Finite Group Databasebeta

← #27648

Subgroups — #27648

Order 256.

463 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 151 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×#484C212yesmaximal
2C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#384C211yesmaximal, characteristic
4#2083C212yesmaximal, complemented
5#2072C212yesmaximal, complemented
6#1947C212yesmaximal, complemented
7#1824C212yesmaximal
8#1803C212yesmaximal, complemented
9#1797C211yesmaximal, characteristic
10C2×C22⋊Q8C2211yescharacteristic
11C2×C4×Q8C2211yescharacteristic
12C22×C4⋊C4—21no—
13C22×C4⋊C4—21no—
14C22×C4⋊C4C2212yes—
15C22×C42C2211yescharacteristic
16#484—28no—
17C2×#107C2214yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2214yes—
20C2×C22⋊C8C2212yes—
21#391C2211yescharacteristic
22#389C2211yescharacteristic
23#387—21no—
24#385—21no—
25#384—21no—
26#384—21no—
27#384C2212yes—
28#384C2212yes—
29#380C2211yescharacteristic
30C4×C4⋊C4—21no—
31C4×C4⋊C4C2211yescharacteristic
32C4×C22⋊C4—21no—
33#340C2214yes—
34#340C2214yes—
35#336C2212yes—
36C22×Q8C2311yescharacteristic
37C23×C4—22no—
38C23×C4D411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4C2311yescharacteristic
41C22×C8C2314yes—
42C22⋊Q8—22no—
43C22⋊Q8D414yes—
44C4×Q8—24no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—24no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—44no—
56C2×C4⋊C4—44no—
57C2×C4⋊C4C2314yes—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C42—24no—
64C2×C42—21no—
65C2×C42—21no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#107—28no—
71C4⋊C8—24no—
72#103—28no—
73C22⋊C8—24no—
74#95—21no—
75#95—21no—
76#95—21no—
77#95—21no—
78#95—21no—
79C24C2×D411yescharacteristic
80C2×Q8—21no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8C2×D412yes—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—24no—
96C22×C4—44no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4Q1614yes—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D412yes—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4#4112yes—
107C22×C4C2411yescharacteristic
108C2×C8—24no—
109C2×C8—24no—
110C2×C8—24no—
111C4⋊C4—24no—
112C4⋊C4—21no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—44no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—44no—
124C4⋊C4C2×D412yes—
125C22⋊C4—24no—
126C22⋊C4—42no—
127C42—22no—
128C42—22no—
129C42—22no—
130C42—42no—
131C42—42no—
132C42#4114yes—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23#12112yes—
138C23C2×Q1612yes—
139C23#13712yes—
140C23C22×D411yescharacteristic
141Q8—24no—
142Q8—42no—
143C2×C4—28no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—28no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—44no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—88no—
171C2×C4#12112yes—
172C2×C4#12312yes—
173C2×C4#12312yes—
174C2×C4C2×Q1612yes—
175C2×C4C2×Q1612yes—
176C2×C4AGL(1,ℤ/8)12yes—
177C2×C4AGL(1,ℤ/8)12yes—
178C2×C4C22×D411yescharacteristic
179C2×C4C2×#4112yes—
180C2×C4C2×#4111yescharacteristic
181C2×C42−1+412yes—
182C8—44no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22#46214yes—
190C22C2×#12111yescharacteristic
191C22C2×#12311yescharacteristic
192C22#53611yescharacteristic
193C22C22×Q1611yescharacteristic
194C22C2×AGL(1,ℤ/8)11yescharacteristic
195C22C2×AGL(1,ℤ/8)11yescharacteristic
196C22C2×#13711yescharacteristic
197C4—22no—
198C4—22no—
199C4—22no—
200C4—22no—
201C4—42no—
202C4—42no—
203C4—81no—
204C4—81no—
205C4—84no—
206C4#48414yescomplemented
207C4#54712yes—
208C4#55212yes—
209C2—24no—
210C2#248611yescharacteristic
211C2C2×#46211yescharacteristic
212C2#303811yescharacteristic
213C2C2×#48411yescharacteristic
214C2#309211yescharacteristic
215C2#323011yescharacteristic
216C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
4. #2083C2semidirect
5. #2072C2semidirect
6. #1947C2semidirect
8. #1803C2semidirect
206. C4#484semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C4×C22⋊Q82
C2×#3842
#20832
#20722
#19472
#18242
#18032
#17972