Finite Group Databasebeta

← #25816

Subgroups — #25816

Order 256.

625 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 625 conjugacy classes of subgroups — too many to draw. Only its 163 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×#501C212yesmaximal, complemented
3#2851C211yesmaximal, characteristic
4#2058C212yesmaximal, complemented
5#2057C212yesmaximal, complemented
6#1993C212yesmaximal, complemented
7#1853C212yesmaximal
8#1852C212yesmaximal
9#1749C211yesmaximal, characteristic
10C23×Q8—21no—
11C2×C4⋊Q8—28no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8C2212yes—
15C2×C4⋊Q8C2212yes—
16C2×#125C2211yescharacteristic
17C22×C4⋊C4—21no—
18C22×C4⋊C4C2212yes—
19C22×C42C2211yescharacteristic
20#501—28no—
21C2×#107C2214yes—
22C2×#106C2214yes—
23C2×C22⋊C8C2212yes—
24C2×#97C2212yes—
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#387—21no—
28#387—21no—
29#385—21no—
30#385—21no—
31#382C2211yescharacteristic
32#380C2211yescharacteristic
33#380C2211yescharacteristic
34#340C2214yes—
35#340C2214yes—
36#336C2212yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—22no—
40C22×Q8—44no—
41C22×Q8—44no—
42C23×C4—22no—
43C23×C4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C22×C8C2314yes—
47C4⋊Q8—28no—
48C4⋊Q8—28no—
49C4⋊Q8—28no—
50C4⋊Q8—24no—
51C4⋊Q8D418yes—
52#125—24no—
53C2×C4⋊C4—28no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—28no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2314yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—24no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—28no—
75#106—28no—
76C22⋊C8Q818yescomplemented
77#97—24no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84C24C2×D411yescharacteristic
85C2×Q8—28no—
86C2×Q8—28no—
87C2×Q8—432no—
88C2×Q8—44no—
89C2×Q8—44no—
90C2×Q8—44no—
91C2×Q8—44no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—28no—
101C22×C4—24no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—44no—
107C22×C4—44no—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×Q814yes—
114C22×C4C2411yescharacteristic
115C2×C8—24no—
116C2×C8—24no—
117C2×C8C2×Q818yescomplemented
118C4⋊C4—28no—
119C4⋊C4—28no—
120C4⋊C4—28no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4C2×D418yes—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C42—22no—
134C42—22no—
135C42—22no—
136C42C2×D414yes—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Q812yes—
142C23AGL(1,ℤ/8)12yes—
143C23#13712yes—
144C23C22×D411yescharacteristic
145Q8—416no—
146Q8—416no—
147C2×C4—28no—
148C2×C4—28no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—28no—
160C2×C4—28no—
161C2×C4—416no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4#12012yes—
177C2×C4#12012yes—
178C2×C4C4⋊Q812yes—
179C2×C4AGL(1,ℤ/8)12yes—
180C2×C4#13712yes—
181C2×C4C22×D412yes—
182C2×C4C22×Q812yes—
183C2×C42−1+412yes—
184C8—28no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22#50114yescomplemented
192C22C2×#12011yescharacteristic
193C22C2×C4⋊Q811yescharacteristic
194C22#53611yescharacteristic
195C22C2×AGL(1,ℤ/8)11yescharacteristic
196C22C2×AGL(1,ℤ/8)11yescharacteristic
197C22C2×#13711yescharacteristic
198C22C2×#13711yescharacteristic
199C4—22no—
200C4—22no—
201C4—22no—
202C4—22no—
203C4—22no—
204C4—48no—
205C4—81no—
206C4—81no—
207C4—81no—
208C4—81no—
209C4D4×Q814yes—
210C2—24no—
211C2#243411yescharacteristic
212C2#300311yescharacteristic
213C2#300411yescharacteristic
214C2C2×#50111yescharacteristic
215C2#316511yescharacteristic
216C2#323811yescharacteristic
217C2#323911yescharacteristic
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
2. C2×#501C2semidirect
4. #2058C2semidirect
5. #2057C2semidirect
6. #1993C2semidirect
76. C22⋊C8Q8semidirect
117. C2×C8C2×Q8semidirect
191. C22#501semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#5012
#28512
#20582
#20572
#19932
#18532
#18522
#17492