Finite Group Databasebeta

← #24201

Subgroups — #24201

Order 256.

534 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#556C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#440C211yesmaximal, characteristic
4#2581C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal
6#2015C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1878C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C22×SD16C2211yescharacteristic
14#556—24no—
15#556—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#126—22no—
18C2×#124C2212yes—
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117—21no—
25#462—24no—
26#440—24no—
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×#97C2211yescharacteristic
36#400—21no—
37#395—21no—
38#391C2212yes—
39#389C2212yes—
40#387—22no—
41#382C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8D411yescharacteristic
46C22×Q8C2312yes—
47C22×Q8C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C2×Q16—24no—
51C2×Q16—24no—
52C2×Q16—42no—
53C2×SD16—22no—
54C2×SD16—22no—
55C2×SD16—22no—
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C4⋊Q8—22no—
60C4⋊Q8D414yes—
61#126—216no—
62#124—24no—
63#124—24no—
64C22⋊Q8—28no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×D4—22no—
72C4×D4D414yes—
73#117—28no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4D412yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42C2312yes—
90C2×C42C2311yescharacteristic
91#107—22no—
92C4⋊C8—22no—
93#103—24no—
94#103—24no—
95#103—22no—
96#103—42no—
97#102—22no—
98#102—42no—
99#97—22no—
100#95—21no—
101#95—21no—
102#95—22no—
103#95—22no—
104#95—22no—
105C24—21no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—42no—
118C2×Q8C2×D414yes—
119C2×Q8#4112yes—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4C2×D412yes—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—44no—
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138Q16—48no—
139SD16—28no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8#4112yes—
149C4⋊C4—28no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—24no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4C2×D412yes—
162C22⋊C4—28no—
163C22⋊C4—28no—
164C22⋊C4—24no—
165C42—28no—
166C42—22no—
167C42—22no—
168C42—42no—
169C42C2×D412yes—
170C42C2×D412yes—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—42no—
175C23—42no—
176C23C22×D411yescharacteristic
177Q8—22no—
178Q8—22no—
179Q8—24no—
180Q8—24no—
181Q8—44no—
182Q8—42no—
183Q8—42no—
184Q8—41no—
185Q8—41no—
186D4—22no—
187D4—22no—
188D4—41no—
189D4—41no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—48no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4#12012yes—
212C2×C4#12012yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C2×#4111yescharacteristic
216C2×C4C2×#4111yescharacteristic
217C2×C4H(5,𝔽2)11yescharacteristic
218C2×C42−1+411yescharacteristic
219C8—22no—
220C8—41no—
221C8—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—82no—
226C22—82no—
227C22C2×#12011yescharacteristic
228C22#54811yescharacteristic
229C22#55011yescharacteristic
230C22#57512yes—
231C22#57812yes—
232C4—22no—
233C4—22no—
234C4—44no—
235C4—42no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—82no—
240C4—81no—
241C4#52511yescharacteristic
242C4#53711yescharacteristic
243C4#54611yescharacteristic
244C4#54711yescharacteristic
245C2—82no—
246C2#240711yescharacteristic
247C2#300511yescharacteristic
248C2#301111yescharacteristic
249C2#337712yes—
250C2#340212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#462C2semidirect
4. #2581C2semidirect
7. #1985C2semidirect
8. #1960C2semidirect
9. #1878C2semidirect
10. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4622
C2×#4402
#25812
#20422
#20152
#19852
#19602
#18782
#18632
#18042