Finite Group Databasebeta

← #30869

Subgroups — #30869

Order 256.

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

normal subgroups only

This group has 459 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
1C2×#471C212yesmaximal, complemented
2#3062C212yesmaximal, complemented
3#3043C212yesmaximal, complemented
4#2063C212yesmaximal, complemented
5#2014C211yesmaximal, characteristic, complemented
6#2007C211yesmaximal, characteristic, complemented
7#2002C212yesmaximal, complemented
8#1866C212yesmaximal, complemented
9#1857C211yesmaximal, characteristic, complemented
10C2×#137C2212yes—
11C2×AGL(1,ℤ/8)C2212yes—
12C22×SD16C2212yes—
13C2×#131C2212yes—
14#537C2211yescharacteristic
15#535C2212yes—
16C2×#127C2211yescharacteristic
17#519C2211yescharacteristic
18#471—28no—
19#469—22no—
20#468—22no—
21#466—22no—
22#465—22no—
23#464—22no—
24#464—22no—
25#460—22no—
26#460—22no—
27C2×#108C2212yes—
28#428C2212yes—
29#424C2212yes—
30C2×#104C2212yes—
31#418C2212yes—
32#416C2212yes—
33C2×#102C2212yes—
34C2×#101C2211yescharacteristic
35C2×#100C2212yes—
36#391C2211yescharacteristic
37#343C2211yescharacteristic
38#339C2212yes—
39#336C2211yescharacteristic
40C2×#41C2312yes—
41C22×Q8C2311yescharacteristic
42C22×D4—41no—
43C22×D4C2312yes—
44#137—24no—
45#137—42no—
46AGL(1,ℤ/8)—24no—
47AGL(1,ℤ/8)—42no—
48C2×Q16—22no—
49C2×SD16—24no—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×SD16—22no—
53C2×SD16—22no—
54C2×SD16—42no—
55C2×D8—22no—
56#131—24no—
57#131—42no—
58C2×M16—22no—
59C2×M16C2312yes—
60C2×M16C2312yes—
61C22×C8—22no—
62C22×C8C2312yes—
63#127—22no—
64#127—42no—
65#124—22no—
66C22⋊Q8—22no—
67#121—22no—
68#121—22no—
69#121—22no—
70#121—41no—
71#120—42no—
72C4×Q8—22no—
73C4×Q8—22no—
74#117—41no—
75#117C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#108—24no—
79#107—22no—
80#106—22no—
81C4⋊C8—22no—
82C4⋊C8—22no—
83#104—42no—
84#103D412yescomplemented
85#103D412yescomplemented
86#102—22no—
87#102—22no—
88#102—22no—
89#102—22no—
90#102D412yescomplemented
91#102D412yescomplemented
92#101D414yescomplemented
93#100—24no—
94#95—41no—
95C24—22no—
96#41—22no—
97#41—22no—
98#41—22no—
99#41—22no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8C2×D412yes—
104C2×Q8C2×D412yes—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—44no—
115C2×D4—41no—
116C2×D4—82no—
117C2×D4—82no—
118C2×D4C2×D412yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4C2411yescharacteristic
124Q16—44no—
125SD16—44no—
126SD16—44no—
127SD16—44no—
128SD16—44no—
129D8—44no—
130M16—22no—
131M16—22no—
132M16—22no—
133M16—44no—
134M16C2×D414yes—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—44no—
140C2×C8C2×D412yes—
141C2×C8C2×D412yes—
142C4⋊C4—22no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4C2×D412yes—
147C4⋊C4C2×D412yes—
148C22⋊C4—22no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—42no—
155C23—22no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—82no—
160C23—82no—
161C23H(5,𝔽2)11yescharacteristic
162Q8—22no—
163Q8—24no—
164Q8—22no—
165Q8—42no—
166D4—22no—
167D4—22no—
168D4—42no—
169D4—42no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—42no—
174D4—88no—
175D4—162no—
176D4—162no—
177D4—161no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4C22×D412yes—
192C2×C4C22×D411yescharacteristic
193C2×C4H(5,𝔽2)12yes—
194C2×C42−1+411yescharacteristic
195C8—22no—
196C8—22no—
197C8—24no—
198C8—42no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—84no—
205C22—82no—
206C22—82no—
207C22—82no—
208C22—162no—
209C22#53711yescharacteristic
210C22D4211yescharacteristic
211C22#56011yescharacteristic
212C4—42no—
213C4—42no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—81no—
218C4#53712yes—
219C4D4212yes—
220C2—21no—
221C2—21no—
222C2—82no—
223C2—162no—
224C2#267511yescharacteristic
225C2#328811yescharacteristic
226C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#471C2semidirect
2. #3062C2semidirect
3. #3043C2semidirect
4. #2063C2semidirect
5. #2014C2semidirect
6. #2007C2semidirect
7. #2002C2semidirect
8. #1866C2semidirect
9. #1857C2semidirect
84. #103D4semidirect
85. #103D4semidirect
90. #102D4semidirect
91. #102D4semidirect
92. #101D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4712
#30622
#30432
#20632
#20142
#20072
#20022
#18662
#18572