Finite Group Databasebeta

← #27898

Subgroups — #27898

Order 256.

365 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 115 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#2620C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic, complemented
3#2035C212yesmaximal, complemented
4#2034C212yesmaximal
5#1959C211yesmaximal, characteristic, complemented
6#1958C211yesmaximal, characteristic, complemented
7#1927C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1844C211yesmaximal, characteristic, complemented
10#1836C211yesmaximal, characteristic, complemented
11#1750C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#106C2212yes—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×C22⋊C8C2212yes—
23C2×#97C2211yescharacteristic
24C2×C4×C8C2211yescharacteristic
25#396—22no—
26#394—22no—
27#391C2211yescharacteristic
28#390C2211yescharacteristic
29#388—22no—
30#387—22no—
31#384C2211yescharacteristic
32#384C2211yescharacteristic
33#382C2211yescharacteristic
34#380C2212yes—
35C4×C4⋊C4C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8C2311yescharacteristic
40C22×D4—21no—
41C22×D4C2311yescharacteristic
42C23×C4C2311yescharacteristic
43C22×C8C2312yes—
44C22×C8C2312yes—
45#124—22no—
46#124—22no—
47C22⋊Q8—42no—
48#121—42no—
49#117—22no—
50#117—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4D412yes—
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4D412yes—
65C2×C42—22no—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#106—42no—
69#106—42no—
70#103—24no—
71#103—42no—
72#102—24no—
73#102—42no—
74C22⋊C8—24no—
75#97—22no—
76C4×C8—22no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81C24—21no—
82C24C2×D411yescharacteristic
83C2×Q8—22no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—22no—
87C2×D4—22no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—22no—
91C2×D4—42no—
92C2×D4—82no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D412yes—
105C22×C4C2411yescharacteristic
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8#4114yes—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—42no—
120C4⋊C4—82no—
121C4⋊C4—82no—
122C22⋊C4—24no—
123C22⋊C4—42no—
124C22⋊C4—42no—
125C22⋊C4—44no—
126C22⋊C4—82no—
127C22⋊C4—82no—
128C42—21no—
129C42—21no—
130C42—82no—
131C42#4112yes—
132C42#4112yes—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—22no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—42no—
143C23C22×D411yescharacteristic
144Q8—42no—
145Q8—42no—
146D4—42no—
147D4—42no—
148D4—84no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4#12314yes—
173C2×C4C2×#4111yescharacteristic
174C2×C4C2×#4112yes—
175C2×C4C2×#4111yescharacteristic
176C2×C4H(5,𝔽2)12yes—
177C8—24no—
178C8—42no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—82no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22C2×#12311yescharacteristic
190C22#53211yescharacteristic
191C22#53511yescharacteristic
192C22#57712yes—
193C22#57712yes—
194C4—24no—
195C4—42no—
196C4—41no—
197C4—82no—
198C4—82no—
199C4—81no—
200C4—81no—
201C4—81no—
202C4#55312yes—
203C4#55512yes—
204C2—81no—
205C2—161no—
206C2#250011yescharacteristic
207C2#309912yes—
208C2#317312yes—
209C2#322111yescharacteristic
210C2#322211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2314C2semidirect
3. #2035C2semidirect
5. #1959C2semidirect
6. #1958C2semidirect
7. #1927C2semidirect
8. #1926C2semidirect
9. #1844C2semidirect
10. #1836C2semidirect
11. #1750C2semidirect

Maximal subgroups

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

subgroupindex
#26202
#23142
#20352
#20342
#19592
#19582
#19272
#19262
#18442
#18362
#17502