Finite Group Databasebeta

← #29854

Subgroups — #29854

Order 256.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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×#489C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic, complemented
3#2857C211yesmaximal, characteristic, complemented
4#2850C211yesmaximal, characteristic
5#2067C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#2036C212yesmaximal, complemented
8#2034C212yesmaximal
9#1993C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1847C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#489—24no—
19#432—24no—
20C2×#107C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2212yes—
28C2×#97C2211yescharacteristic
29C2×#97C2211yescharacteristic
30#391—21no—
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#389—21no—
34#389C2212yes—
35#388—21no—
36#388—21no—
37#386—21no—
38#386—21no—
39#384C2211yescharacteristic
40#382C2211yescharacteristic
41#380—21no—
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56#124—22no—
57#124—22no—
58C22⋊Q8—42no—
59C22⋊Q8—44no—
60#121—42no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#107—42no—
84#106—42no—
85C4⋊C8—24no—
86#103—24no—
87#103—42no—
88#102—24no—
89#102—42no—
90C22⋊C8—24no—
91#97—22no—
92#97D414yescomplemented
93#95—22no—
94#95—22no—
95#95—21no—
96#95—21no—
97#95—21no—
98#95—21no—
99C24—21no—
100C24C2×D411yescharacteristic
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—42no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8C2×D414yes—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—44no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C42—22no—
161C42—22no—
162C42—21no—
163C42—21no—
164C42C2×D412yes—
165C42#4112yes—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23#13712yes—
180C23C22×D411yescharacteristic
181Q8—42no—
182Q8—42no—
183Q8—84no—
184D4—42no—
185D4—42no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4#12112yes—
219C2×C4#12112yes—
220C2×C4AGL(1,ℤ/8)12yes—
221C2×C4C22×D411yescharacteristic
222C2×C4C2×#4111yescharacteristic
223C2×C4H(5,𝔽2)12yes—
224C2×C42−1+412yes—
225C8—24no—
226C8—42no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22C2×#12111yescharacteristic
242C22#53511yescharacteristic
243C22#53611yescharacteristic
244C22C2×AGL(1,ℤ/8)11yescharacteristic
245C22C2×#13711yescharacteristic
246C22#57511yescharacteristic
247C22#57511yescharacteristic
248C4—22no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—41no—
253C4—81no—
254C4—81no—
255C4—82no—
256C4—81no—
257C4—82no—
258C4#53712yes—
259C4#54212yes—
260C2—42no—
261C2—161no—
262C2#262011yescharacteristic
263C2#305511yescharacteristic
264C2#305611yescharacteristic
265C2#321411yescharacteristic
266C2#321511yescharacteristic
267C2#323911yescharacteristic
268C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#489C2semidirect
2. C2×#432C2semidirect
3. #2857C2semidirect
5. #2067C2semidirect
6. #2066C2semidirect
7. #2036C2semidirect
9. #1993C2semidirect
10. #1959C2semidirect
11. #1847C2semidirect
92. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4322
#28572
#28502
#20672
#20662
#20362
#20342
#19932
#19592
#18472