Finite Group Databasebeta

← #24749

Subgroups — #24749

Order 256.

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

normal subgroups only

This group has 522 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×#463C212yesmaximal, complemented
2C2×#459C211yesmaximal, characteristic
3C2×#440C211yesmaximal, characteristic, complemented
4#2548C211yesmaximal, characteristic, complemented
5#2349C211yesmaximal, characteristic, complemented
6#2067C212yesmaximal, complemented
7#2042C212yesmaximal, complemented
8#1961C211yesmaximal, characteristic, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1879C212yesmaximal, complemented
11#1805C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×D8C2212yes—
14C2×#127C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2212yes—
18C2×C4×D4C2211yescharacteristic
19#463—28no—
20#459—24no—
21#440—24no—
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2211yescharacteristic
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2211yescharacteristic
30C2×#102C2212yes—
31C2×#97C2211yescharacteristic
32#394—21no—
33#394—21no—
34#390—22no—
35#390C2212yes—
36#389C2212yes—
37#385—22no—
38#382C2211yescharacteristic
39#380—21no—
40C4×C4⋊C4C2211yescharacteristic
41C2×#95—21no—
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2311yescharacteristic
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4—22no—
51C2×SD16—22no—
52C2×SD16—22no—
53C2×SD16D414yescomplemented
54C2×D8—24no—
55C2×D8—24no—
56C2×D8—24no—
57C22×C8C2312yes—
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60#127—22no—
61#127—22no—
62#124—24no—
63#124—42no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×D4—24no—
67C4×D4—22no—
68C4×D4—42no—
69C4×D4D414yes—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D412yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C42—21no—
83C2×C42C2312yes—
84C2×C42C2311yescharacteristic
85#107—22no—
86C4⋊C8—24no—
87C4⋊C8—22no—
88#103—24no—
89#103—22no—
90#102—22no—
91#102—24no—
92#102—24no—
93#102—42no—
94#97—22no—
95#95—21no—
96#95—22no—
97#95—21no—
98#95D411yescharacteristic
99#95D411yescharacteristic
100C24—21no—
101C24—22no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8C2×D412yes—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—44no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—82no—
121C2×D4C2×D412yes—
122C2×D4C2×D412yes—
123C2×D4#4112yes—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138SD16—28no—
139D8—48no—
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×C8C2×D412yes—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—82no—
157C4⋊C4C2×D412yes—
158C22⋊C4—24no—
159C22⋊C4—44no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—82no—
163C22⋊C4—82no—
164C42—22no—
165C42—22no—
166C42—42no—
167C42C2×D412yes—
168C42#4112yes—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23C22×D411yescharacteristic
182Q8—22no—
183Q8—22no—
184Q8—41no—
185Q8—41no—
186D4—22no—
187D4—22no—
188D4—22no—
189D4—42no—
190D4—42no—
191D4—42no—
192D4—42no—
193D4—84no—
194D4—84no—
195D4#12114yes—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—84no—
215C2×C4—88no—
216C2×C4#12112yes—
217C2×C4#12112yes—
218C2×C4C22×D411yescharacteristic
219C2×C4C22×D411yescharacteristic
220C2×C4C2×#4111yescharacteristic
221C2×C4C2×#4111yescharacteristic
222C2×C4H(5,𝔽2)11yescharacteristic
223C2×C42−1+411yescharacteristic
224C8—22no—
225C8—42no—
226C8—41no—
227C22—24no—
228C22—44no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—82no—
237C22—82no—
238C22—82no—
239C22C2×#12111yescharacteristic
240C22D4211yescharacteristic
241C22#54811yescharacteristic
242C22#57512yes—
243C22#57612yes—
244C4—22no—
245C4—22no—
246C4—42no—
247C4—42no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—81no—
252C4C2×#12111yescharacteristic
253C4#53711yescharacteristic
254C4#54711yescharacteristic
255C4#55311yescharacteristic
256C2—44no—
257C2—162no—
258C2#241611yescharacteristic
259C2#304011yescharacteristic
260C2#304111yescharacteristic
261C2#326812yes—
262C2#337012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
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
1. C2×#463C2semidirect
3. C2×#440C2semidirect
4. #2548C2semidirect
5. #2349C2semidirect
6. #2067C2semidirect
7. #2042C2semidirect
8. #1961C2semidirect
9. #1940C2semidirect
10. #1879C2semidirect
11. #1805C2semidirect
53. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4592
C2×#4402
#25482
#23492
#20672
#20422
#19612
#19402
#18792
#18052