Finite Group Databasebeta

← #29654

Subgroups — #29654

Order 256.

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

normal subgroups only

This group has 474 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#3458C211yesmaximal, characteristic, complemented
2#3094C212yesmaximal
3#2500C211yesmaximal, characteristic
4#2061C212yesmaximal, complemented
5#2060C212yesmaximal, complemented
6#2027C212yesmaximal, complemented
7#2027C212yesmaximal, complemented
8#1826C212yesmaximal, complemented
9#1822C211yesmaximal, characteristic, complemented
10#544—22no—
11#544—22no—
12#544C2212yes—
13#544C2212yes—
14#543—22no—
15#533—24no—
16#533—22no—
17C2×#121C2211yescharacteristic
18C2×#117C2212yes—
19C2×#117C2211yescharacteristic
20#482—24no—
21#481—22no—
22#480—22no—
23#428C2214yes—
24#418C2214yes—
25#407C2212yes—
26#394—22no—
27#390C2211yescharacteristic
28#390C2211yescharacteristic
29#388—22no—
30#384C2211yescharacteristic
31#382C2211yescharacteristic
32C4×C22⋊C4C2211yescharacteristic
33#343C2212yes—
34#343C2212yes—
35#339C2214yes—
36#339C2214yes—
37C22×D4—21no—
38C22×D4C2311yescharacteristic
39C23×C4C2311yescharacteristic
40C2×M16C2314yes—
41C4⋊Q8—22no—
42C4⋊Q8—22no—
43C4⋊Q8—24no—
44C4⋊Q8—24no—
45#125—24no—
46#125—24no—
47#125—24no—
48C22⋊Q8—28no—
49C22⋊Q8—28no—
50C22⋊Q8—24no—
51C22⋊Q8—24no—
52#121—22no—
53#121—22no—
54C4×Q8—24no—
55C4×Q8—24no—
56#117—22no—
57#117—22no—
58#117—22no—
59#117—22no—
60#117D414yes—
61#117C2314yes—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D412yes—
64C2×C4⋊C4D412yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4D412yes—
74C2×C42—22no—
75C2×C42D412yes—
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—24no—
79#106—24no—
80#102—24no—
81#102—24no—
82C22⋊C8—22no—
83C22⋊C8—22no—
84#95—21no—
85#95—22no—
86C24—21no—
87C24C2×D411yescharacteristic
88C2×Q8—24no—
89C2×Q8—24no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—42no—
94C2×D4—82no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D412yes—
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2411yescharacteristic
108M16—24no—
109M16—24no—
110C2×C8—24no—
111C4⋊C4—28no—
112C4⋊C4—28no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—82no—
121C4⋊C4C2×D414yes—
122C4⋊C4C2×D414yes—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—44no—
127C22⋊C4—42no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—82no—
131C42—24no—
132C42—22no—
133C42—22no—
134C42—24no—
135C42—22no—
136C42—22no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—42no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23#12012yes—
147C23C22×D411yescharacteristic
148Q8—84no—
149Q8—84no—
150D4—44no—
151D4—84no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4#12011yescharacteristic
173C2×C4#12011yescharacteristic
174C2×C4C22×D412yes—
175C2×C42−1+412yes—
176C2×C42−1+412yes—
177C8—44no—
178C22—21no—
179C22—21no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—81no—
184C22—81no—
185C22—82no—
186C22—82no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22C2×#12011yescharacteristic
191C22#53611yescharacteristic
192C22#53611yescharacteristic
193C4—21no—
194C4—21no—
195C4—42no—
196C4—42no—
197C4—44no—
198C4—42no—
199C4—44no—
200C4—81no—
201C4—81no—
202C4#53614yes—
203C2—21no—
204C2—21no—
205C2—42no—
206C2—161no—
207C2#261411yescharacteristic
208C2#301811yescharacteristic
209C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3458C2semidirect
4. #2061C2semidirect
5. #2060C2semidirect
6. #2027C2semidirect
7. #2027C2semidirect
8. #1826C2semidirect
9. #1822C2semidirect

Maximal subgroups

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

subgroupindex
#34582
#30942
#25002
#20612
#20602
#20272
#18262
#18222