Finite Group Databasebeta

← #28554

Subgroups — #28554

Order 256.

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

normal subgroups only

This group has 451 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×#467C211yesmaximal, characteristic, complemented
2C2×C8×D4C211yesmaximal, characteristic
3#2415C212yesmaximal
4#2086C211yesmaximal, characteristic, complemented
5#2075C211yesmaximal, characteristic, complemented
6#2056C211yesmaximal, characteristic, complemented
7#2001C211yesmaximal, characteristic, complemented
8#1933C212yesmaximal, complemented
9#1892C212yesmaximal, complemented
10#1873C212yesmaximal, complemented
11#1735C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×C22⋊Q8C2212yes—
17C2×C4×D4C2211yescharacteristic
18C2×#117C2212yes—
19#467—24no—
20C8×D4—24no—
21C2×#107C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#397—22no—
30#394—22no—
31#391C2212yes—
32#389—22no—
33#389C2212yes—
34#388—22no—
35#380C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41C22×Q8C2312yes—
42C22×D4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C2×Q16—22no—
46C2×Q16—42no—
47C22×C8—22no—
48C22×C8—42no—
49C22×C8—42no—
50C22×C8D414yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C22⋊Q8—24no—
58C22⋊Q8—42no—
59C4×D4—22no—
60C4×D4—22no—
61#117—24no—
62#117—42no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D411yescharacteristic
75C2×C22⋊C4D411yescharacteristic
76C2×C42—22no—
77C2×C42C2311yescharacteristic
78#107—22no—
79C4⋊C8—22no—
80#103—24no—
81#103—24no—
82#103—42no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85C4×C8—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90C24C2×D411yescharacteristic
91C24C2×D411yescharacteristic
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4#4112yes—
117C22×C4C2411yescharacteristic
118Q16—44no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—44no—
126C2×C8—44no—
127C2×C8C2×D412yes—
128C2×C8C2×D412yes—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—44no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4#4114yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—44no—
142C22⋊C4—44no—
143C22⋊C4—44no—
144C22⋊C4—44no—
145C42—21no—
146C42—21no—
147C42—44no—
148C23—21no—
149C23—22no—
150C23—22no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—22no—
155C23—22no—
156C23#12112yes—
157C23#13512yes—
158C23C22×D411yescharacteristic
159Q8—42no—
160Q8—42no—
161Q8—42no—
162Q8—42no—
163D4—28no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4#12112yes—
190C2×C4#13512yes—
191C2×C4C22×D411yescharacteristic
192C2×C4C2×#4112yes—
193C2×C4C2×#4111yescharacteristic
194C2×C4H(5,𝔽2)12yes—
195C8—22no—
196C8—22no—
197C8—41no—
198C8—41no—
199C22—24no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—41no—
208C22—41no—
209C22C2×#12111yescharacteristic
210C22#54612yes—
211C22C2×#13511yescharacteristic
212C22C2×#13511yescharacteristic
213C22#57812yes—
214C4—22no—
215C4—22no—
216C4—44no—
217C4—41no—
218C4—41no—
219C4—82no—
220C4—82no—
221C4—82no—
222C4#53211yescharacteristic
223C4#53411yescharacteristic
224C4#55312yes—
225C2—42no—
226C2—42no—
227C2#254311yescharacteristic
228C2#305011yescharacteristic
229C2#305911yescharacteristic
230C2#330012yes—
231C2#330212yes—
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×#467C2semidirect
4. #2086C2semidirect
5. #2075C2semidirect
6. #2056C2semidirect
7. #2001C2semidirect
8. #1933C2semidirect
9. #1892C2semidirect
10. #1873C2semidirect
11. #1735C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×C8×D42
#24152
#20862
#20752
#20562
#20012
#19332
#18922
#18732
#17352