Finite Group Databasebeta

← #32957

Subgroups — #32957

Order 256.

666 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C22×C4×Q8C211yesmaximal, characteristic
2C2×#477C212yesmaximal, complemented
3#2857C211yesmaximal, characteristic
4#1863C214yesmaximal, complemented
5#1852C212yesmaximal
6#1803C212yesmaximal, complemented
7#1797C211yesmaximal, characteristic
8#1787C212yesmaximal
9C23×Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C22⋊Q8—21no—
12C2×C4×Q8—28no—
13C2×C4×Q8—21no—
14C2×C4×Q8—21no—
15C2×C4×Q8C2212yes—
16C2×C4×Q8C2212yes—
17C22×C4⋊C4—21no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C42—21no—
20C22×C42C2211yescharacteristic
21#477—28no—
22C2×#107C2214yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2214yes—
25C2×#103C2214yes—
26C2×C22⋊C8C2212yes—
27#391—21no—
28#389—21no—
29#389C2212yes—
30#384C2211yescharacteristic
31#380—21no—
32#380C2212yes—
33#340C2214yes—
34#336C2212yes—
35C22×Q8—24no—
36C22×Q8—24no—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8—22no—
40C22×Q8Q814yescomplemented
41C22×Q8C2312yes—
42C23×C4—21no—
43C23×C4—21no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2314yes—
48C4⋊Q8—24no—
49C22⋊Q8—44no—
50C4×Q8—28no—
51C4×Q8—28no—
52C4×Q8—24no—
53C4×Q8—24no—
54C4×Q8—24no—
55C4×Q8D418yes—
56C2×C4⋊C4—28no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C42—28no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—24no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—28no—
78C4⋊C8—24no—
79#103—28no—
80#103—28no—
81C22⋊C8—24no—
82#95—22no—
83#95—22no—
84C24C2×D411yescharacteristic
85C2×Q8—24no—
86C2×Q8—216no—
87C2×Q8—24no—
88C2×Q8—216no—
89C2×Q8—24no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8C2×D414yes—
97C2×Q8C2×Q814yes—
98C2×Q8#4114yes—
99C22×C4—24no—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4Q1614yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×Q812yes—
123C22×C4#4112yes—
124C22×C4C2411yescharacteristic
125C2×C8—24no—
126C2×C8—24no—
127C2×C8—24no—
128C4⋊C4—28no—
129C4⋊C4—24no—
130C4⋊C4—28no—
131C4⋊C4—22no—
132C4⋊C4—24no—
133C4⋊C4—24no—
134C4⋊C4—42no—
135C4⋊C4—44no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4C2×D414yes—
139C22⋊C4—44no—
140C42—28no—
141C42—28no—
142C42—22no—
143C42—22no—
144C42—22no—
145C42C2×D414yes—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23C22⋊Q812yes—
151C23C2×Q1612yes—
152C23AGL(1,ℤ/8)12yes—
153C23C22×D411yescharacteristic
154Q8—216no—
155Q8—24no—
156Q8—24no—
157Q8—24no—
158Q8—84no—
159Q8C22⋊Q818yescomplemented
160C2×C4—216no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—28no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—44no—
178C2×C4—48no—
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#12012yes—
188C2×C4#12012yes—
189C2×C4C22⋊Q812yes—
190C2×C4C2×Q1612yes—
191C2×C4C2×Q1612yes—
192C2×C4#13712yes—
193C2×C4C22×D412yes—
194C2×C4C22×Q811yescharacteristic
195C2×C4C2×#4112yes—
196C2×C4C2×#4111yescharacteristic
197C8—44no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22#47714yescomplemented
205C22C2×#12011yescharacteristic
206C22C2×C22⋊Q811yescharacteristic
207C22#52511yescharacteristic
208C22C22×Q1611yescharacteristic
209C22C2×AGL(1,ℤ/8)11yescharacteristic
210C22C2×#13711yescharacteristic
211C22#57511yescharacteristic
212C4—28no—
213C4—22no—
214C4—22no—
215C4—22no—
216C4—41no—
217C4—44no—
218C4—82no—
219C4—82no—
220C4#45114yes—
221C4C2×C22⋊Q812yes—
222C4#52512yes—
223C2—24no—
224C2#284811yescharacteristic
225C2C2×#45111yescharacteristic
226C2#300611yescharacteristic
227C2C2×#47711yescharacteristic
228C2#307811yescharacteristic
229C2#310411yescharacteristic
230C2#311211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#477C2semidirect
4. #1863C2semidirect
6. #1803C2semidirect
40. C22×Q8Q8semidirect
159. Q8C22⋊Q8semidirect
204. C22#477semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q82
C2×#4772
#28572
#18632
#18522
#18032
#17972
#17872