Finite Group Databasebeta

← #28620

Subgroups — #28620

Order 256.

439 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 127 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×#497C211yesmaximal, characteristic
2C2×#489C211yesmaximal, characteristic
3#2862C211yesmaximal, characteristic, complemented
4C4×#124C211yesmaximal, characteristic
5#2056C212yesmaximal, complemented
6#2055C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1878C212yesmaximal, complemented
10#1877C212yesmaximal
11#1767C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#497—24no—
20#489—24no—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#97C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#391—21no—
29#391—21no—
30#391C2212yes—
31#391C2211yescharacteristic
32#391C2212yes—
33#390C2211yescharacteristic
34#388—21no—
35#388—21no—
36#382—21no—
37#382—21no—
38#382C2211yescharacteristic
39C4×C22⋊C4—21no—
40C4×C22⋊C4—21no—
41C43C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8C2312yes—
47C22×Q8C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C2×Q16—24no—
51C2×Q16—24no—
52C2×Q16—42no—
53C2×SD16—24no—
54C2×SD16—24no—
55C2×SD16—42no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62#124—22no—
63#124D414yes—
64C4×Q8—42no—
65C4×D4—42no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C42—21no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86C4⋊C8—24no—
87#103—24no—
88#103—42no—
89#103—42no—
90#102—24no—
91#97—22no—
92#97—22no—
93#95—21no—
94#95—21no—
95#95—21no—
96#95—21no—
97#95—22no—
98#95—22no—
99#95—22no—
100#95—22no—
101C24—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8C2×D412yes—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—22no—
118C2×D4C2×D412yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2411yescharacteristic
134Q16—48no—
135SD16—48no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8#4114yes—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—24no—
151C22⋊C4—24no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C42—22no—
156C42—22no—
157C42—22no—
158C42—22no—
159C42—41no—
160C42—41no—
161C42—42no—
162C42C2×D412yes—
163C42#4112yes—
164C23—22no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22×D411yescharacteristic
172Q8—24no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176Q8—42no—
177Q8—42no—
178Q8—84no—
179D4—24no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—84no—
211C2×C4#12112yes—
212C2×C4#12112yes—
213C2×C4#13712yes—
214C2×C4#13712yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C2×#4112yes—
217C2×C4C2×#4111yescharacteristic
218C2×C4H(5,𝔽2)12yes—
219C8—24no—
220C8—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22C2×#12111yescharacteristic
229C22#53211yescharacteristic
230C22#53511yescharacteristic
231C22C2×#13711yescharacteristic
232C22C2×#13711yescharacteristic
233C22#57511yescharacteristic
234C22#57511yescharacteristic
235C4—22no—
236C4—22no—
237C4—22no—
238C4—22no—
239C4—41no—
240C4—41no—
241C4—42no—
242C4—81no—
243C4—81no—
244C4—82no—
245C4—82no—
246C4#54612yes—
247C4#55912yes—
248C2—82no—
249C2#254411yescharacteristic
250C2#303911yescharacteristic
251C2#304011yescharacteristic
252C2#317211yescharacteristic
253C2#318111yescharacteristic
254C2#321511yescharacteristic
255C2#321611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2862C2semidirect
5. #2056C2semidirect
7. #1993C2semidirect
8. #1960C2semidirect
9. #1878C2semidirect
11. #1767C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4892
#28622
C4×#1242
#20562
#20552
#19932
#19602
#18782
#18772
#17672