Finite Group Databasebeta

← #29906

Subgroups — #29906

Order 256.

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

normal subgroups only

This group has 442 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#2854C211yesmaximal, characteristic
2C2×#384C211yesmaximal, characteristic, complemented
3#2099C212yesmaximal, complemented
4#2093C212yesmaximal, complemented
5#2083C212yesmaximal, complemented
6#2071C212yesmaximal, complemented
7#1874C212yesmaximal
8#1852C212yesmaximal, complemented
9#1817C211yesmaximal, characteristic, complemented
10C2×#123—21no—
11C2×#121C2211yescharacteristic
12C22×C4⋊C4—21no—
13C22×C4⋊C4—21no—
14C22×C4⋊C4C2212yes—
15C22×C42C2211yescharacteristic
16C2×#107C2214yes—
17C2×#102C2214yes—
18C2×C22⋊C8C2212yes—
19#390C2211yescharacteristic
20#390C2211yescharacteristic
21#389C2211yescharacteristic
22#389C2211yescharacteristic
23#388—21no—
24#388—21no—
25#386—21no—
26#384—21no—
27#384—21no—
28#384C2212yes—
29#384C2212yes—
30#380C2211yescharacteristic
31#340C2214yes—
32#340C2214yes—
33#336C2212yes—
34#336C2212yes—
35C22×D4—21no—
36C22×D4C2311yescharacteristic
37C23×C4—22no—
38C23×C4D411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4C2311yescharacteristic
41C22×C8C2314yes—
42#123—44no—
43#121—24no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—44no—
52C2×C4⋊C4—44no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4C2314yes—
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C42—24no—
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#107—44no—
70#102—44no—
71C22⋊C8—24no—
72#95—21no—
73#95—21no—
74#95—21no—
75C24—21no—
76C24C2×D411yescharacteristic
77C2×D4—22no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—42no—
81C2×D4—42no—
82C2×D4—42no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—24no—
94C22×C4—44no—
95C22×C4—44no—
96C22×C4—44no—
97C22×C4D814yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4#4114yes—
104C22×C4C2411yescharacteristic
105C2×C8—24no—
106C2×C8—24no—
107C2×C8—24no—
108C4⋊C4—22no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—24no—
112C4⋊C4—22no—
113C4⋊C4—44no—
114C4⋊C4—44no—
115C4⋊C4—44no—
116C4⋊C4—44no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—44no—
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—42no—
125C22⋊C4—42no—
126C42—22no—
127C42—22no—
128C42—22no—
129C42—22no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23#12312yes—
142C23C2×D812yes—
143C23#13712yes—
144C23C22×D411yescharacteristic
145D4—44no—
146D4—84no—
147C2×C4—28no—
148C2×C4—21no—
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—28no—
159C2×C4—28no—
160C2×C4—41no—
161C2×C4—44no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—44no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—88no—
173C2×C4#12312yes—
174C2×C4C2×D812yes—
175C2×C4C2×D812yes—
176C2×C4AGL(1,ℤ/8)12yes—
177C2×C4#13712yes—
178C2×C4C2×#4112yes—
179C2×C4H(5,𝔽2)12yes—
180C2×C42−1+412yes—
181C8—44no—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22#48014yes—
196C22C2×#12311yescharacteristic
197C22#53511yescharacteristic
198C22#53611yescharacteristic
199C22C22×D811yescharacteristic
200C22C2×AGL(1,ℤ/8)11yescharacteristic
201C22C2×#13711yescharacteristic
202C22C2×#13711yescharacteristic
203C4—22no—
204C4—22no—
205C4—22no—
206C4—22no—
207C4—22no—
208C4—81no—
209C4—81no—
210C4—84no—
211C4—81no—
212C4#53914yes—
213C2—24no—
214C2—161no—
215C2#262311yescharacteristic
216C2C2×#48011yescharacteristic
217C2#309211yescharacteristic
218C2#320511yescharacteristic
219C2#321811yescharacteristic
220C2#322611yescharacteristic
221C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. C2×#384C2semidirect
3. #2099C2semidirect
4. #2093C2semidirect
5. #2083C2semidirect
6. #2071C2semidirect
8. #1852C2semidirect
9. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28542
C2×#3842
#20992
#20932
#20832
#20712
#18742
#18522
#18172