristic
234C4#54711yescharacteristic
235C4#55311yescharacteristic
236C2—44no—
237C2#242211yescharacteristic
238C2C2×#48411yescharacteristic
239C2#309811yescharacteristic
240C2#327411yescharacteristic
241C2D4×Q1611yescharacteristic
242C2#333811yescharacteristic
243C2#335311yescharacteristic
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. #2076C2semidirect
5. #2036C2semidirect
7. #1947C2semidirect
8. #1852C2semidirect
9. #1845C2semidirect
10. #1803C2semidirect
11. C4×#103C2semidirect
84. #103D4semidirect
231. C4#484semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
#25382
D4×C4⋊C42
#20762
#20362
#19542
#19472
#18522
#18452
#18032
C4×#1032