Finite Group Databasebeta

← #24743

Subgroups — #24743

Order 256.

466 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 135 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×#464C212yesmaximal
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#440C211yesmaximal, characteristic
4#2547C211yesmaximal, characteristic
5#2351C211yesmaximal, characteristic, complemented
6#2068C212yesmaximal, complemented
7#2042C212yesmaximal
8#1960C211yesmaximal, characteristic, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1878C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C22×SD16C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#464—28no—
20#462—24no—
21#440—24no—
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2211yescharacteristic
28C2×#103C2212yes—
29C2×#102C2212yes—
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32#397—21no—
33#397—21no—
34#391C2212yes—
35#389—22no—
36#389C2212yes—
37#382—21no—
38#382C2211yescharacteristic
39#380—22no—
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×Q16—24no—
49C2×Q16—24no—
50C2×Q16—24no—
51C2×SD16—22no—
52C2×SD16—22no—
53C2×SD16D414yes—
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59#124—24no—
60#124—42no—
61C4×Q8—22no—
62C4×Q8—24no—
63C4×Q8—42no—
64C4×Q8D414yes—
65C4×D4—22no—
66C4×D4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D412yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C42—21no—
80C2×C42—22no—
81C2×C42C2312yes—
82C2×C42C2311yescharacteristic
83#107—22no—
84C4⋊C8—24no—
85C4⋊C8—22no—
86#103—22no—
87#103—24no—
88#103—24no—
89#103—42no—
90#102—24no—
91#102—22no—
92#97—22no—
93#95—21no—
94#95—22no—
95#95—22no—
96#95—22no—
97#95—21no—
98#95D411yescharacteristic
99#95D411yescharacteristic
100C24—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8C2×D412yes—
111C2×Q8C2×D412yes—
112C2×Q8#4112yes—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4C2×D412yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4C2×D412yes—
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2411yescharacteristic
131Q16—48no—
132SD16—28no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8C2×D412yes—
142C4⋊C4—24no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4C2×D412yes—
154C22⋊C4—44no—
155C22⋊C4—44no—
156C22⋊C4—42no—
157C42—24no—
158C42—22no—
159C42—22no—
160C42—44no—
161C42C2×D412yes—
162C42#4112yes—
163C23—22no—
164C23—41no—
165C23—41no—
166C23—42no—
167C23—42no—
168C23C22×D411yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—22no—
172Q8—42no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176Q8#12114yes—
177D4—22no—
178D4—22no—
179D4—41no—
180D4—41no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—84no—
203C2×C4#12112yes—
204C2×C4#12112yes—
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C2×C4C2×#4111yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C2×C4H(5,𝔽2)11yescharacteristic
210C2×C42−1+411yescharacteristic
211C8—22no—
212C8—42no—
213C8—41no—
214C22—42no—
215C22—44no—
216C22—82no—
217C22—81no—
218C22—81no—
219C22C2×#12111yescharacteristic
220C22D4211yescharacteristic
221C22#54811yescharacteristic
222C22#57512yes—
223C22#57812yes—
224C4—24no—
225C4—22no—
226C4—22no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4—82no—
232C4—81no—
233C4C2×#12111yescharacteristic
234C4#53711yescharacteristic
235C4#54711yescharacteristic
236C4#55311yescharacteristic
237C2—82no—
238C2#241611yescharacteristic
239C2#304011yescharacteristic
240C2#304411yescharacteristic
241C2#326912yes—
242C2#337712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#462C2semidirect
5. #2351C2semidirect
6. #2068C2semidirect
8. #1960C2semidirect
9. #1940C2semidirect
10. #1878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
C2×#4622
C2×#4402
#25472
#23512
#20682
#20422
#19602
#19402
#18782
#18042