Finite Group Databasebeta

← #25394

Subgroups — #25394

Order 256.

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

normal subgroups only

This group has 493 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×#470C212yesmaximal, complemented
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#436C211yesmaximal, characteristic
4#2542C211yesmaximal, characteristic
5#2538C211yesmaximal, characteristic
6#2035C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1894C212yesmaximal, complemented
11C4×#103C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×M16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×C22⋊Q8C2212yes—
17C2×C22⋊Q8C2212yes—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#470—28no—
21#462—24no—
22#436—24no—
23C2×#106C2212yes—
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×C22⋊C8C2212yes—
31C2×C4×C8C2211yescharacteristic
32#399—22no—
33#396—22no—
34#390—21no—
35#390—21no—
36#389—21no—
37#384—21no—
38#384C2211yescharacteristic
39#380C2212yes—
40C4×C4⋊C4C2211yescharacteristic
41#340C2211yescharacteristic
42#340C2212yes—
43#340C2211yescharacteristic
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4D411yescharacteristic
47C23×C4C2312yes—
48C2×Q16—24no—
49C2×Q16—24no—
50C2×Q16—42no—
51C2×M16—24no—
52C2×M16—24no—
53C2×M16—42no—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22×C8C2312yes—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—42no—
63C22⋊Q8—42no—
64C22⋊Q8—42no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×D4—22no—
68C4×D4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4D412yes—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42C2311yescharacteristic
90#106—24no—
91C4⋊C8D414yes—
92#103—24no—
93#103—24no—
94#103—22no—
95#103—42no—
96#103D414yescomplemented
97C22⋊C8—24no—
98C4×C8—22no—
99#95—21no—
100#95—22no—
101C24C2×D412yes—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8C2×D412yes—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4C2×D412yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133Q16—48no—
134M16—48no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8C2×D414yes—
143C2×C8C2×D412yes—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—81no—
159C4⋊C4C2×D412yes—
160C4⋊C4#4112yes—
161C22⋊C4—24no—
162C22⋊C4—44no—
163C22⋊C4—44no—
164C22⋊C4—44no—
165C22⋊C4—82no—
166C42—42no—
167C42—42no—
168C42C2×D412yes—
169C42#4112yes—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23C22×D411yescharacteristic
178Q8—22no—
179Q8—22no—
180Q8—41no—
181Q8—41no—
182Q8—42no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186D4—44no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4#12112yes—
215C2×C4#12112yes—
216C2×C4#13712yes—
217C2×C4#13712yes—
218C2×C4C22×D411yescharacteristic
219C2×C4C22×D411yescharacteristic
220C2×C4C2×#4111yescharacteristic
221C2×C4C2×#4111yescharacteristic
222C2×C4H(5,𝔽2)11yescharacteristic
223C2×C4H(5,𝔽2)11yescharacteristic
224C8—22no—
225C8—24no—
226C8—41no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22C2×#12111yescharacteristic
235C22D4211yescharacteristic
236C22#54611yescharacteristic
237C22C2×#13711yescharacteristic
238C22C2×#13711yescharacteristic
239C22#57711yescharacteristic
240C22#57811yescharacteristic
241C4—22no—
242C4—42no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—82no—
248C4—81no—
249C4#47014yescomplemented
250C4#53211yescharacteristic
251C4#54011yescharacteristic
252C4D4211yescharacteristic
253C4#55011yescharacteristic
254C2—82no—
255C2#242811yescharacteristic
256C2C2×#47011yescharacteristic
257C2#306311yescharacteristic
258C2#327611yescharacteristic
259C2#327811yescharacteristic
260C2#332811yescharacteristic
261C2#333111yescharacteristic
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×#470C2semidirect
2. C2×#462C2semidirect
6. #2035C2semidirect
7. #2001C2semidirect
8. #1956C2semidirect
9. #1948C2semidirect
10. #1894C2semidirect
11. C4×#103C2semidirect
96. #103D4semidirect
249. C4#470semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4622
C2×#4362
#25422
#25382
#20352
#20012
#19562
#19482
#18942
C4×#1032