Finite Group Databasebeta

← #28591

Subgroups — #28591

Order 256.

397 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#3056C212yesmaximal
2C2×#467C211yesmaximal, characteristic, complemented
3#2929C211yesmaximal, characteristic
4#2089C211yesmaximal, characteristic, complemented
5#2078C211yesmaximal, characteristic, complemented
6#2047C211yesmaximal, characteristic, complemented
7#2020C211yesmaximal, characteristic, complemented
8#1930C212yesmaximal, complemented
9#1908C212yesmaximal, complemented
10#1901C212yesmaximal, complemented
11C2×#341C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C22×Q16C2211yescharacteristic
14C2×#131C2211yescharacteristic
15C22×M16C2212yes—
16C23×C8C2211yescharacteristic
17#525C2211yescharacteristic
18C2×C22⋊Q8C2212yes—
19#470—22no—
20#468—22no—
21#467—21no—
22#467—21no—
23#467—21no—
24#467—21no—
25#467—22no—
26#465—22no—
27#435—21no—
28#435—21no—
29C8×D4—21no—
30C8×D4—21no—
31#428C2212yes—
32C2×#107C2211yescharacteristic
33#424C2211yescharacteristic
34#419C2212yes—
35#417C2212yes—
36C2×#103C2212yes—
37C2×#101C2212yes—
38C2×#101C2212yes—
39#408C2211yescharacteristic
40#407C2211yescharacteristic
41#405C2211yescharacteristic
42#343C2212yes—
43#343C2212yes—
44#341C2211yescharacteristic
45#341C2211yescharacteristic
46#341C2212yes—
47C2×#41C2311yescharacteristic
48C22×Q8C2312yes—
49C23×C4C2311yescharacteristic
50#137—24no—
51#137—42no—
52C2×Q16—22no—
53C2×Q16—21no—
54C2×Q16—21no—
55C2×Q16—42no—
56C2×SD16—22no—
57#131—42no—
58C2×M16—22no—
59C2×M16—21no—
60C2×M16—24no—
61C2×M16—42no—
62C2×M16C2312yes—
63C2×M16C2311yescharacteristic
64C2×M16C2311yescharacteristic
65C2×M16C2312yes—
66C22×C8—22no—
67C22×C8—21no—
68C22×C8—42no—
69C22×C8D411yescharacteristic
70C22×C8D411yescharacteristic
71C22×C8D411yescharacteristic
72C22×C8D411yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#123—21no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8D411yescharacteristic
81C22⋊Q8D411yescharacteristic
82#121D411yescharacteristic
83#121D411yescharacteristic
84#120—21no—
85C4×D4—21no—
86C4×D4—21no—
87#117C2311yescharacteristic
88C2×C4⋊C4—22no—
89C2×C4⋊C4C2312yes—
90C2×C22⋊C4—22no—
91#107—21no—
92#107—21no—
93#107—22no—
94#106—22no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97#103—22no—
98#103—22no—
99#103—22no—
100#103—22no—
101#103—22no—
102#102—22no—
103#101—24no—
104#101—42no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109#97—21no—
110C4×C8—21no—
111C24#4111yescharacteristic
112#41—22no—
113#41—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8C2×D411yescharacteristic
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4C2×D411yescharacteristic
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4#4111yescharacteristic
135C22×C4C2411yescharacteristic
136Q16—44no—
137Q16—42no—
138Q16—42no—
139SD16—44no—
140M16—22no—
141M16—22no—
142M16—22no—
143M16—22no—
144M16—22no—
145M16—44no—
146M16—42no—
147M16#4114yes—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—44no—
159C2×C8—42no—
160C2×C8C2×D411yescharacteristic
161C2×C8C2×D411yescharacteristic
162C2×C8C2×D411yescharacteristic
163C2×C8C2×D411yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4C2×D411yescharacteristic
171C4⋊C4C2×D411yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C42—21no—
180C23—22no—
181C23—22no—
182C23—21no—
183C23—41no—
184C23#12111yescharacteristic
185C23#12111yescharacteristic
186C23C2×#4111yescharacteristic
187Q8—22no—
188Q8—42no—
189Q8—42no—
190Q8—42no—
191Q8—42no—
192D4—22no—
193D4—42no—
194D4—42no—
195D4—42no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4#12111yescharacteristic
214C2×C4#12111yescharacteristic
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C2×#4112yes—
218C2×C4H(5,𝔽2)12yes—
219C8—21no—
220C8—21no—
221C8—21no—
222C8—21no—
223C8—24no—
224C8—41no—
225C8—41no—
226C8—42no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22#47312yes—
239C22C2×#12111yescharacteristic
240C22#55312yes—
241C4—21no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—41no—
246C4—82no—
247C4—82no—
248C4#53211yescharacteristic
249C4#53411yescharacteristic
250C4#54612yes—
251C2—22no—
252C2—22no—
253C2—41no—
254C2—81no—
255C2#254311yescharacteristic
256C2C2×#47311yescharacteristic
257C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#467C2semidirect
4. #2089C2semidirect
5. #2078C2semidirect
6. #2047C2semidirect
7. #2020C2semidirect
8. #1930C2semidirect
9. #1908C2semidirect
10. #1901C2semidirect
11. C2×#341C2semidirect

Maximal subgroups

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

subgroupindex
#30562
C2×#4672
#29292
#20892
#20782
#20472
#20202
#19302
#19082
#19012
C2×#3412