Finite Group Databasebeta

← #33186

Subgroups — #33186

Order 256.

558 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 159 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×#497C211yesmaximal, characteristic, complemented
2C2×#470C212yesmaximal, complemented
3C2×#468C212yesmaximal
4C2×C4×M16C211yesmaximal, characteristic, complemented
5#2857C211yesmaximal, characteristic, complemented
6#2850C211yesmaximal, characteristic
7#1969C212yesmaximal, complemented
8#1968C212yesmaximal, complemented
9#1960C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1847C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C22×M16C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2212yes—
19C2×#121C2211yescharacteristic
20C22×C42C2211yescharacteristic
21#497—24no—
22#470—28no—
23#468—28no—
24C2×#107C2212yes—
25C2×#106C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2212yes—
30C4×M16—24no—
31C2×#97C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2212yes—
34#391C2211yescharacteristic
35#391C2212yes—
36#390C2211yescharacteristic
37#389—22no—
38#388—22no—
39#386—22no—
40#384C2211yescharacteristic
41#382C2211yescharacteristic
42#380—22no—
43C22×Q8C2311yescharacteristic
44C22×Q8C2312yes—
45C22×D4—21no—
46C22×D4C2311yescharacteristic
47C23×C4D412yes—
48C23×C4C2311yescharacteristic
49C2×Q16—24no—
50C2×Q16—24no—
51C2×Q16—24no—
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16—24no—
55C2×M16—24no—
56C2×M16—24no—
57C2×M16D418yes—
58C22×C8C2312yes—
59C22×C8C2312yes—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62#124—22no—
63#124—22no—
64C22⋊Q8—24no—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67#121—24no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42D412yes—
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85#107—24no—
86#106—24no—
87#103—24no—
88#103—24no—
89#103—24no—
90#102—24no—
91#97—22no—
92#97D414yescomplemented
93C4×C8—24no—
94#95—24no—
95#95—22no—
96#95—22no—
97C24—21no—
98C24C2×D411yescharacteristic
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×D4—22no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—44no—
114C2×D4—42no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D414yes—
129C22×C4C2×D412yes—
130C22×C4C2411yescharacteristic
131Q16—48no—
132SD16—48no—
133M16—216no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8C2×D414yes—
137C2×C8C2×D414yes—
138C2×C8C2×D414yes—
139C2×C8#4114yes—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—42no—
147C4⋊C4—44no—
148C22⋊C4—44no—
149C22⋊C4—42no—
150C22⋊C4—44no—
151C22⋊C4—42no—
152C22⋊C4—44no—
153C42—24no—
154C42—21no—
155C42—21no—
156C42C2×D412yes—
157C42#4112yes—
158C23—21no—
159C23—21no—
160C23—22no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—42no—
166C23—42no—
167C23#12712yes—
168C23C22×D411yescharacteristic
169Q8—42no—
170Q8—42no—
171Q8—42no—
172Q8—42no—
173Q8—42no—
174Q8—42no—
175D4—42no—
176D4—42no—
177D4—84no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—48no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4#12114yes—
200C2×C4#12712yes—
201C2×C4#13714yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C22×D412yes—
204C2×C4C2×#4112yes—
205C2×C4C2×#4111yescharacteristic
206C8—24no—
207C8#12118yes—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—82no—
217C22—82no—
218C22C2×#12111yescharacteristic
219C22C2×#12711yescharacteristic
220C22#53211yescharacteristic
221C22C2×#13712yes—
222C22#57512yes—
223C4—22no—
224C4—24no—
225C4—24no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4—82no—
230C4—81no—
231C4C2×#12112yes—
232C4#53212yes—
233C2—42no—
234C2—161no—
235C2#285611yescharacteristic
236C2#305612yes—
237C2#315211yescharacteristic
238C2#315311yescharacteristic
239C2#317212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
1. C2×#497C2semidirect
2. C2×#470C2semidirect
4. C2×C4×M16C2semidirect
5. #2857C2semidirect
7. #1969C2semidirect
8. #1968C2semidirect
9. #1960C2semidirect
10. #1959C2semidirect
11. #1847C2semidirect
92. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4702
C2×#4682
C2×C4×M162
#28572
#28502
#19692
#19682
#19602
#19592
#18472