Finite Group Databasebeta

← #33019

Subgroups — #33019

Order 256.

608 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 143 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×#519C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic
4#2625C211yesmaximal, characteristic
5#1878C212yesmaximal, complemented
6#1878C212yesmaximal, complemented
7#1865C212yesmaximal, complemented
8#1863C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic
10#1818C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C2×C22⋊Q8C2211yescharacteristic
15#519—24no—
16#519—24no—
17C2×C4×Q8—21no—
18C2×C4×Q8—21no—
19C2×C4×Q8C2212yes—
20C2×C4×Q8C2212yes—
21C2×#117—22no—
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#485—24no—
25#451—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#400—21no—
36#398—21no—
37#397—22no—
38#391C2212yes—
39#389C2212yes—
40#380C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—24no—
46C22×Q8—24no—
47C22×Q8—21no—
48C22×Q8—22no—
49C22×Q8D414yes—
50C22×Q8C2311yescharacteristic
51C22×Q8C2312yes—
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C2×Q16—24no—
55C2×Q16—24no—
56C2×Q16—24no—
57C22×C8C2312yes—
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C4×Q8—28no—
63C4×Q8—28no—
64C4×Q8—24no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8—24no—
68#117—216no—
69#117—22no—
70#117—22no—
71#117—22no—
72#117D414yes—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4D412yes—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42C2312yes—
91#107—22no—
92#106—22no—
93#103—24no—
94#103—24no—
95#103—24no—
96#103—22no—
97#103—22no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100#95—21no—
101#95—22no—
102#95—22no—
103#95—22no—
104C24C2×D411yescharacteristic
105C2×Q8—28no—
106C2×Q8—28no—
107C2×Q8—28no—
108C2×Q8—216no—
109C2×Q8—24no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8C2×D414yes—
118C2×Q8#4114yes—
119C2×Q8#4114yes—
120C22×C4—28no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C2×D412yes—
131C22×C4C2×D412yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2411yescharacteristic
136Q16—48no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C4⋊C4—28no—
145C4⋊C4—28no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—82no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—42no—
161C22⋊C4—81no—
162C42—28no—
163C42—28no—
164C42—24no—
165C42—22no—
166C42—22no—
167C23—22no—
168C23—21no—
169C23—22no—
170C23—21no—
171C23#12012yes—
172C23C22×D411yescharacteristic
173Q8—216no—
174Q8—24no—
175Q8—24no—
176Q8—24no—
177Q8—42no—
178Q8—41no—
179Q8—41no—
180Q8#12318yes—
181C2×C4—216no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4#12012yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C2×#4112yes—
211C2×C4C2×#4112yes—
212C8—42no—
213C8—41no—
214C8—41no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—42no—
219C22—42no—
220C22C2×#12011yescharacteristic
221C22#52512yes—
222C22#57711yescharacteristic
223C22#57812yes—
224C22#57811yescharacteristic
225C4—28no—
226C4—22no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—81no—
232C4—82no—
233C4—81no—
234C4C2×#12312yes—
235C4#52511yescharacteristic
236C4#52511yescharacteristic
237C2—42no—
238C2#284911yescharacteristic
239C2#301111yescharacteristic
240C2#301311yescharacteristic
241C2#312112yes—
242C2#312712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #1878C2semidirect
6. #1878C2semidirect
7. #1865C2semidirect
8. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5192
C2×#4852
C2×#4512
#26252
#18782
#18652
#18632
#18252
#18182
#17872