Finite Group Databasebeta

← #29578

Subgroups — #29578

Order 256.

366 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 117 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#2859C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic
5#2084C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic
7#1944C212yesmaximal, complemented
8#1936C212yesmaximal
9#1921C212yesmaximal, complemented
10#1864C212yesmaximal
11#1745C211yesmaximal, characteristic, complemented
12C2×#125—21no—
13C2×C4×D4C2211yescharacteristic
14C2×#117—21no—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×#102C2212yes—
18C2×#102C2212yes—
19C2×#97C2212yes—
20#390C2211yescharacteristic
21#390C2211yescharacteristic
22#389—21no—
23#389—21no—
24#389C2211yescharacteristic
25#389C2212yes—
26#389C2212yes—
27#389C2211yescharacteristic
28#388—21no—
29#388—21no—
30#384C2211yescharacteristic
31#384C2211yescharacteristic
32#382—21no—
33C4×C4⋊C4C2211yescharacteristic
34C4×C22⋊C4—21no—
35C43C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C22×C8C2312yes—
46C22×C8C2312yes—
47#125—44no—
48C4×D4—22no—
49C4×D4—22no—
50#117—44no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—24no—
78#106—24no—
79#102—24no—
80#102—24no—
81#97—24no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—22no—
86#95—22no—
87C24—21no—
88C2×D4—22no—
89C2×D4—21no—
90C2×D4—22no—
91C2×D4#4112yes—
92C22×C4—24no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2411yescharacteristic
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8#4114yes—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—44no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4#4112yes—
129C22⋊C4—24no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C42—22no—
135C42—22no—
136C42—22no—
137C42—22no—
138C42—21no—
139C42—21no—
140C42—22no—
141C42—41no—
142C42—42no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23C22×D411yescharacteristic
151D4—24no—
152D4—42no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—48no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4AGL(1,ℤ/8)12yes—
185C2×C4AGL(1,ℤ/8)12yes—
186C2×C4#13712yes—
187C2×C4C2×#4111yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C2×C4C2×#4112yes—
190C2×C42−1+412yes—
191C8—24no—
192C8—42no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22#52511yescharacteristic
201C22#53211yescharacteristic
202C22#53611yescharacteristic
203C22C2×AGL(1,ℤ/8)11yescharacteristic
204C22C2×AGL(1,ℤ/8)11yescharacteristic
205C22C2×#13711yescharacteristic
206C22#57511yescharacteristic
207C4—22no—
208C4—22no—
209C4—22no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—82no—
215C4—81no—
216C4—82no—
217C4—81no—
218C4#55212yes—
219C4#55212yes—
220C2—82no—
221C2#261111yescharacteristic
222C2#311011yescharacteristic
223C2#311511yescharacteristic
224C2#317211yescharacteristic
225C2#318411yescharacteristic
226C2#323411yescharacteristic
227C2#323811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2859C2semidirect
3. #2099C2semidirect
5. #2084C2semidirect
7. #1944C2semidirect
9. #1921C2semidirect
11. #1745C2semidirect

Maximal subgroups

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

subgroupindex
#28592
#22932
#20992
#20932
#20842
#20712
#19442
#19362
#19212
#18642
#17452