Finite Group Databasebeta

← #33010

Subgroups — #33010

Order 256.

657 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×C4×Q8C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic
4#2855C211yesmaximal, characteristic
5#1878C214yesmaximal, complemented
6#1863C214yesmaximal, complemented
7#1824C211yesmaximal, characteristic
8#1817C211yesmaximal, characteristic
9#1787C211yesmaximal, characteristic
10C23×Q8C2211yescharacteristic
11C22×Q16C2212yes—
12C2×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8—24no—
14C2×C4×Q8—24no—
15C2×C4×Q8—22no—
16C2×C4×Q8C2214yes—
17C22×C4⋊C4—21no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C42—21no—
20C22×C42C2211yescharacteristic
21#484—24no—
22#451—24no—
23C2×#107C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2214yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#391C2212yes—
30#389C2212yes—
31#387—22no—
32#385—21no—
33#381—21no—
34#380C2211yescharacteristic
35#340C2214yes—
36#336C2214yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—21no—
40C22×Q8D414yes—
41C22×Q8C2311yescharacteristic
42C22×Q8C2312yes—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C2×Q16—24no—
49C2×Q16—28no—
50C22×C8C2312yes—
51C22×C8C2312yes—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C4×Q8—216no—
55C4×Q8—28no—
56C4×Q8—28no—
57C2×C4⋊C4—28no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D414yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—28no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—24no—
73C2×C42D412yes—
74C2×C42C2312yes—
75#107—24no—
76#103—24no—
77#103—28no—
78#103—24no—
79C22⋊C8—22no—
80C22⋊C8—22no—
81#95—21no—
82#95—22no—
83#95—22no—
84#95—22no—
85C24C2×D411yescharacteristic
86C2×Q8—216no—
87C2×Q8—28no—
88C2×Q8—216no—
89C2×Q8—24no—
90C2×Q8—22no—
91C2×Q8—24no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—24no—
95C2×Q8C2×D414yes—
96C2×Q8#4118yes—
97C22×C4—28no—
98C22×C4—22no—
99C22×C4—24no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—42no—
110C22×C4—44no—
111C22×C4Q1614yescomplemented
112C22×C4C2×D412yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D412yes—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2411yescharacteristic
118Q16—48no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—24no—
122C2×C8—24no—
123C4⋊C4—216no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—24no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4C2×D414yes—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C42—216no—
135C42—24no—
136C42—22no—
137C42—22no—
138C23—22no—
139C23—22no—
140C23#12012yes—
141C23C2×Q1612yes—
142C23#13712yes—
143C23C22×D411yescharacteristic
144Q8—216no—
145Q8—24no—
146Q8—28no—
147Q8—42no—
148Q8—42no—
149Q8#12318yes—
150C2×C4—216no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—28no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—28no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—44no—
164C2×C4—48no—
165C2×C4—44no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4#12012yes—
174C2×C4C2×Q1614yes—
175C2×C4C22×D411yescharacteristic
176C2×C4C22×D411yescharacteristic
177C2×C4C2×#4114yes—
178C8—42no—
179C8—42no—
180C22—24no—
181C22—22no—
182C22—24no—
183C22—22no—
184C22#45114yes—
185C22C2×#12011yescharacteristic
186C22#52512yes—
187C22C22×Q1611yescharacteristic
188C22C2×#13711yescharacteristic
189C22#57512yes—
190C4—28no—
191C4—22no—
192C4—24no—
193C4—42no—
194C4—44no—
195C4—81no—
196C4—82no—
197C4—81no—
198C4C2×#12312yes—
199C4#52512yes—
200C2—24no—
201C2#284911yescharacteristic
202C2C2×#45111yescharacteristic
203C2#300511yescharacteristic
204C2#310412yes—
205C2#311712yes—
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
5. #1878C2semidirect
6. #1863C2semidirect
111. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q82
C2×#4842
C2×#4512
#28552
#18782
#18632
#18242
#18172
#17872