Finite Group Databasebeta

← #30435

Subgroups — #30435

Order 256.

436 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 105 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#3069C212yesmaximal, complemented
2#3004C212yesmaximal, complemented
3#2091C211yesmaximal, characteristic, complemented
4#2082C212yesmaximal, complemented
5#2049C212yesmaximal, complemented
6#2010C211yesmaximal, characteristic
7#1902C212yesmaximal, complemented
8#1891C212yesmaximal, complemented
9#1784C211yesmaximal, characteristic, complemented
10C23×Q8C2211yescharacteristic
11#575C2211yescharacteristic
12C2×#137C2212yes—
13C2×#137C2212yes—
14#568C2212yes—
15#525C2212yes—
16#473—24no—
17#471—24no—
18#451—24no—
19#448—24no—
20#430C2212yes—
21#430C2211yescharacteristic
22#421C2212yes—
23#419C2212yes—
24#419C2212yes—
25#413C2212yes—
26C2×#101C2212yes—
27C2×#101C2212yes—
28#407C2212yes—
29#407C2211yescharacteristic
30#407C2211yescharacteristic
31#344C2212yes—
32#344C2212yes—
33#338C2211yescharacteristic
34H(3,ℤ/4)C2211yescharacteristic
35C2×#41C2312yes—
36C22×Q8—21no—
37C22×Q8—24no—
38C22×Q8—41no—
39C22×Q8—41no—
40C22×Q8—42no—
41C22×Q8C2312yes—
42C23×C4—21no—
43C23×C4C2311yescharacteristic
44#137—24no—
45#137—22no—
46#137—42no—
47#137—42no—
48#137—42no—
49AGL(1,ℤ/8)—22no—
50#135—41no—
51C2×Q16—22no—
52C2×Q16D414yescomplemented
53C2×SD16—22no—
54C2×SD16D414yescomplemented
55#131—24no—
56#131—42no—
57C2×M16—42no—
58C2×M16C2312yes—
59C2×M16C2312yes—
60C2×M16C2311yescharacteristic
61C2×M16C2311yescharacteristic
62C2×M16C2312yes—
63#123—22no—
64C22⋊Q8—22no—
65#121—22no—
66#120—22no—
67C4×D4—42no—
68#117C2312yes—
69#108—24no—
70#108—41no—
71#104—42no—
72#103—24no—
73#103—24no—
74#101—24no—
75#101—22no—
76#101—42no—
77#100—22no—
78C22⋊C8—24no—
79C22⋊C8—41no—
80C22⋊C8—41no—
81C24#4111yescharacteristic
82#41—22no—
83#41—22no—
84#41—22no—
85#41—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—44no—
92C2×Q8—42no—
93C2×Q8—44no—
94C2×Q8—48no—
95C2×Q8—44no—
96C2×Q8C2×D414yes—
97C2×Q8C2×D412yes—
98C2×D4—22no—
99C2×D4—42no—
100C2×D4C2×D412yes—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—44no—
108C22×C4—42no—
109C22×C4#4111yescharacteristic
110C22×C4C2411yescharacteristic
111Q16—44no—
112Q16—44no—
113Q16—41no—
114SD16—44no—
115SD16—44no—
116SD16—42no—
117D8—41no—
118M16—22no—
119M16—22no—
120M16—22no—
121M16—22no—
122M16—22no—
123M16—22no—
124M16—22no—
125M16—22no—
126M16—44no—
127C2×C8—22no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8—44no—
132C2×C8C2×D414yes—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C22⋊C4—22no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C42—22no—
139C23—21no—
140C23—22no—
141C23—22no—
142C23—42no—
143C23C2×#4111yescharacteristic
144Q8—24no—
145Q8—22no—
146Q8—22no—
147Q8—24no—
148Q8—48no—
149Q8—42no—
150Q8—42no—
151D4—22no—
152D4—22no—
153D4—42no—
154D4—42no—
155D4—84no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—48no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4C22×D412yes—
170C2×C4H(5,𝔽2)11yescharacteristic
171C2×C4H(5,𝔽2)12yes—
172C2×C4H(5,𝔽2)11yescharacteristic
173C8—24no—
174C8—42no—
175C8—41no—
176C8—42no—
177C8—41no—
178C22—24no—
179C22—22no—
180C22—41no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—42no—
185C22—82no—
186C22#53811yescharacteristic
187C22#54612yes—
188C4—41no—
189C4—44no—
190C4—42no—
191C4—82no—
192C4#53412yes—
193C4D4211yescharacteristic
194C4#56011yescharacteristic
195C2—22no—
196C2—21no—
197C2—42no—
198C2—82no—
199C2#266811yescharacteristic
subgroupG/H
center, socleC2#2668
commutator, Frattini, cosocleC22×C4C24
prime radical#2010C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3069C2semidirect
2. #3004C2semidirect
3. #2091C2semidirect
4. #2082C2semidirect
5. #2049C2semidirect
7. #1902C2semidirect
8. #1891C2semidirect
9. #1784C2semidirect
52. C2×Q16D4semidirect
54. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
#30692
#30042
#20912
#20822
#20492
#20102
#19022
#18912
#17842