Finite Group Databasebeta

← #14786

Subgroups — #14786

Order 256.

1095 conjugacy classes of subgroups; only the 247 normal subgroups are shown.

normal subgroups only

This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 247 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×C4⋊C4C211yesmaximal, characteristic
2C2×#398C212yesmaximal, complemented
3C2×#385C211yesmaximal, characteristic
4C2×#385C212yesmaximal
5#1445C212yesmaximal, complemented
6#1444C212yesmaximal, complemented
7#1441C212yesmaximal, complemented
8C22⋊#95C212yesmaximal
9C22⋊#95C211yesmaximal, characteristic
10C24×C4C2211yescharacteristic
11C24×C4C2212yes—
12C22×C4⋊C4—24no—
13C22×C4⋊C4—28no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2214yes—
18C22×C4⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22#398—28no—
23#385—24no—
24#385—28no—
25C2×#95C2214yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2214yes—
29C2×#95C2211yescharacteristic
30C2×#95C2211yescharacteristic
31C2×#95C2214yes—
32C25C2311yescharacteristic
33C23×C4—24no—
34C23×C4—24no—
35C23×C4—24no—
36C23×C4—24no—
37C23×C4—28no—
38C23×C4—24no—
39C23×C4—24no—
40C23×C4—44no—
41C23×C4D414yes—
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2314yes—
47C23×C4C2314yes—
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C2×C4⋊C4—232no—
51C2×C4⋊C4—28no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—28no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—28no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—44no—
62C2×C4⋊C4D418yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4D414yes—
70C2×C22⋊C4Q818yes—
71#95—28no—
72#95—22no—
73#95—22no—
74#95—24no—
75#95—28no—
76#95—24no—
77#95—44no—
78C24—22no—
79C24—22no—
80C24—22no—
81C24—22no—
82C24C2×D412yes—
83C24C2×D412yes—
84C24C2×Q814yes—
85C24#4112yes—
86C24#4114yes—
87C24C2411yescharacteristic
88C22×C4—232no—
89C22×C4—28no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—28no—
95C22×C4—28no—
96C22×C4—216no—
97C22×C4—24no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—416no—
126C22×C4—416no—
127C22×C4—48no—
128C22×C4—48no—
129C22×C4C2×D412yes—
130C22×C4C2×D418yes—
131C22×C4C2×D412yes—
132C22×C4C2×D412yes—
133C22×C4C2×D414yes—
134C22×C4C2×D412yes—
135C22×C4C2×Q818yes—
136C22×C4#4114yes—
137C4⋊C4—216no—
138C4⋊C4—216no—
139C4⋊C4—416no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C22⋊C4—28no—
143C22⋊C4—216no—
144C22⋊C4—44no—
145C23—22no—
146C23—22no—
147C23—28no—
148C23—22no—
149C23—22no—
150C23—28no—
151C23—24no—
152C23—24no—
153C23—24no—
154C23—28no—
155C23—24no—
156C23—28no—
157C23#12114yes—
158C23C22⋊Q818yes—
159C23#12314yes—
160C23#12518yes—
161C23C4⋊Q814yes—
162C23C22×D411yescharacteristic
163C23C22×D412yes—
164C23C22×D411yescharacteristic
165C23C22×Q812yes—
166C23C2×#4111yescharacteristic
167C23C2×#4112yes—
168C23C2×#4112yes—
169C23H(5,𝔽2)12yes—
170C232−1+412yes—
171C2×C4—232no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—28no—
177C2×C4—28no—
178C2×C4—28no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—28no—
182C2×C4—24no—
183C2×C4—28no—
184C2×C4—28no—
185C2×C4—24no—
186C2×C4—28no—
187C2×C4—28no—
188C2×C4—432no—
189C2×C4—41no—
190C2×C4—44no—
191C2×C4—41no—
192C2×C4—44no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—41no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—44no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—44no—
211C2×C4#12014yes—
212C2×C4#12314yes—
213C22—24no—
214C22—24no—
215C22—24no—
216C22—24no—
217C22—216no—
218C22—216no—
219C22—24no—
220C22—24no—
221C22#39818yescomplemented
222C22C2×#12011yescharacteristic
223C22C2×#12111yescharacteristic
224C22C2×C22⋊Q812yes—
225C22C2×#12311yescharacteristic
226C22C2×#12311yescharacteristic
227C22#52514yes—
228C22C2×#12512yes—
229C22C2×C4⋊Q811yescharacteristic
230C22#53511yescharacteristic
231C22#53611yescharacteristic
232C22#53914yes—
233C22D4212yes—
234C22#54612yes—
235C22#54712yes—
236C22D4×Q814yes—
237C22#55212yes—
238C22#55414yes—
239C4—24no—
240C4—24no—
241C4—42no—
242C4—48no—
243C4—48no—
244C4—42no—
245C4—81no—
246C4—81no—
247C2—28no—
248C2C2×#39811yescharacteristic
249C2#240811yescharacteristic
250C2#243411yescharacteristic
251C2#244914yes—
252C2#247314yes—
253C2#248611yescharacteristic
254C2#250211yescharacteristic
255C2#262311yescharacteristic
256C2#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#398C2semidirect
5. #1445C2semidirect
6. #1444C2semidirect
7. #1441C2semidirect
221. C22#398semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C42
C2×#3982
C2×#3852
#14452
#14442
#14412
C22⋊#952