Finite Group Databasebeta

← #33100

Subgroups — #33100

Order 256.

429 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 131 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#2855C211yesmaximal, characteristic
2C2×C4×C4⋊C4C211yesmaximal, characteristic
3#1928C212yesmaximal, complemented
4#1927C212yesmaximal, complemented
5#1919C212yesmaximal, complemented
6#1918C212yesmaximal, complemented
7#1872C211yesmaximal, characteristic
8#1872C211yesmaximal, characteristic
9#1853C211yesmaximal, characteristic
10#1852C211yesmaximal, characteristic
11#1817C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C42—21no—
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23#391C2211yescharacteristic
24#391C2211yescharacteristic
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#387—21no—
28#387—21no—
29#385—21no—
30#381—21no—
31#380C2211yescharacteristic
32C4×C4⋊C4—21no—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2212yes—
35C4×C4⋊C4C2212yes—
36C2×#95—21no—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42—22no—
71C2×C42—44no—
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—24no—
77#106—24no—
78#103—24no—
79#103—24no—
80C22⋊C8—22no—
81C22⋊C8—22no—
82#95—22no—
83#95—22no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91C24C2×D411yescharacteristic
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4#4112yes—
122C22×C4#4112yes—
123C22×C4C2411yescharacteristic
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4#4114yes—
142C4⋊C4#4114yes—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C42—22no—
146C42—22no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—44no—
152C42—44no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23#12312yes—
158C23#13512yes—
159C23#13712yes—
160C23C22×D411yescharacteristic
161Q8—42no—
162Q8—42no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—48no—
181C2×C4—48no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4#12312yes—
200C2×C4AGL(1,ℤ/8)12yes—
201C2×C4#13712yes—
202C2×C4C2×#4111yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4C2×#4112yes—
205C2×C4C2×#4112yes—
206C8—42no—
207C8—42no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22#48514yes—
215C22C2×#12311yescharacteristic
216C22#52511yescharacteristic
217C22#52511yescharacteristic
218C22C2×#13511yescharacteristic
219C22C2×AGL(1,ℤ/8)11yescharacteristic
220C22C2×#13711yescharacteristic
221C22C2×#13711yescharacteristic
222C4—22no—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—44no—
228C4—44no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4—81no—
233C4#52912yes—
234C4#52912yes—
235C2—24no—
236C2#285311yescharacteristic
237C2C2×#48511yescharacteristic
238C2#309211yescharacteristic
239C2#310511yescharacteristic
240C2#310811yescharacteristic
241C2#311011yescharacteristic
242C2#311311yescharacteristic
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
3. #1928C2semidirect
4. #1927C2semidirect
5. #1919C2semidirect
6. #1918C2semidirect

Maximal subgroups

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

subgroupindex
#28552
C2×C4×C4⋊C42
#19282
#19272
#19192
#19182
#18722
#18532
#18522
#18172