Finite Group Databasebeta

← #33023

Subgroups — #33023

Order 256.

657 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 135 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#3428C211yesmaximal, characteristic
2#2484C212yesmaximal, complemented
3#2406C211yesmaximal, characteristic
4#1890C214yesmaximal, complemented
5#1867C214yesmaximal, complemented
6#1820C212yesmaximal, complemented
7#1782C211yesmaximal, characteristic
8C22×#41C2211yescharacteristic
9C2×#123C2212yes—
10#520—22no—
11#520—22no—
12#519—21no—
13#518—21no—
14#518C2214yes—
15C4×#41—24no—
16C2×#117—22no—
17C2×#117C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C2×#99C2214yes—
20#400—22no—
21#398—22no—
22#395—21no—
23#394—21no—
24#387—24no—
25#386C2212yes—
26#385—22no—
27#385C2214yes—
28#381C2214yes—
29#381C2211yescharacteristic
30#381C2212yes—
31#342C2218yes—
32C2×#41—28no—
33C2×#41—22no—
34C2×#41—22no—
35C22×Q8D411yescharacteristic
36C22×D4D411yescharacteristic
37C22×D4C2312yes—
38C23×C4D412yes—
39C23×C4C2311yescharacteristic
40C23×C4C2311yescharacteristic
41C23×C4C2311yescharacteristic
42#123—28no—
43C4×Q8—24no—
44C4×Q8—24no—
45C4×D4—24no—
46C4×D4—24no—
47C4×D4—28no—
48C4×D4—28no—
49#117—24no—
50#117—24no—
51#117—22no—
52#117—24no—
53#117—22no—
54#117—22no—
55#117—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4D412yes—
65C2×C4⋊C4D412yes—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4C2312yes—
70C2×C22⋊C4C2314yes—
71C2×C22⋊C4C2314yes—
72C2×C42—24no—
73C2×C42D414yes—
74#99—28no—
75#95—21no—
76#95—22no—
77#95—24no—
78#95—24no—
79#95—21no—
80#95—22no—
81#95—22no—
82#95—24no—
83C24C2×D412yes—
84C24C2411yescharacteristic
85#41—216no—
86#41—216no—
87C2×Q8—22no—
88C2×Q8—24no—
89C2×D4—24no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—22no—
93C2×D4#4118yes—
94C22×C4—24no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—24no—
108C22×C4—42no—
109C22×C4C2×D412yes—
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2×D414yes—
115C22×C4C2×D414yes—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—48no—
123C4⋊C4—41no—
124C4⋊C4—42no—
125C4⋊C4—41no—
126C4⋊C4—44no—
127C22⋊C4—28no—
128C22⋊C4—24no—
129C22⋊C4—24no—
130C22⋊C4—28no—
131C22⋊C4—24no—
132C22⋊C4—44no—
133C22⋊C4—44no—
134C42—28no—
135C42—24no—
136C42—24no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—42no—
143C23#12314yes—
144C23C22×D411yescharacteristic
145C23C22×D412yes—
146C23C2×#4114yes—
147Q8—28no—
148D4—28no—
149D4—216no—
150C2×C4—28no—
151C2×C4—28no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—42no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—84no—
177C2×C4—84no—
178C2×C4#12011yescharacteristic
179C2×C4#12012yes—
180C2×C4#12011yescharacteristic
181C2×C4#12314yes—
182C22—22no—
183C22—21no—
184C22—22no—
185C22—21no—
186C22—24no—
187C22—44no—
188C22—44no—
189C22—42no—
190C22—42no—
191C22—82no—
192C22C2×#12011yescharacteristic
193C22C2×#12312yes—
194C22#52514yes—
195C4—21no—
196C4—21no—
197C4—22no—
198C4—24no—
199C4—42no—
200C4—44no—
201C4—44no—
202C4—82no—
203C4—84no—
204C4—81no—
205C2—22no—
206C2—24no—
207C2—44no—
208C2#284911yescharacteristic
209C2#302511yescharacteristic
210C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radicalC22×#41C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2484C2semidirect
4. #1890C2semidirect
5. #1867C2semidirect
6. #1820C2semidirect

Maximal subgroups

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

subgroupindex
#34282
#24842
#24062
#18902
#18672
#18202
#17822