Finite Group Databasebeta

← #21873

Subgroups — #21873

Order 256.

511 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C2×#541C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic
3#2317C211yesmaximal, characteristic
4#1992C211yesmaximal, characteristic, complemented
5#1991C211yesmaximal, characteristic, complemented
6#1960C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1873C212yesmaximal
9#1872C212yesmaximal
10#1805C212yesmaximal, complemented
11#1803C212yesmaximal, complemented
12#541—24no—
13#541C418yescomplemented
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—22no—
16C2×#125—22no—
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8C2212yes—
19C2×C4×Q8C2212yes—
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#433—24no—
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×C22⋊C8C2212yes—
30C2×#97C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#389—22no—
33#385—22no—
34#384C2211yescharacteristic
35#384C2211yescharacteristic
36#382C2211yescharacteristic
37#382C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42C22×Q8C2312yes—
43C23×C4—21no—
44C23×C4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8C2×C414yescomplemented
49#126—216no—
50#125—216no—
51#125—22no—
52#125C2×C414yescomplemented
53C22⋊Q8—24no—
54C22⋊Q8C2×C418yes—
55C4×Q8—24no—
56C4×Q8C2×C418yescomplemented
57#117—42no—
58#117C2×C414yes—
59#117D414yes—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4D412yes—
77C2×C42—22no—
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81C4⋊C8—24no—
82#103—42no—
83#103—42no—
84#103—42no—
85#103—42no—
86C22⋊C8—24no—
87#97—22no—
88C4×C8—22no—
89#95—22no—
90#95—22no—
91C24C2×D411yescharacteristic
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8C22×C414yes—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—42no—
107C22×C4—82no—
108C22×C4C22×C412yes—
109C22×C4C2×D412yes—
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D412yes—
113C22×C4C2411yescharacteristic
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—28no—
121C4⋊C4—28no—
122C4⋊C4—28no—
123C4⋊C4—28no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—42no—
130C4⋊C4—82no—
131C4⋊C4—82no—
132C4⋊C4C22⋊C418yes—
133C4⋊C4C22×C414yes—
134C22⋊C4—28no—
135C22⋊C4—42no—
136C22⋊C4C22⋊C418yes—
137C42—28no—
138C42—42no—
139C42C22×C412yes—
140C42C22×C412yes—
141C42C2×D412yes—
142C42C2×D412yes—
143C23—21no—
144C23—21no—
145C23—22no—
146C23—22no—
147C23C2×C22⋊C412yes—
148C23C22×D411yescharacteristic
149Q8—24no—
150Q8—24no—
151Q8—42no—
152Q8—42no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—84no—
176C2×C4C2×C22⋊C414yes—
177C2×C4C2×C22⋊C414yes—
178C2×C4C2×C22⋊C412yes—
179C2×C4C23×C411yescharacteristic
180C2×C4C22×D411yescharacteristic
181C2×C4H(5,𝔽2)12yes—
182C2×C42−1+412yes—
183C8—42no—
184C8—42no—
185C22—22no—
186C22—24no—
187C22—41no—
188C22—41no—
189C22—42no—
190C22C22×C22⋊C411yescharacteristic
191C22#53511yescharacteristic
192C22#53611yescharacteristic
193C22#57712yes—
194C22#57812yes—
195C4—22no—
196C4—24no—
197C4—24no—
198C4—44no—
199C4—44no—
200C4—81no—
201C4—81no—
202C4—81no—
203C4—81no—
204C4#52012yes—
205C4#53712yes—
206C2—42no—
207C2#231111yescharacteristic
208C2#289812yes—
209C2#322211yescharacteristic
210C2#322311yescharacteristic
211C2#324212yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1992C2semidirect
5. #1991C2semidirect
6. #1960C2semidirect
7. #1956C2semidirect
10. #1805C2semidirect
11. #1803C2semidirect
13. #541C4semidirect
48. C4⋊Q8C2×C4semidirect
52. #125C2×C4semidirect
56. C4×Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4332
#23172
#19922
#19912
#19602
#19562
#18732
#18722
#18052
#18032