Finite Group Databasebeta

← #32039

Subgroups — #32039

Order 256.

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

normal subgroups only

This group has 404 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#2542C211yesmaximal, characteristic
2#2486C211yesmaximal, characteristic
3#2043C212yesmaximal, complemented
4#2036C212yesmaximal, complemented
5#2036C212yesmaximal, complemented
6#1918C212yesmaximal, complemented
7#1894C212yesmaximal, complemented
8#1872C212yesmaximal, complemented
9#1870C211yesmaximal, characteristic
10C22×M16C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×#123—21no—
13C2×C22⋊Q8C2212yes—
14C2×C22⋊Q8C2212yes—
15C2×C4×D4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C2×#107C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×C22⋊C8C2212yes—
23C2×C22⋊C8C2211yescharacteristic
24#399—21no—
25#399—21no—
26#398—21no—
27#397—21no—
28#390—21no—
29#389C2212yes—
30#384—21no—
31#381—21no—
32#380C2212yes—
33#380C2211yescharacteristic
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8C2312yes—
40C22×D4D411yescharacteristic
41C23×C4—21no—
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C2×M16—42no—
45C2×M16—41no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—41no—
49C4⋊Q8—41no—
50#123—44no—
51C22⋊Q8—24no—
52C22⋊Q8—24no—
53C22⋊Q8—42no—
54C22⋊Q8—42no—
55C4×D4—41no—
56C4×D4—41no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4D411yescharacteristic
74C2×C22⋊C4D411yescharacteristic
75C2×C42—21no—
76C2×C42C2311yescharacteristic
77#107—24no—
78#103—42no—
79#103—42no—
80#103—42no—
81#103—42no—
82C22⋊C8—41no—
83C22⋊C8—42no—
84#95—21no—
85#95—22no—
86#95—21no—
87#95—22no—
88C24C2×D411yescharacteristic
89C24C2×D411yescharacteristic
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—41no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119M16—44no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—41no—
132C4⋊C4—44no—
133C4⋊C4—44no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—44no—
138C4⋊C4#4114yes—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C22⋊C4—44no—
142C22⋊C4—44no—
143C22⋊C4—42no—
144C22⋊C4—44no—
145C42—21no—
146C42—21no—
147C42—42no—
148C23—21no—
149C23—22no—
150C23—21no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—21no—
156C23—22no—
157C23#13712yes—
158C23C22×D411yescharacteristic
159Q8—42no—
160Q8—42no—
161Q8—42no—
162Q8—42no—
163D4—44no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
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—42no—
190C2×C4—84no—
191C2×C4AGL(1,ℤ/8)12yes—
192C2×C4C2×#4112yes—
193C2×C4H(5,𝔽2)11yescharacteristic
194C2×C4H(5,𝔽2)12yes—
195C2×C4H(5,𝔽2)11yescharacteristic
196C8—42no—
197C8—42no—
198C22—22no—
199C22—24no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—42no—
208C22#53411yescharacteristic
209C22#54612yes—
210C22C2×AGL(1,ℤ/8)11yescharacteristic
211C22C2×#13711yescharacteristic
212C22#57812yes—
213C4—22no—
214C4—44no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4#54012yes—
222C4#55911yescharacteristic
223C4#56011yescharacteristic
224C2—42no—
225C2—81no—
226C2#270911yescharacteristic
227C2#319311yescharacteristic
228C2#320211yescharacteristic
229C2#331212yes—
230C2#333012yes—
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. #2043C2semidirect
4. #2036C2semidirect
5. #2036C2semidirect
6. #1918C2semidirect
7. #1894C2semidirect
8. #1872C2semidirect

Maximal subgroups

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

subgroupindex
#25422
#24862
#20432
#20362
#19182
#18942
#18722
#18702