Finite Group Databasebeta

← #28230

Subgroups — #28230

Order 256.

517 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 143 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×#468C212yesmaximal, complemented
2C2×C4×SD16C211yesmaximal, characteristic, complemented
3C2×#436C211yesmaximal, characteristic, complemented
4#2538C211yesmaximal, characteristic, complemented
5#2537C211yesmaximal, characteristic
6#2036C212yesmaximal, complemented
7#2033C212yesmaximal, complemented
8#1845C211yesmaximal, characteristic, complemented
9#1809C212yesmaximal, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1802C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×M16C2212yes—
14C2×C22⋊Q8C2212yes—
15C2×#121—21no—
16C2×#121C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#468—28no—
21C4×SD16—24no—
22#436—24no—
23C2×#107C2212yes—
24C2×#107C2212yes—
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2212yes—
28C2×#103C2211yescharacteristic
29C2×#102C2212yes—
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2212yes—
32C2×C4×C8C2211yescharacteristic
33#396—22no—
34#394—22no—
35#390—21no—
36#389—21no—
37#384C2211yescharacteristic
38#384C2211yescharacteristic
39#380—21no—
40#380C2212yes—
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44C22×Q8C2311yescharacteristic
45C22×D4—22no—
46C22×D4D411yescharacteristic
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C23×C4C2312yes—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×SD16D414yescomplemented
53C2×M16—24no—
54C2×M16—24no—
55C2×M16—24no—
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C22⋊Q8—24no—
60C22⋊Q8—42no—
61#121—24no—
62#121—42no—
63#121—44no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×D4—22no—
67C4×D4—22no—
68C4×D4—22no—
69C4×D4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2312yes—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4D412yes—
87C2×C42—21no—
88C2×C42C2311yescharacteristic
89#107—24no—
90#107—42no—
91#106—22no—
92C4⋊C8—22no—
93#103—24no—
94#103—22no—
95#102—24no—
96#102—22no—
97C22⋊C8—24no—
98C4×C8D414yescomplemented
99#95—22no—
100#95—22no—
101C24—21no—
102C24C2×D412yes—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8C2×D412yes—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—82no—
120C2×D4C2×D412yes—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4C2×D412yes—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2411yescharacteristic
139SD16—28no—
140M16—48no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8C2×D412yes—
147C2×C8C2×D412yes—
148C2×C8C2×D412yes—
149C2×C8#4112yes—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C22⋊C4—24no—
166C22⋊C4—44no—
167C22⋊C4—44no—
168C22⋊C4—44no—
169C22⋊C4—42no—
170C22⋊C4—82no—
171C42—42no—
172C42C2×D412yes—
173C42#4112yes—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23C22×D411yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—41no—
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—44no—
198D4—88no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—84no—
224C2×C4#12112yes—
225C2×C4#12112yes—
226C2×C4AGL(1,ℤ/8)12yes—
227C2×C4#13712yes—
228C2×C4C22×D411yescharacteristic
229C2×C4C22×D411yescharacteristic
230C2×C4C2×#4111yescharacteristic
231C2×C4C2×#4111yescharacteristic
232C2×C4H(5,𝔽2)11yescharacteristic
233C2×C42−1+411yescharacteristic
234C8—22no—
235C8—42no—
236C8#12114yescomplemented
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22C2×#12111yescharacteristic
252C22D4211yescharacteristic
253C22#54711yescharacteristic
254C22C2×AGL(1,ℤ/8)11yescharacteristic
255C22C2×#13711yescharacteristic
256C22#57611yescharacteristic
257C22#57811yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—82no—
262C4—82no—
263C4—81no—
264C4—81no—
265C4#46814yescomplemented
266C4C2×#12111yescharacteristic
267C4#53711yescharacteristic
268C4#54711yescharacteristic
269C4#55511yescharacteristic
270C2—82no—
271C2—82no—
272C2#253711yescharacteristic
273C2C2×#46811yescharacteristic
274C2#306111yescharacteristic
275C2#327411yescharacteristic
276C2#327611yescharacteristic
277C2#335311yescharacteristic
278C2#335711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#468C2semidirect
2. C2×C4×SD16C2semidirect
3. C2×#436C2semidirect
4. #2538C2semidirect
6. #2036C2semidirect
7. #2033C2semidirect
8. #1845C2semidirect
9. #1809C2semidirect
10. #1803C2semidirect
11. #1802C2semidirect
52. C2×SD16D4semidirect
98. C4×C8D4semidirect
236. C8#121semidirect
265. C4#468semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×C4×SD162
C2×#4362
#25382
#25372
#20362
#20332
#18452
#18092
#18032
#18022