Finite Group Databasebeta

← #32029

Subgroups — #32029

Order 256.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#2487C211yesmaximal, characteristic
3#2043C212yesmaximal, complemented
4#2036C212yesmaximal, complemented
5#2036C212yesmaximal, complemented
6#1921C212yesmaximal, complemented
7#1892C212yesmaximal, complemented
8#1872C212yesmaximal, complemented
9#1869C211yesmaximal, characteristic
10C23×C8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C22⋊Q8C2212yes—
13C2×C22⋊Q8C2212yes—
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C2×#107C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×C22⋊C8C2212yes—
22C2×C22⋊C8C2211yescharacteristic
23#400—21no—
24#399—21no—
25#399—21no—
26#399—21no—
27#390—21no—
28#389C2212yes—
29#386—21no—
30#385—21no—
31#384—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
44C22×C8—42no—
45C22×C8—41no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—41no—
49C4⋊Q8—41no—
50C22⋊Q8—24no—
51C22⋊Q8—24no—
52C22⋊Q8—42no—
53C22⋊Q8—42no—
54C4×D4—41no—
55C4×D4—41no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4D411yescharacteristic
73C2×C22⋊C4D411yescharacteristic
74C2×C42—21no—
75C2×C42C2311yescharacteristic
76#107—24no—
77#103—42no—
78#103—42no—
79#103—42no—
80#103—42no—
81C22⋊C8—41no—
82C22⋊C8—42no—
83#95—21no—
84#95—22no—
85#95—21no—
86#95—22no—
87C24C2×D411yescharacteristic
88C24C2×D411yescharacteristic
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×D4—41no—
96C2×D4—41no—
97C2×D4Q1614yes—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4—82no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—44no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—41no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C4⋊C4#4114yes—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C22⋊C4—42no—
142C22⋊C4—82no—
143C42—21no—
144C42—21no—
145C42—42no—
146C23—21no—
147C23—21no—
148C23—22no—
149C23—22no—
150C23—21no—
151C23—21no—
152C23—22no—
153C23—22no—
154C23C2×Q1612yes—
155C23C2×Q1612yes—
156C23C22×D411yescharacteristic
157Q8—42no—
158Q8—42no—
159Q8—42no—
160Q8—42no—
161D4—44no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
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—84no—
189C2×C4C2×Q1612yes—
190C2×C4C2×#4112yes—
191C2×C4H(5,𝔽2)11yescharacteristic
192C2×C4H(5,𝔽2)12yes—
193C2×C4H(5,𝔽2)11yescharacteristic
194C8—42no—
195C8—42no—
196C22—22no—
197C22—22no—
198C22—24no—
199C22—24no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—42no—
205C22—42no—
206C22#53411yescharacteristic
207C22#54612yes—
208C22C22×Q1611yescharacteristic
209C22#57511yescharacteristic
210C22#57712yes—
211C4—22no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—44no—
216C4—82no—
217C4—82no—
218C4—82no—
219C4#54012yes—
220C4#55911yescharacteristic
221C4#56011yescharacteristic
222C2—42no—
223C2—42no—
224C2#270911yescharacteristic
225C2#318911yescharacteristic
226C2#320011yescharacteristic
227C2#329912yes—
228C2#330712yes—
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. #1921C2semidirect
7. #1892C2semidirect
8. #1872C2semidirect

Maximal subgroups

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

subgroupindex
#25422
#24872
#20432
#20362
#19212
#18922
#18722
#18692