Finite Group Databasebeta

← #33049

Subgroups — #33049

Order 256.

457 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 147 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×#487C211yesmaximal, characteristic, complemented
2C2×C4×M16C211yesmaximal, characteristic
3#2857C211yesmaximal, characteristic
4#2857C211yesmaximal, characteristic
5#1990C211yesmaximal, characteristic, complemented
6#1956C211yesmaximal, characteristic, complemented
7#1935C214yesmaximal, complemented
8#1894C214yesmaximal
9#1834C211yesmaximal, characteristic, complemented
10C22×M16C2212yes—
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C2212yes—
15C22×C42C2211yescharacteristic
16#487—24no—
17C2×#103C2214yes—
18C2×#103C2214yes—
19C4×M16—24no—
20C2×#97C2212yes—
21C2×C4×C8C2211yescharacteristic
22C2×C4×C8C2211yescharacteristic
23#391—21no—
24#391C2212yes—
25#389—21no—
26#389—21no—
27#389—21no—
28#389C2212yes—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#380—21no—
32#380—21no—
33#380—21no—
34#380C2212yes—
35#340C2214yes—
36#340C2214yes—
37C22×Q8—21no—
38C22×Q8C2312yes—
39C23×C4D411yescharacteristic
40C23×C4D411yescharacteristic
41C23×C4C2311yescharacteristic
42C2×M16—24no—
43C2×M16—24no—
44C2×M16—24no—
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C22⋊Q8—28no—
52C22⋊Q8—44no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#103—28no—
74#103—28no—
75#97—24no—
76C4×C8—22no—
77C4×C8D414yes—
78#95—22no—
79#95—22no—
80#95—22no—
81#95—22no—
82C24C2×D411yescharacteristic
83C2×Q8—22no—
84C2×Q8—24no—
85C2×Q8—24no—
86C2×Q8—22no—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×Q8—42no—
90C22×C4—21no—
91C22×C4—21no—
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—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4#4114yes—
109C22×C4C2411yescharacteristic
110M16—216no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8C2×D414yes—
116C2×C8#4114yes—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—22no—
121C4⋊C4—44no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C22⋊C4—44no—
131C22⋊C4—44no—
132C42—22no—
133C42—22no—
134C42—21no—
135C42—21no—
136C42C2×D412yes—
137C42#4112yes—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23#12412yes—
145C23C22×D411yescharacteristic
146Q8—44no—
147Q8—44no—
148Q8—84no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—44no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4#12112yes—
174C2×C4#12112yes—
175C2×C4#12412yes—
176C2×C4AGL(1,ℤ/8)12yes—
177C2×C4#13712yes—
178C2×C4#13712yes—
179C2×C4#13712yes—
180C2×C4C22×D411yescharacteristic
181C2×C4C2×#4112yes—
182C2×C4C2×#4112yes—
183C2×C4C2×#4111yescharacteristic
184C8—24no—
185C8—24no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—41no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22C2×#12111yescharacteristic
194C22C2×#12411yescharacteristic
195C22#53211yescharacteristic
196C22C2×AGL(1,ℤ/8)11yescharacteristic
197C22C2×#13711yescharacteristic
198C22C2×#13711yescharacteristic
199C22C2×#13711yescharacteristic
200C4—22no—
201C4—41no—
202C4—41no—
203C4—82no—
204C4—82no—
205C4—82no—
206C4—82no—
207C4#46814yes—
208C4#47014yes—
209C4#52512yes—
210C4#52912yes—
211C2—42no—
212C2#285011yescharacteristic
213C2C2×#46811yescharacteristic
214C2C2×#47011yescharacteristic
215C2#313911yescharacteristic
216C2#314111yescharacteristic
217C2#318211yescharacteristic
218C2#318511yescharacteristic
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
1. C2×#487C2semidirect
5. #1990C2semidirect
6. #1956C2semidirect
7. #1935C2semidirect
9. #1834C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
C2×C4×M162
#28572
#19902
#19562
#19352
#18942
#18342