Finite Group Databasebeta

← #32836

Subgroups — #32836

Order 256.

316 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 107 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#2860C211yesmaximal, characteristic
2#2859C211yesmaximal, characteristic, complemented
3#2101C212yesmaximal, complemented
4#2092C212yesmaximal
5#2084C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2071C211yesmaximal, characteristic
8#2071C211yesmaximal, characteristic
9#2059C212yesmaximal, complemented
10#2025C212yesmaximal
11#1839C211yesmaximal, characteristic, complemented
12C2×#126—21no—
13C2×#125C2212yes—
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×#102C2212yes—
17C2×#102C2212yes—
18#390C2211yescharacteristic
19#390C2211yescharacteristic
20#390C2211yescharacteristic
21#389—21no—
22#389—21no—
23#389—21no—
24#389—21no—
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#388—21no—
29#388—21no—
30#388—21no—
31#384C2211yescharacteristic
32#384C2211yescharacteristic
33#382C2212yes—
34C43C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#336C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×D4C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46#126—44no—
47#125—42no—
48#125—42no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—42no—
77#106—42no—
78#102—42no—
79#102—42no—
80#95—22no—
81#95—22no—
82#95—21no—
83#95—21no—
84#95—21no—
85C24—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—22no—
89C2×D4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2411yescharacteristic
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C22⋊C4—42no—
126C22⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C42—22no—
132C42—22no—
133C42—21no—
134C42—21no—
135C42—22no—
136C42—22no—
137C42—41no—
138C42—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23C22×D411yescharacteristic
147D4—42no—
148D4—42no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4#13512yes—
181C2×C4AGL(1,ℤ/8)12yes—
182C2×C4H(5,𝔽2)11yescharacteristic
183C2×C4H(5,𝔽2)11yescharacteristic
184C2×C42−1+412yes—
185C2×C42−1+412yes—
186C8—42no—
187C8—42no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22#53511yescharacteristic
196C22#53611yescharacteristic
197C22#53611yescharacteristic
198C22C2×#13511yescharacteristic
199C22C2×AGL(1,ℤ/8)11yescharacteristic
200C22#57511yescharacteristic
201C22#57511yescharacteristic
202C4—22no—
203C4—22no—
204C4—41no—
205C4—41no—
206C4—41no—
207C4—41no—
208C4—82no—
209C4—82no—
210C4—81no—
211C4—81no—
212C4—81no—
213C4#56312yes—
214C4#56312yes—
215C2—161no—
216C2#282111yescharacteristic
217C2#321211yescharacteristic
218C2#321411yescharacteristic
219C2#322811yescharacteristic
220C2#323211yescharacteristic
221C2#323411yescharacteristic
222C2#324011yescharacteristic
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
2. #2859C2semidirect
3. #2101C2semidirect
5. #2084C2semidirect
6. #2083C2semidirect
9. #2059C2semidirect
11. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28602
#28592
#21012
#20922
#20842
#20832
#20712
#20592
#20252
#18392