Finite Group Databasebeta

← #28595

Subgroups — #28595

Order 256.

416 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 115 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#3055C212yesmaximal, complemented
2C2×#469C211yesmaximal, characteristic, complemented
3#2930C211yesmaximal, characteristic, complemented
4#2090C211yesmaximal, characteristic, complemented
5#2079C211yesmaximal, characteristic, complemented
6#2045C211yesmaximal, characteristic, complemented
7#2018C211yesmaximal, characteristic, complemented
8#1931C212yesmaximal, complemented
9#1907C212yesmaximal, complemented
10#1900C212yesmaximal, complemented
11#1738C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)C2212yes—
13C22×D8C2211yescharacteristic
14C2×#131C2211yescharacteristic
15C22×M16C2212yes—
16C22×M16C2211yescharacteristic
17#525C2211yescharacteristic
18C2×#121C2212yes—
19#469—22no—
20#469—22no—
21#469—22no—
22#468—22no—
23#466—22no—
24#465—22no—
25#436—22no—
26#435—22no—
27#428C2212yes—
28C2×#106C2211yescharacteristic
29#424C2211yescharacteristic
30#418C2212yes—
31#417C2212yes—
32C2×#102C2212yes—
33C2×#100C2212yes—
34C2×#100C2212yes—
35#408C2211yescharacteristic
36#407C2211yescharacteristic
37#405C2211yescharacteristic
38#343C2212yes—
39#343C2212yes—
40#341C2211yescharacteristic
41#341C2211yescharacteristic
42#341C2212yes—
43C2×#41C2311yescharacteristic
44C22×D4—22no—
45C22×D4C2312yes—
46C23×C4C2311yescharacteristic
47AGL(1,ℤ/8)—24no—
48AGL(1,ℤ/8)—42no—
49C2×SD16—22no—
50C2×D8—22no—
51C2×D8—22no—
52C2×D8—42no—
53#131—42no—
54C2×M16—22no—
55C2×M16—21no—
56C2×M16—24no—
57C2×M16—42no—
58C2×M16—42no—
59C2×M16D412yes—
60C2×M16D412yes—
61C2×M16C2312yes—
62C2×M16C2311yescharacteristic
63C2×M16C2311yescharacteristic
64C2×M16C2312yes—
65C22×C8—22no—
66C22×C8—21no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69#123—21no—
70C22⋊Q8D412yes—
71#121—22no—
72#121—22no—
73#121—22no—
74#121—22no—
75#121D412yes—
76#120—21no—
77C4×D4—22no—
78#117C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C22⋊C4—22no—
81#107—22no—
82#106—22no—
83#106—22no—
84C4⋊C8—22no—
85#103—22no—
86#102—22no—
87#102—22no—
88#102—22no—
89#102—22no—
90#102—22no—
91#100—24no—
92#100—42no—
93C22⋊C8—22no—
94C22⋊C8—22no—
95#97—21no—
96C4×C8—21no—
97C24—22no—
98C24#4111yescharacteristic
99#41—22no—
100#41—22no—
101C2×Q8C2×D411yescharacteristic
102C2×D4—21no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—84no—
113C2×D4C2×D411yescharacteristic
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4#4111yescharacteristic
121C22×C4C2411yescharacteristic
122SD16—44no—
123D8—44no—
124D8—44no—
125M16—22no—
126M16—22no—
127M16—22no—
128M16—22no—
129M16—22no—
130M16—42no—
131M16—42no—
132M16—44no—
133M16—42no—
134M16#4114yes—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8—44no—
142C2×C8—42no—
143C2×C8C2×D411yescharacteristic
144C2×C8C2×D411yescharacteristic
145C2×C8C2×D412yes—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4C2×D412yes—
150C22⋊C4—22no—
151C22⋊C4—21no—
152C22⋊C4—22no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C42—21no—
156C23—22no—
157C23—21no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—82no—
163C23—82no—
164C23#12112yes—
165C23C2×#4111yescharacteristic
166Q8—22no—
167D4—22no—
168D4—42no—
169D4—42no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—84no—
175D4—84no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4#12112yes—
188C2×C4C22×D411yescharacteristic
189C2×C4C22×D411yescharacteristic
190C2×C4C2×#4112yes—
191C2×C4H(5,𝔽2)12yes—
192C8—21no—
193C8—21no—
194C8—22no—
195C8—24no—
196C8—41no—
197C8—41no—
198C8—42no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—41no—
204C22—42no—
205C22—82no—
206C22—82no—
207C22—82no—
208C22—82no—
209C22—162no—
210C22—162no—
211C22C2×#12111yescharacteristic
212C22#55312yes—
213C4—22no—
214C4—41no—
215C4—42no—
216C4—82no—
217C4#53211yescharacteristic
218C4#53411yescharacteristic
219C4#54612yes—
220C2—22no—
221C2—42no—
222C2—81no—
223C2—162no—
224C2#254311yescharacteristic
225C2#306812yes—
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
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
1. #3055C2semidirect
2. C2×#469C2semidirect
3. #2930C2semidirect
4. #2090C2semidirect
5. #2079C2semidirect
6. #2045C2semidirect
7. #2018C2semidirect
8. #1931C2semidirect
9. #1907C2semidirect
10. #1900C2semidirect
11. #1738C2semidirect

Maximal subgroups

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

subgroupindex
#30552
C2×#4692
#29302
#20902
#20792
#20452
#20182
#19312
#19072
#19002
#17382