Finite Group Databasebeta

← #32271

Subgroups — #32271

Order 256.

378 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#2547C211yesmaximal, characteristic
2#2486C211yesmaximal, characteristic
3#2077C212yesmaximal, complemented
4#2076C212yesmaximal, complemented
5#2057C212yesmaximal, complemented
6#1919C212yesmaximal, complemented
7#1870C211yesmaximal, characteristic
8#1852C212yesmaximal, complemented
9#1809C212yesmaximal, complemented
10C22×M16C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×#124—21no—
13C2×#123—21no—
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C2×#107C2212yes—
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C22⋊C8C2212yes—
20C2×C22⋊C8C2211yescharacteristic
21#398—21no—
22#397—21no—
23#397—21no—
24#397—21no—
25#391—21no—
26#389C2212yes—
27#381—21no—
28#380C2212yes—
29#380C2212yes—
30#380C2211yescharacteristic
31#380C2212yes—
32C4×C4⋊C4C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×D4D411yescharacteristic
41C23×C4—21no—
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C2×M16—24no—
45C2×M16—41no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—41no—
49C4⋊Q8—41no—
50#124—44no—
51#123—44no—
52C4×D4—41no—
53C4×D4—41no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4D411yescharacteristic
69C2×C22⋊C4D411yescharacteristic
70C2×C42—21no—
71C2×C42C2311yescharacteristic
72#107—42no—
73#107—42no—
74#106—24no—
75C22⋊C8—24no—
76C22⋊C8—41no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95—22no—
81#95—21no—
82#95—22no—
83C24C2×D411yescharacteristic
84C24C2×D411yescharacteristic
85C2×Q8—42no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4—41no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111M16—44no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8#4114yes—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—41no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—44no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—42no—
133C22⋊C4—44no—
134C22⋊C4—44no—
135C22⋊C4—44no—
136C42—21no—
137C42—21no—
138C42—42no—
139C23—21no—
140C23—22no—
141C23—21no—
142C23—22no—
143C23—21no—
144C23—22no—
145C23—21no—
146C23—21no—
147C23—22no—
148C23AGL(1,ℤ/8)12yes—
149C23C22×D411yescharacteristic
150Q8—84no—
151D4—44no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—84no—
179C2×C4AGL(1,ℤ/8)12yes—
180C2×C4C2×#4112yes—
181C2×C4H(5,𝔽2)11yescharacteristic
182C2×C4H(5,𝔽2)11yescharacteristic
183C2×C42−1+412yes—
184C8—24no—
185C8—42no—
186C22—24no—
187C22—22no—
188C22—42no—
189C22—41no—
190C22—41no—
191C22—42no—
192C22—42no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22#53411yescharacteristic
197C22#54712yes—
198C22C2×AGL(1,ℤ/8)11yescharacteristic
199C22C2×AGL(1,ℤ/8)11yescharacteristic
200C22#57812yes—
201C4—22no—
202C4—41no—
203C4—41no—
204C4—41no—
205C4—82no—
206C4—82no—
207C4—82no—
208C4—82no—
209C4#54212yes—
210C4#55111yescharacteristic
211C4#56311yescharacteristic
212C2—42no—
213C2—81no—
214C2#271711yescharacteristic
215C2#319211yescharacteristic
216C2#320211yescharacteristic
217C2#334312yes—
218C2#335312yes—
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
3. #2077C2semidirect
4. #2076C2semidirect
5. #2057C2semidirect
6. #1919C2semidirect
8. #1852C2semidirect
9. #1809C2semidirect

Maximal subgroups

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

subgroupindex
#25472
#24862
#20772
#20762
#20572
#19192
#18702
#18522
#18092