Finite Group Databasebeta

← #32906

Subgroups — #32906

Order 256.

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

normal subgroups only

This group has 718 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×#462C212yesmaximal, complemented
3C2×#451C212yesmaximal
4#2857C211yesmaximal, characteristic
5#1878C214yesmaximal, complemented
6#1872C212yesmaximal
7#1803C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic
9C23×Q8C2211yescharacteristic
10C22×Q16C2214yes—
11C2×C4⋊Q8C2211yescharacteristic
12C2×C22⋊Q8C2212yes—
13C2×C4×Q8—28no—
14C2×C4×Q8—21no—
15C2×C4×Q8—21no—
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2212yes—
18C22×C4⋊C4—21no—
19C22×C4⋊C4C2211yescharacteristic
20C22×C42—21no—
21C22×C42C2211yescharacteristic
22#462—28no—
23#451—28no—
24C2×C4⋊C8C2212yes—
25C2×#103C2214yes—
26C2×#103C2214yes—
27C2×C22⋊C8C2212yes—
28#391C2212yes—
29#389—21no—
30#389—21no—
31#384C2211yescharacteristic
32#380—21no—
33#380—21no—
34#340C2214yes—
35#336C2212yes—
36C22×Q8—24no—
37C22×Q8—24no—
38C22×Q8—21no—
39C22×Q8—22no—
40C22×Q8D414yes—
41C22×Q8C2312yes—
42C22×Q8C2312yes—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C2×Q16—28no—
49C2×Q16—28no—
50C2×Q16—28no—
51C22×C8C2314yes—
52C4⋊Q8—24no—
53C22⋊Q8—24no—
54C22⋊Q8—24no—
55C4×Q8—28no—
56C4×Q8—28no—
57C4×Q8—24no—
58C4×Q8—24no—
59C4×Q8—24no—
60C4×Q8D418yes—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C42—28no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—24no—
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82C4⋊C8—24no—
83#103—28no—
84#103—28no—
85C22⋊C8—24no—
86#95—22no—
87#95—22no—
88C24C2×D411yescharacteristic
89C2×Q8—24no—
90C2×Q8—216no—
91C2×Q8—24no—
92C2×Q8—216no—
93C2×Q8—24no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—24no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—24no—
101C2×Q8C2×D414yes—
102C2×Q8C2×D414yes—
103C2×Q8#4114yes—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4Q1614yescomplemented
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4#4112yes—
129C22×C4C2411yescharacteristic
130Q16—416no—
131C2×C8—24no—
132C2×C8—24no—
133C2×C8—24no—
134C4⋊C4—28no—
135C4⋊C4—24no—
136C4⋊C4—28no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4C2×D414yes—
146C22⋊C4—44no—
147C42—28no—
148C42—28no—
149C42—22no—
150C42—22no—
151C42—22no—
152C42C2×D414yes—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23#12112yes—
158C23C2×Q1612yes—
159C23#13712yes—
160C23C22×D411yescharacteristic
161Q8—216no—
162Q8—24no—
163Q8—24no—
164Q8—24no—
165Q8—44no—
166Q8—44no—
167Q8#12118yes—
168C2×C4—216no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—28no—
181C2×C4—21no—
182C2×C4—24no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—44no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4#12012yes—
196C2×C4#12012yes—
197C2×C4#12112yes—
198C2×C4C2×Q1612yes—
199C2×C4C2×Q1612yes—
200C2×C4#13712yes—
201C2×C4C22×D411yescharacteristic
202C2×C4C22×D412yes—
203C2×C4C2×#4112yes—
204C2×C4C2×#4111yescharacteristic
205C8—44no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22#46214yescomplemented
213C22C2×#12011yescharacteristic
214C22C2×#12111yescharacteristic
215C22#52511yescharacteristic
216C22C22×Q1611yescharacteristic
217C22C2×#13711yescharacteristic
218C22C2×#13711yescharacteristic
219C22#57511yescharacteristic
220C4—28no—
221C4—22no—
222C4—22no—
223C4—22no—
224C4—41no—
225C4—44no—
226C4—82no—
227C4—82no—
228C4#45114yescomplemented
229C4C2×#12112yes—
230C4#52512yes—
231C2—24no—
232C2#284711yescharacteristic
233C2C2×#45111yescharacteristic
234C2#300711yescharacteristic
235C2C2×#46211yescharacteristic
236C2#303711yescharacteristic
237C2#310411yescharacteristic
238C2#311311yescharacteristic
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×#462C2semidirect
5. #1878C2semidirect
7. #1803C2semidirect
121. C22×C4Q16semidirect
212. C22#462semidirect
228. C4#451semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q82
C2×#4622
C2×#4512
#28572
#18782
#18722
#18032
#17972