Finite Group Databasebeta

← #33008

Subgroups — #33008

Order 256.

657 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 155 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×#483C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic
4#2855C211yesmaximal, characteristic
5#1878C214yesmaximal, complemented
6#1865C214yesmaximal, complemented
7#1823C211yesmaximal, 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#483—24no—
22#451—24no—
23C2×#106C2212yes—
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#106—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—24no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D412yes—
118C22×C4C2411yescharacteristic
119Q16—48no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—24no—
123C2×C8—24no—
124C4⋊C4—216no—
125C4⋊C4—24no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—24no—
130C4⋊C4—42no—
131C4⋊C4—44no—
132C4⋊C4C2×D414yes—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C42—216no—
136C42—24no—
137C42—22no—
138C42—22no—
139C23—22no—
140C23—22no—
141C23#12012yes—
142C23#13512yes—
143C23#13712yes—
144C23C22×D411yescharacteristic
145Q8—216no—
146Q8—24no—
147Q8—28no—
148Q8—42no—
149Q8—42no—
150Q8#12318yes—
151C2×C4—28no—
152C2×C4—216no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—44no—
167C2×C4—48no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4#12012yes—
176C2×C4#13714yes—
177C2×C4C22×D411yescharacteristic
178C2×C4C22×D411yescharacteristic
179C2×C4C2×#4114yes—
180C8—42no—
181C8—42no—
182C22—22no—
183C22—24no—
184C22—24no—
185C22—22no—
186C22#45214yes—
187C22C2×#12011yescharacteristic
188C22#52512yes—
189C22C2×#13511yescharacteristic
190C22C2×#13712yes—
191C22C2×#13711yescharacteristic
192C4—28no—
193C4—22no—
194C4—24no—
195C4—42no—
196C4—44no—
197C4—81no—
198C4—82no—
199C4—81no—
200C4C2×#12312yes—
201C4#52512yes—
202C2—24no—
203C2#284911yescharacteristic
204C2C2×#45211yescharacteristic
205C2#300411yescharacteristic
206C2#310812yes—
207C2#311312yes—
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. #1865C2semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q82
C2×#4832
C2×#4512
#28552
#18782
#18652
#18232
#18172
#17872