Finite Group Databasebeta

← #24360

Subgroups — #24360

Order 256.

691 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 691 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C23×Q16C211yesmaximal, characteristic
2C2×#462C212yesmaximal, complemented
3C2×#451C211yesmaximal, characteristic
4#2434C211yesmaximal, characteristic
5Q8×C22⋊C4C211yesmaximal, characteristic
6#2056C212yesmaximal, complemented
7#2016C212yesmaximal, complemented
8#1892C211yesmaximal, characteristic
9#1878C212yesmaximal, complemented
10#1850C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C22×Q16—24no—
15C22×Q16—28no—
16C22×Q16C2212yes—
17C22×Q16C2211yescharacteristic
18C22×Q16C2212yes—
19C22×Q16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C4⋊Q8C2212yes—
22C2×C22⋊Q8C2211yescharacteristic
23C2×C4×Q8C2212yes—
24#462—28no—
25#451—24no—
26C2×C4⋊C8C2212yes—
27C2×#103C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33#395—22no—
34#391—22no—
35#391C2212yes—
36#391C2212yes—
37#385—21no—
38#385—21no—
39#380—21no—
40#380C2211yescharacteristic
41C4×C22⋊C4—21no—
42C4×C22⋊C4C2211yescharacteristic
43#340C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—24no—
49C22×Q8—24no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—42no—
53C22×Q8—42no—
54C22×Q8D414yes—
55C22×Q8C2311yescharacteristic
56C22×Q8C2312yes—
57C22×Q8C2311yescharacteristic
58C22×Q8C2311yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C2×Q16—232no—
63C2×Q16—24no—
64C2×Q16—24no—
65C2×Q16—24no—
66C2×Q16—24no—
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—24no—
70C2×Q16—24no—
71C2×Q16—42no—
72C2×Q16D418yes—
73C22×C8—22no—
74C22×C8—24no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2312yes—
78C4⋊Q8—24no—
79C4⋊Q8—42no—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C4×Q8—24no—
83C4×Q8—42no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4C2312yes—
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4D411yescharacteristic
93C2×C22⋊C4D411yescharacteristic
94C2×C42—22no—
95C2×C42C2312yes—
96C4⋊C8—24no—
97#103—22no—
98#103—22no—
99#103—24no—
100#103—24no—
101#103—42no—
102C22⋊C8—22no—
103#95—22no—
104#95—22no—
105#95—21no—
106#95—22no—
107#95—22no—
108#95D412yes—
109C24C2×D411yescharacteristic
110C2×Q8—24no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—28no—
116C2×Q8—28no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—48no—
131C2×Q8—48no—
132C2×Q8—44no—
133C2×Q8—42no—
134C2×Q8C2×D414yes—
135C2×Q8C2×D412yes—
136C2×Q8C2×D412yes—
137C2×Q8C2×D414yes—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4C2×D412yes—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D412yes—
153C22×C4C2×D412yes—
154C22×C4#4112yes—
155C22×C4C2411yescharacteristic
156Q16—216no—
157Q16—216no—
158Q16—44no—
159Q16—44no—
160C2×C8—28no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8C2×D414yes—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C22⋊C4—22no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4Q1614yescomplemented
180C42—22no—
181C42—22no—
182C42—44no—
183C23—22no—
184C23—22no—
185C23#12112yes—
186C23C2×Q1612yes—
187C23#13712yes—
188C23C22×D411yescharacteristic
189Q8—22no—
190Q8—24no—
191Q8—22no—
192Q8—24no—
193Q8—22no—
194Q8—24no—
195Q8—48no—
196Q8—48no—
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—41no—
201Q8—41no—
202Q8—42no—
203Q8#12014yes—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—48no—
218C2×C4—48no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4#12112yes—
230C2×C4C2×Q1614yescomplemented
231C2×C4C22×D411yescharacteristic
232C2×C4C22×D412yes—
233C2×C4C2×#4111yescharacteristic
234C2×C4H(5,𝔽2)12yes—
235C8—24no—
236C8—41no—
237C8—41no—
238C22—22no—
239C22—24no—
240C22—24no—
241C22—22no—
242C22#46214yescomplemented
243C22C2×#12111yescharacteristic
244C22D4212yes—
245C22C22×Q1611yescharacteristic
246C22C2×#13711yescharacteristic
247C22#57812yes—
248C4—22no—
249C4—24no—
250C4—24no—
251C4—42no—
252C4—44no—
253C4—42no—
254C4—81no—
255C4—82no—
256C4—81no—
257C4C2×#12011yescharacteristic
258C4#53411yescharacteristic
259C4#54612yes—
260C2—24no—
261C2#241011yescharacteristic
262C2C2×#46211yescharacteristic
263C2#304411yescharacteristic
264C2#327012yes—
265C2D4×Q1612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#462C2semidirect
6. #2056C2semidirect
7. #2016C2semidirect
9. #1878C2semidirect
179. C22⋊C4Q16semidirect
230. C2×C4C2×Q16semidirect
242. C22#462semidirect

Maximal subgroups

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

subgroupindex
C23×Q162
C2×#4622
C2×#4512
#24342
Q8×C22⋊C42
#20562
#20162
#18922
#18782
#18502
#17872