Finite Group Databasebeta

← #30197

Subgroups — #30197

Order 256.

465 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 119 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#2857C211yesmaximal, characteristic, complemented
2#2856C211yesmaximal, characteristic
3#2100C212yesmaximal, complemented
4#2071C212yesmaximal, complemented
5#2041C212yesmaximal, complemented
6#2023C212yesmaximal, complemented
7#1895C214yesmaximal, complemented
8#1728C211yesmaximal, characteristic, complemented
9C22×M16C2212yes—
10C2×C4⋊Q8C2211yescharacteristic
11C2×#127C2211yescharacteristic
12C2×C22⋊Q8—21no—
13C2×#121—21no—
14C2×#121—21no—
15C2×#121C2212yes—
16C22×C42C2211yescharacteristic
17C2×#102C2214yes—
18C2×#102C2214yes—
19#391—21no—
20#390—21no—
21#390—21no—
22#390C2212yes—
23#389—21no—
24#389C2212yes—
25#384C2211yescharacteristic
26#384C2211yescharacteristic
27#380—21no—
28#380C2212yes—
29#340C2214yes—
30#340C2214yes—
31#336C2212yes—
32#336C2212yes—
33C22×Q8—21no—
34C22×D4—22no—
35C22×D4—21no—
36C22×D4—21no—
37C22×D4C2312yes—
38C23×C4D411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4C2311yescharacteristic
41C2×M16—24no—
42C2×M16—42no—
43C2×M16—42no—
44C22×C8C2314yes—
45C4⋊Q8—42no—
46#127—42no—
47C22⋊Q8—44no—
48#121—28no—
49#121—44no—
50#121—44no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C42—42no—
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#102—44no—
70#102—44no—
71#95—22no—
72#95—22no—
73C24—22no—
74C24C2×D411yescharacteristic
75C2×Q8—42no—
76C2×Q8—42no—
77C2×Q8—42no—
78C2×D4—24no—
79C2×D4—22no—
80C2×D4—24no—
81C2×D4—22no—
82C2×D4—42no—
83C2×D4—42no—
84C2×D4—42no—
85C2×D4—42no—
86C2×D4—44no—
87C2×D4—44no—
88C2×D4—44no—
89C2×D4—42no—
90C2×D4—42no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4#4114yes—
109C22×C4C2411yescharacteristic
110M16—48no—
111C2×C8—24no—
112C2×C8—24no—
113C2×C8—24no—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—44no—
124C22⋊C4—44no—
125C22⋊C4—44no—
126C22⋊C4—44no—
127C22⋊C4—44no—
128C42—22no—
129C42—22no—
130C42—41no—
131C42—41no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—42no—
139C23—42no—
140C23—42no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23#12412yes—
146C23C22×D411yescharacteristic
147Q8—84no—
148D4—44no—
149D4—44no—
150D4—84no—
151D4—84no—
152D4—88no—
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—24no—
164C2×C4—24no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4#12412yes—
177C2×C4AGL(1,ℤ/8)12yes—
178C2×C4AGL(1,ℤ/8)12yes—
179C2×C4AGL(1,ℤ/8)12yes—
180C2×C4#13712yes—
181C2×C4C2×#4112yes—
182C2×C4H(5,𝔽2)11yescharacteristic
183C2×C4H(5,𝔽2)12yes—
184C2×C42−1+411yescharacteristic
185C8—44no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—41no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—82no—
194C22—82no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22—82no—
200C22C2×#12411yescharacteristic
201C22#53411yescharacteristic
202C22#53711yescharacteristic
203C22C2×AGL(1,ℤ/8)11yescharacteristic
204C22C2×AGL(1,ℤ/8)11yescharacteristic
205C22C2×AGL(1,ℤ/8)11yescharacteristic
206C22C2×#13711yescharacteristic
207C4—22no—
208C4—22no—
209C4—22no—
210C4—22no—
211C4—22no—
212C4—82no—
213C4—82no—
214C4—82no—
215C4#53812yes—
216C4#53912yes—
217C2—42no—
218C2—162no—
219C2#263911yescharacteristic
220C2#313811yescharacteristic
221C2#314111yescharacteristic
222C2#319211yescharacteristic
223C2#319311yescharacteristic
224C2#325311yescharacteristic
225C2#325411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2856C2
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. #2857C2semidirect
3. #2100C2semidirect
4. #2071C2semidirect
5. #2041C2semidirect
6. #2023C2semidirect
7. #1895C2semidirect
8. #1728C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#28562
#21002
#20712
#20412
#20232
#18952
#17282