Finite Group Databasebeta

← #29953

Subgroups — #29953

Order 256.

454 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×#391C211yesmaximal, characteristic
3#2095C212yesmaximal, complemented
4#2068C214yesmaximal, complemented
5#2024C212yesmaximal, complemented
6#1852C212yesmaximal
7#1797C211yesmaximal, characteristic
8#1787C212yesmaximal
9C23×Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C22⋊Q8—21no—
12C22×C4⋊C4C2211yescharacteristic
13C22×C42C2211yescharacteristic
14C2×#107C2214yes—
15C2×C4⋊C8C2212yes—
16C2×#103C2214yes—
17C2×#103C2214yes—
18C2×C22⋊C8C2212yes—
19#391—21no—
20#391—21no—
21#391—21no—
22#391C2212yes—
23#391C2212yes—
24#389—21no—
25#389C2212yes—
26#384C2211yescharacteristic
27#380—21no—
28#380C2212yes—
29C2×#95—21no—
30C2×#95—21no—
31#340C2214yes—
32#336C2212yes—
33C22×Q8—21no—
34C22×Q8—21no—
35C22×Q8—22no—
36C22×Q8—42no—
37C22×Q8—42no—
38C22×Q8Q814yescomplemented
39C22×Q8C2312yes—
40C23×C4—21no—
41C23×C4—21no—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8C2314yes—
46C4⋊Q8—42no—
47C22⋊Q8—44no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C22⋊C4—22no—
58C2×C42—42no—
59C2×C42D411yescharacteristic
60C2×C42D411yescharacteristic
61C2×C42C2311yescharacteristic
62C2×C42C2311yescharacteristic
63#107—44no—
64C4⋊C8—42no—
65#103—44no—
66#103—44no—
67C22⋊C8—24no—
68#95—22no—
69#95—22no—
70#95—22no—
71#95—22no—
72#95—22no—
73#95—22no—
74C24C2×D411yescharacteristic
75C2×Q8—24no—
76C2×Q8—22no—
77C2×Q8—24no—
78C2×Q8—24no—
79C2×Q8—48no—
80C2×Q8—42no—
81C2×Q8—42no—
82C2×Q8—48no—
83C2×Q8—42no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×Q8C2×Q814yes—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—24no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4Q1614yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×Q812yes—
111C22×C4#4112yes—
112C22×C4C2411yescharacteristic
113C2×C8—24no—
114C2×C8—24no—
115C2×C8—24no—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—42no—
124C22⋊C4—44no—
125C42—22no—
126C42—22no—
127C42—41no—
128C42—41no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23C22⋊Q812yes—
134C23C2×Q1612yes—
135C23AGL(1,ℤ/8)12yes—
136C23C22×D411yescharacteristic
137Q8—28no—
138Q8—48no—
139Q8—44no—
140Q8—84no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—44no—
156C2×C4—44no—
157C2×C4—48no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—84no—
170C2×C4C22⋊Q812yes—
171C2×C4C2×Q1612yes—
172C2×C4C2×Q1612yes—
173C2×C4#13712yes—
174C2×C4C22×Q811yescharacteristic
175C2×C4C2×#4111yescharacteristic
176C2×C4H(5,𝔽2)12yes—
177C2×C4H(5,𝔽2)12yes—
178C8—44no—
179C22—22no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—22no—
185C22#47714yes—
186C22C2×C22⋊Q811yescharacteristic
187C22#53411yescharacteristic
188C22#53511yescharacteristic
189C22C22×Q1611yescharacteristic
190C22C2×AGL(1,ℤ/8)11yescharacteristic
191C22C2×#13711yescharacteristic
192C22#57511yescharacteristic
193C4—22no—
194C4—22no—
195C4—22no—
196C4—22no—
197C4—44no—
198C4—41no—
199C4—82no—
200C4—82no—
201C4—82no—
202C4#53812yes—
203C4#54312yes—
204C2—24no—
205C2#262611yescharacteristic
206C2C2×#47711yescharacteristic
207C2#307811yescharacteristic
208C2#318911yescharacteristic
209C2#319311yescharacteristic
210C2#320911yescharacteristic
211C2#321311yescharacteristic
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. #2095C2semidirect
4. #2068C2semidirect
5. #2024C2semidirect
38. C22×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
C2×#3912
#20952
#20682
#20242
#18522
#17972
#17872