Finite Group Databasebeta

← #28138

Subgroups — #28138

Order 256.

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

normal subgroups only

This group has 430 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#3093C211yesmaximal, characteristic
2#3004C211yesmaximal, characteristic
3#2930C211yesmaximal, characteristic
4C2×#419C211yesmaximal, characteristic, complemented
5#2064C212yesmaximal, complemented
6#2052C212yesmaximal, complemented
7#1930C212yesmaximal, complemented
8#1908C212yesmaximal, complemented
9#1822C211yesmaximal, characteristic, complemented
10#1811C211yesmaximal, characteristic, complemented
11#1784C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C2×#137C2212yes—
14C2×#131C2211yescharacteristic
15C22×M16C2211yescharacteristic
16#525C2211yescharacteristic
17C2×#117C2211yescharacteristic
18#485—22no—
19#482—22no—
20#451—22no—
21#448—22no—
22#436—22no—
23#435—22no—
24C2×#108C2212yes—
25#428C2212yes—
26#424C2211yescharacteristic
27#419—22no—
28#419—22no—
29#419C2212yes—
30#417C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#101C2212yes—
34C2×#101C2212yes—
35#408C2211yescharacteristic
36#407C2211yescharacteristic
37#407C2211yescharacteristic
38#407C2211yescharacteristic
39#405C2211yescharacteristic
40#343C2212yes—
41#343C2212yes—
42#341C2212yes—
43C2×#41C2311yescharacteristic
44C22×Q8—21no—
45C22×Q8—24no—
46C22×Q8—22no—
47C22×Q8—22no—
48C22×Q8—42no—
49C22×Q8C2312yes—
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52#137—24no—
53#137—42no—
54C2×Q16—22no—
55C2×SD16—22no—
56#131—42no—
57C2×M16—21no—
58C2×M16—42no—
59C2×M16D412yes—
60C2×M16D412yes—
61C2×M16C2311yescharacteristic
62C2×M16C2312yes—
63C2×M16C2312yes—
64C2×M16C2311yescharacteristic
65C22×C8—21no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#123—21no—
69C22⋊Q8D412yes—
70#121D412yes—
71#120—21no—
72C4×D4—22no—
73#117—22no—
74#117—22no—
75#117C2311yescharacteristic
76C2×C4⋊C4C2312yes—
77C2×C22⋊C4—21no—
78C2×C42—21no—
79#108—42no—
80#107—22no—
81#106—22no—
82C4⋊C8—22no—
83#103—22no—
84#103—22no—
85#103—22no—
86#103—24no—
87#103—24no—
88#102—22no—
89#101—24no—
90#101—42no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93C22⋊C8—22no—
94C22⋊C8—22no—
95#97—21no—
96C4×C8—21no—
97C24#4111yescharacteristic
98#41—22no—
99#41—22no—
100C2×Q8—22no—
101C2×Q8—28no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—44no—
107C2×Q8—48no—
108C2×Q8—44no—
109C2×Q8—42no—
110C2×Q8C2×D411yescharacteristic
111C2×Q8#4114yes—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4C2×D411yescharacteristic
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—44no—
121C22×C4—42no—
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4#4111yescharacteristic
125C22×C4C2411yescharacteristic
126Q16—44no—
127SD16—44no—
128M16—22no—
129M16—22no—
130M16—22no—
131M16—22no—
132M16—22no—
133M16—22no—
134M16—42no—
135M16—42no—
136M16—42no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—21no—
143C2×C8—42no—
144C2×C8C2×D411yescharacteristic
145C2×C8C2×D411yescharacteristic
146C2×C8C2×D412yes—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4C2×D412yes—
151C22⋊C4—22no—
152C22⋊C4—21no—
153C22⋊C4—22no—
154C22⋊C4—42no—
155C42—21no—
156C42—42no—
157C23—22no—
158C23—21no—
159C23—42no—
160C23#12112yes—
161C23C2×#4111yescharacteristic
162Q8—24no—
163Q8—24no—
164Q8—22no—
165Q8—48no—
166Q8—42no—
167Q8—42no—
168D4—22no—
169D4—42no—
170D4—82no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—48no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—82no—
185C2×C4#12112yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C22×D411yescharacteristic
188C2×C4C2×#4112yes—
189C2×C4H(5,𝔽2)12yes—
190C8—21no—
191C8—22no—
192C8—21no—
193C8—41no—
194C8—41no—
195C8—42no—
196C8—42no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—44no—
202C22—41no—
203C22—42no—
204C22C2×#12111yescharacteristic
205C22#55112yes—
206C4—22no—
207C4—44no—
208C4—41no—
209C4—42no—
210C4—82no—
211C4#52511yescharacteristic
212C4#53411yescharacteristic
213C4#54612yes—
214C2—22no—
215C2—42no—
216C2—81no—
217C2#253411yescharacteristic
218C2#306912yes—
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
4. C2×#419C2semidirect
5. #2064C2semidirect
6. #2052C2semidirect
7. #1930C2semidirect
8. #1908C2semidirect
9. #1822C2semidirect
10. #1811C2semidirect
11. #1784C2semidirect

Maximal subgroups

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

subgroupindex
#30932
#30042
#29302
C2×#4192
#20642
#20522
#19302
#19082
#18222
#18112
#17842