Finite Group Databasebeta

← #28939

Subgroups — #28939

Order 256.

381 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#3137C212yesmaximal, complemented
2#3063C212yesmaximal
3#2060C211yesmaximal, characteristic, complemented
4#2039C211yesmaximal, characteristic
5#2005C212yesmaximal, complemented
6#1956C211yesmaximal, characteristic, complemented
7#1930C212yesmaximal, complemented
8#1908C212yesmaximal, complemented
9#1764C212yesmaximal, complemented
10C2×#137C2212yes—
11C2×#131C2212yes—
12#544C2211yescharacteristic
13#537C2211yescharacteristic
14#536C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16#491—22no—
17#489—22no—
18#488—22no—
19#487—22no—
20#470—22no—
21#468—22no—
22#467—22no—
23#465—22no—
24#428C2212yes—
25C2×#104C2212yes—
26#419C2212yes—
27#417C2212yes—
28#416C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#101C2212yes—
32#405C2212yes—
33C2×C4×C8C2211yescharacteristic
34#384C2211yescharacteristic
35#343C2211yescharacteristic
36#343C2211yescharacteristic
37#341C2212yes—
38#339C2212yes—
39C2×#41C2312yes—
40C22×Q8C2312yes—
41#137—24no—
42#137—42no—
43C2×Q16—22no—
44C2×SD16—22no—
45#131—24no—
46#131—42no—
47C2×M16—22no—
48C2×M16C2312yes—
49C2×M16C2312yes—
50C22×C8—22no—
51C22×C8C2312yes—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—41no—
56#125—22no—
57#124—22no—
58#123—42no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—41no—
63#121—22no—
64#121—41no—
65#117C2312yes—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—22no—
72#106—22no—
73#104—24no—
74#103—22no—
75#103—22no—
76#103—22no—
77#103—22no—
78#103—22no—
79#103—22no—
80#103—22no—
81#103D412yes—
82#103D412yes—
83#102—22no—
84#102D412yes—
85#102D412yes—
86#101—24no—
87#97—22no—
88C4×C8—22no—
89C4×C8—41no—
90#41—22no—
91#41—22no—
92#41—22no—
93#41—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—41no—
101C2×Q8C2×D412yes—
102C2×D4—22no—
103C2×D4—41no—
104C2×D4C2×D412yes—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4C2411yescharacteristic
111Q16—44no—
112SD16—44no—
113M16—22no—
114M16—22no—
115M16—22no—
116M16—44no—
117M16#4114yes—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—44no—
123C2×C8C2×D412yes—
124C2×C8C2×D412yes—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—41no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—41no—
131C4⋊C4—42no—
132C4⋊C4—44no—
133C4⋊C4—82no—
134C4⋊C4C2×D412yes—
135C4⋊C4C2×D412yes—
136C22⋊C4—22no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—22no—
141C42—21no—
142C42#4112yes—
143C23—22no—
144C23H(5,𝔽2)11yescharacteristic
145Q8—22no—
146Q8—22no—
147Q8—42no—
148Q8—42no—
149Q8—42no—
150Q8—42no—
151Q8—161no—
152D4—22no—
153D4—22no—
154D4—42no—
155D4—42no—
156D4—161no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4C22×D412yes—
173C2×C4C2×#4111yescharacteristic
174C2×C4C2×#4112yes—
175C2×C4H(5,𝔽2)11yescharacteristic
176C8—22no—
177C8—22no—
178C8—24no—
179C8—42no—
180C22—21no—
181C22—21no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22#53811yescharacteristic
186C22D4211yescharacteristic
187C22#55511yescharacteristic
188C4—24no—
189C4—42no—
190C4—42no—
191C4—42no—
192C4—81no—
193C4—81no—
194C4—82no—
195C4#53212yes—
196C4#54612yes—
197C2—21no—
198C2—21no—
199C2—82no—
200C2#255511yescharacteristic
201C2#329111yescharacteristic
202C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3137C2semidirect
3. #2060C2semidirect
5. #2005C2semidirect
6. #1956C2semidirect
7. #1930C2semidirect
8. #1908C2semidirect
9. #1764C2semidirect

Maximal subgroups

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

subgroupindex
#31372
#30632
#20602
#20392
#20052
#19562
#19302
#19082
#17642