Finite Group Databasebeta

← #24785

Subgroups — #24785

Order 256.

470 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 135 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×#470C211yesmaximal, characteristic, complemented
2C2×#467C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic
4C2×#427C211yesmaximal, characteristic, complemented
5#2415C212yesmaximal
6#2068C212yesmaximal, complemented
7#2036C212yesmaximal, complemented
8#1971C211yesmaximal, characteristic, complemented
9#1944C211yesmaximal, characteristic, complemented
10#1873C212yesmaximal, complemented
11#1809C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×C22⋊Q8C2212yes—
17C2×C4×D4C2211yescharacteristic
18C2×#117C2212yes—
19#470—24no—
20#467—24no—
21#435—24no—
22#427—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2212yes—
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#397—22no—
35#394—22no—
36#391C2212yes—
37#389—22no—
38#389C2212yes—
39#388—22no—
40#380C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2312yes—
44C22×D4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C2×Q16—22no—
48C2×Q16—22no—
49C2×Q16—22no—
50C2×M16—22no—
51C2×M16—22no—
52C2×M16—22no—
53C22×C8—22no—
54C22×C8—22no—
55C22×C8D414yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—24no—
63C22⋊Q8—42no—
64C4×D4—22no—
65C4×D4—22no—
66#117—24no—
67#117—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2312yes—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4D411yescharacteristic
80C2×C22⋊C4D411yescharacteristic
81C2×C42—22no—
82C2×C42C2311yescharacteristic
83#107—22no—
84#107—24no—
85#106—22no—
86#106D414yescomplemented
87C4⋊C8—22no—
88#103—24no—
89#103—24no—
90#103—42no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93#97—22no—
94#95—22no—
95#95—22no—
96#95—22no—
97#95—22no—
98C24C2×D411yescharacteristic
99C24C2×D411yescharacteristic
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×D4—22no—
107C2×D4—42no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4#4112yes—
128C22×C4C2411yescharacteristic
129Q16—44no—
130M16—44no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—28no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8C2×D412yes—
141C2×C8C2×D412yes—
142C2×C8C2×D412yes—
143C2×C8#4112yes—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—44no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4C2×D414yes—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—44no—
157C22⋊C4—44no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C42—21no—
161C42—21no—
162C42—44no—
163C23—21no—
164C23—21no—
165C23—22no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23—22no—
172C23#12112yes—
173C23C22×D411yescharacteristic
174Q8—42no—
175Q8—42no—
176Q8—42no—
177Q8—42no—
178D4—44no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4#12112yes—
209C2×C4C22×D411yescharacteristic
210C2×C4C22×D411yescharacteristic
211C2×C4C2×#4111yescharacteristic
212C2×C4C2×#4111yescharacteristic
213C2×C4H(5,𝔽2)11yescharacteristic
214C2×C42−1+411yescharacteristic
215C8—22no—
216C8—41no—
217C8—41no—
218C8#12114yes—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22C2×#12111yescharacteristic
230C22D4211yescharacteristic
231C22#54711yescharacteristic
232C22#57512yes—
233C22#57812yes—
234C4—22no—
235C4—41no—
236C4—44no—
237C4—41no—
238C4—41no—
239C4—82no—
240C4—82no—
241C4—82no—
242C4C2×#12111yescharacteristic
243C4#53711yescharacteristic
244C4#54811yescharacteristic
245C4#55311yescharacteristic
246C2—42no—
247C2—81no—
248C2#241611yescharacteristic
249C2#305411yescharacteristic
250C2#305911yescharacteristic
251C2#326912yes—
252C2#334912yes—
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×#467C2semidirect
4. C2×#427C2semidirect
6. #2068C2semidirect
7. #2036C2semidirect
8. #1971C2semidirect
9. #1944C2semidirect
10. #1873C2semidirect
11. #1809C2semidirect
86. #106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4672
C2×#4352
C2×#4272
#24152
#20682
#20362
#19712
#19442
#18732
#18092