Finite Group Databasebeta

← #28203

Subgroups — #28203

Order 256.

420 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 133 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×#475C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic
3C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4C4×#124C211yesmaximal, characteristic
5#2085C212yesmaximal, complemented
6#2042C212yesmaximal
7#1928C212yesmaximal, complemented
8#1926C212yesmaximal
9#1831C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1802C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8—21no—
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#475—24no—
18#474—24no—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27#391—21no—
28#391C2211yescharacteristic
29#390C2211yescharacteristic
30#389—21no—
31#389C2212yes—
32#388—21no—
33#388—21no—
34#384C2211yescharacteristic
35#382C2211yescharacteristic
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2212yes—
38C4×C22⋊C4—21no—
39C4×C22⋊C4—21no—
40C43C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8D414yes—
53#124—22no—
54#124—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—44no—
58C4×D4—22no—
59C4×D4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—22no—
75C2×C42—21no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#107—24no—
84#106—24no—
85C4⋊C8—22no—
86C4⋊C8—22no—
87#103—24no—
88#103—24no—
89#102—24no—
90#102—24no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—22no—
96#95—22no—
97C24—21no—
98C2×Q8—24no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8#4112yes—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4#4112yes—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4C2×D414yes—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—41no—
160C42—44no—
161C42—42no—
162C42C2×D412yes—
163C42#4112yes—
164C23—22no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22×D411yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—48no—
177D4—22no—
178D4—22no—
179D4—41no—
180D4—41no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—84no—
210C2×C4#12112yes—
211C2×C4#12112yes—
212C2×C4#13512yes—
213C2×C4AGL(1,ℤ/8)12yes—
214C2×C4#13712yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C2×#4111yescharacteristic
217C2×C4C2×#4111yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C42−1+411yescharacteristic
220C2×C42−1+411yescharacteristic
221C8—42no—
222C8—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22C2×#12111yescharacteristic
231C22#52511yescharacteristic
232C22#53611yescharacteristic
233C22C2×#13511yescharacteristic
234C22C2×AGL(1,ℤ/8)11yescharacteristic
235C22C2×#13711yescharacteristic
236C22#57511yescharacteristic
237C4—22no—
238C4—22no—
239C4—41no—
240C4—41no—
241C4—41no—
242C4—44no—
243C4—42no—
244C4—82no—
245C4—81no—
246C4—81no—
247C4#46314yes—
248C4#54811yescharacteristic
249C4#54811yescharacteristic
250C4#55211yescharacteristic
251C4#55211yescharacteristic
252C2—82no—
253C2#253611yescharacteristic
254C2C2×#46311yescharacteristic
255C2#303711yescharacteristic
256C2#310711yescharacteristic
257C2#311511yescharacteristic
258C2#323311yescharacteristic
259C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#475C2semidirect
3. C4×C4⋊Q8C2semidirect
5. #2085C2semidirect
7. #1928C2semidirect
9. #1831C2semidirect
10. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4742
C4×C4⋊Q82
C4×#1242
#20852
#20422
#19282
#19262
#18312
#18032
#18022