Finite Group Databasebeta

← #33691

Subgroups — #33691

Order 256.

262 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 93 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#3173C211yesmaximal, characteristic, complemented
2#2955C211yesmaximal, characteristic, complemented
3#2239C211yesmaximal, characteristic, complemented
4#2238C211yesmaximal, characteristic, complemented
5#2230C212yesmaximal, complemented
6#2212C212yesmaximal, complemented
7#2191C211yesmaximal, characteristic, complemented
8#2190C211yesmaximal, characteristic
9AGL(1,ℤ/16)C212yesmaximal, complemented
10#2180C212yesmaximal, complemented
11#2166C211yesmaximal, characteristic, complemented
12#575C2211yescharacteristic
13C2×#131—21no—
14#542—21no—
15#510C2212yes—
16#509C2212yes—
17#498C2211yescharacteristic, complemented
18#495C2211yescharacteristic, complemented
19#489—21no—
20#470C2211yescharacteristic
21#469C2211yescharacteristic
22#465—21no—
23#444—21no—
24#444C2212yes—
25#443—21no—
26#443C2212yes—
27#442C2211yescharacteristic, complemented
28#441C2211yescharacteristic, complemented
29C4×SD16—21no—
30#430C2211yescharacteristic
31#421—21no—
32#405C2211yescharacteristic
33#365C2212yescomplemented
34#365C2212yescomplemented
35#364C2212yescomplemented
36#362C2212yescomplemented
37#361C2212yescomplemented
38#360C2212yes—
39#360C2212yes—
40#349C2211yescharacteristic
41#349C2211yescharacteristic
42#347C2212yescomplemented
43C2×#41—21no—
44#137—22no—
45AGL(1,ℤ/8)—22no—
46#135—21no—
47#135C2312yes—
48C2×Q16C2311yescharacteristic
49C2×SD16—21no—
50C2×D8C2311yescharacteristic
51#131—22no—
52#131—22no—
53#131—41no—
54#131—41no—
55C2×M16—21no—
56C2×M16—41no—
57C2×M16C2311yescharacteristic
58C22×C8—41no—
59C22×C8D411yescharacteristic
60C4⋊Q8—21no—
61#126—41no—
62#124—21no—
63#124—41no—
64#123—41no—
65C22⋊Q8—21no—
66C22⋊Q8—41no—
67#121—21no—
68C4×Q8—21no—
69C4×D4—21no—
70#117D411yescharacteristic
71Q32—22no—
72SD32—22no—
73SD32—22no—
74D16—22no—
75M32C2312yes—
76M32C2312yes—
77#108—21no—
78#108C2312yes—
79#106—21no—
80#106C2311yescharacteristic
81#106C2311yescharacteristic
82#104—22no—
83#104—22no—
84#103—21no—
85#103—21no—
86#102—21no—
87#102—21no—
88#97D411yescharacteristic
89#97C2311yescharacteristic
90C4×C8D411yescharacteristic
91C4×C8C2311yescharacteristic
92#41—22no—
93#41—22no—
94#41—41no—
95#41—41no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—41no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4C2×D411yescharacteristic
106Q16—21no—
107Q16—22no—
108Q16#4112yes—
109SD16—22no—
110SD16—22no—
111SD16—41no—
112SD16—41no—
113D8—21no—
114D8—22no—
115D8#4112yes—
116M16—22no—
117M16—22no—
118M16—42no—
119M16—41no—
120M16—42no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—42no—
124C2×C8—41no—
125C2×C8—42no—
126C2×C8C2×D411yescharacteristic
127C2×C8C2×D411yescharacteristic
128C2×C8C2×D411yescharacteristic
129C2×C8C2411yescharacteristic
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C22⋊C4—21no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C42—41no—
143C42C2×D411yescharacteristic
144C42C2×D411yescharacteristic
145C16—22no—
146C16—22no—
147C23—41no—
148C23—41no—
149C23AGL(1,ℤ/8)11yescharacteristic
150Q8—22no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154Q8—161no—
155D4—22no—
156D4—41no—
157D4—42no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—161no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—161no—
173C2×C4AGL(1,ℤ/8)11yescharacteristic
174C2×C4C22×D411yescharacteristic
175C8—21no—
176C8—21no—
177C8—42no—
178C8AGL(1,ℤ/8)11yescharacteristic
179C8AGL(1,ℤ/8)11yescharacteristic
180C8C2×#4112yes—
181C22—21no—
182C22—41no—
183C22—42no—
184C22—81no—
185C22—81no—
186C22—82no—
187C22—81no—
188C22C2×AGL(1,ℤ/8)11yescharacteristic
189C4—21no—
190C4—41no—
191C4—41no—
192C4—42no—
193C4—81no—
194C4—81no—
195C4—81no—
196C4#52511yescharacteristic
197C4C2×AGL(1,ℤ/8)11yescharacteristic
198C2—21no—
199C2—41no—
200C2—82no—
201C2—161no—
202C2#310911yescharacteristic
subgroupG/H
center, socleC2#3109
commutator, Frattini, cosocleC2×C8C24
prime radical#2190C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3173C2semidirect
2. #2955C2semidirect
3. #2239C2semidirect
4. #2238C2semidirect
5. #2230C2semidirect
6. #2212C2semidirect
7. #2191C2semidirect
9. AGL(1,ℤ/16)C2semidirect
10. #2180C2semidirect
11. #2166C2semidirect
17. #498C22semidirect
18. #495C22semidirect
27. #442C22semidirect
28. #441C22semidirect
33. #365C22semidirect
34. #365C22semidirect
35. #364C22semidirect
36. #362C22semidirect
37. #361C22semidirect
42. #347C22semidirect

Maximal subgroups

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

subgroupindex
#31732
#29552
#22392
#22382
#22302
#22122
#21912
#21902
AGL(1,ℤ/16)2
#21802
#21662