Finite Group Databasebeta

← #28105

Subgroups — #28105

Order 256.

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

normal subgroups only

This group has 502 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×#435C211yesmaximal, characteristic
2C22×#103C211yesmaximal, characteristic, complemented
3#2486C211yesmaximal, characteristic
4#2409C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2036C212yesmaximal, complemented
7#1935C212yesmaximal, complemented
8#1894C212yesmaximal, complemented
9#1823C211yesmaximal, characteristic, complemented
10#1808C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×#123—21no—
16C2×C22⋊Q8C2212yes—
17C2×C4×D4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19#435—24no—
20C2×#107C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103—24no—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×#97C2211yescharacteristic
30#398—21no—
31#397—21no—
32#395—21no—
33#393—21no—
34#391C2212yes—
35#389C2212yes—
36#386—21no—
37#385—21no—
38#381—21no—
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8—24no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—42no—
50C22×Q8C2312yes—
51C22×D4D411yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C2×M16—22no—
57C2×M16—42no—
58C22×C8—42no—
59C22×C8D414yes—
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64#123—44no—
65C22⋊Q8—24no—
66C22⋊Q8—42no—
67C4×D4—22no—
68C4×D4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2312yes—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4D411yescharacteristic
83C2×C22⋊C4D411yescharacteristic
84C2×C42C2311yescharacteristic
85#107—42no—
86C4⋊C8—22no—
87#103—24no—
88#103—24no—
89#103—24no—
90#103—42no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93#97—22no—
94#95—22no—
95#95—21no—
96#95—21no—
97#95—22no—
98#95—22no—
99#95—22no—
100C24C2×D411yescharacteristic
101C24C2×D411yescharacteristic
102C2×Q8—22no—
103C2×Q8—28no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—44no—
109C2×Q8—48no—
110C2×Q8—44no—
111C2×Q8—42no—
112C2×Q8#4114yes—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—44no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4#4112yes—
137C22×C4C2411yescharacteristic
138M16—44no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—44no—
146C2×C8C2×D412yes—
147C2×C8C2×D412yes—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—42no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C42—21no—
165C42—21no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—22no—
173C23#12112yes—
174C23AGL(1,ℤ/8)12yes—
175C23#13712yes—
176C23C22×D411yescharacteristic
177Q8—24no—
178Q8—24no—
179Q8—48no—
180Q8—42no—
181Q8—42no—
182D4—44no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—48no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—84no—
211C2×C4#12112yes—
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4112yes—
214C2×C4C2×#4111yescharacteristic
215C2×C4H(5,𝔽2)12yes—
216C8—22no—
217C8—22no—
218C8—41no—
219C8—41no—
220C22—22no—
221C22—24no—
222C22—24no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22—41no—
229C22#46814yes—
230C22C2×#12111yescharacteristic
231C22#54612yes—
232C22C2×AGL(1,ℤ/8)11yescharacteristic
233C22C2×#13711yescharacteristic
234C22#57812yes—
235C4—22no—
236C4—41no—
237C4—44no—
238C4—41no—
239C4—41no—
240C4—82no—
241C4—82no—
242C4—82no—
243C4#52511yescharacteristic
244C4#53411yescharacteristic
245C4#55112yes—
246C2—24no—
247C2—81no—
248C2#253411yescharacteristic
249C2C2×#46811yescharacteristic
250C2#305911yescharacteristic
251C2#330812yes—
252C2#331212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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
2. C22×#103C2semidirect
5. #2068C2semidirect
6. #2036C2semidirect
7. #1935C2semidirect
8. #1894C2semidirect
9. #1823C2semidirect
10. #1808C2semidirect
11. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C22×#1032
#24862
#24092
#20682
#20362
#19352
#18942
#18232
#18082
#17872