Finite Group Databasebeta

← #43244403

Subgroups — #43244403

Order 28096.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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×#20296330C211yesmaximal, characteristic, complemented
2#41×D439C211yesmaximal, characteristic, complemented
3#28380775C212yesmaximal, complemented
4#28380774C211yesmaximal, characteristic, complemented
5#28380771C211yesmaximal, characteristic, complemented
6#28380770C212yesmaximal, complemented
7#28380768C211yesmaximal, characteristic, complemented
8#28380766C212yesmaximal, complemented
9#28380764C212yesmaximal, complemented
10M16×D439C211yesmaximal, characteristic, complemented
11#137×C439C211yesmaximal, characteristic, complemented
12#20296330—21no—
13#20296330—21no—
14#20296330C2212yes—
15#20296330C2212yescomplemented
16#20296330C2211yescharacteristic, complemented
17#20296329—21no—
18#20296329C2211yescharacteristic
19#20296328—21no—
20#20296328C2211yescharacteristic, complemented
21Q8×D439C2211yescharacteristic, complemented
22C2×D1756—21no—
23C2×D1756C2211yescharacteristic, complemented
24D4×D439—21no—
25D4×D439C2211yescharacteristic, complemented
26C2×C4×D439—21no—
27C2×C4×D439—21no—
28C2×C4×D439C2211yescharacteristic
29#20296319C2212yescomplemented
30#20296318C2212yescomplemented
31#20296316C2212yescomplemented
32#20296316C2212yescomplemented
33D3512C2212yescomplemented
34#20296314C2212yescomplemented
35#20296313C2211yescharacteristic, complemented
36#20296312C2212yescomplemented
37C8×D439C2212yescomplemented
38#41×C439C2211yescharacteristic
39C2×Q8×C439C2211yescharacteristic, complemented
40Q16×C439C2212yescomplemented
41SD16×C439C2212yescomplemented
42M16×C439C2211yescharacteristic, complemented
43C22×D439—21no—
44C22×D439—21no—
45C22×D439D411yescharacteristic
46Dic878—21no—
47Dic878C2311yescharacteristic
48D1756—21no—
49D1756—22no—
50D1756—22no—
51D1756—21no—
52D1756C2312yes—
53D1756C2311yescharacteristic
54C22⋊D439—21no—
55C22⋊D439—21no—
56C22⋊D439—21no—
57C2×Dic439—21no—
58C2×Dic439D411yescharacteristic
59C4×D439—21no—
60C4×D439—21no—
61C4×D439—21no—
62C4×D439—22no—
63C4×D439—22no—
64C4×D439D411yescharacteristic
65C4×D439D411yescharacteristic
66C4×D439C2311yescharacteristic
67C4×D439C2311yescharacteristic
68C439⋊C8C2312yes—
69Q8×C439—21no—
70Q8×C439C2312yes—
71Q8×C439C2311yescharacteristic
72D4×C439—21no—
73D4×C439C2311yescharacteristic
74C2×C4×C439—21no—
75C2×C4×C439—21no—
76C2×C4×C439C2311yescharacteristic
77C8×C439C2312yes—
78C22×C439—21no—
79C22×C439C2×D411yescharacteristic
80C2×D439—21no—
81C2×D439—22no—
82C2×D439—41no—
83C2×D439—42no—
84C2×D439—42no—
85C2×D439C2×D411yescharacteristic
86C2×D439C2×D411yescharacteristic
87C4×C439—21no—
88C4×C439—22no—
89C4×C439C2×D411yescharacteristic
90C4×C439C2411yescharacteristic
91Dic439—21no—
92Dic439C2×D411yescharacteristic
93Dic439C2×D411yescharacteristic
94C2×C439—21no—
95C2×C439—41no—
96C2×C439C22×D411yescharacteristic
97D439—21no—
98D439—21no—
99D439—41no—
100D439—42no—
101C439#57511yescharacteristic, complemented, Sylow
102#575—4391nomaximal, Sylow
103C2×#41—4391no—
104C2×#41—4391no—
105#137—4391no—
106#137—4392no—
107#137D43911yescharacteristic, complemented
108AGL(1,ℤ/8)—4391no—
109AGL(1,ℤ/8)—4392no—
110AGL(1,ℤ/8)—4391no—
111#135—4392no—
112#135—4392no—
113C2×M16—4391no—
114#41—8781no—
115#41—8781no—
116#41—8781no—
117#41—8781no—
118#41—4391no—
119#41—4392no—
120#41—4392no—
121#41—4391no—
122#41—4391no—
123#41C2×D43911yescharacteristic
124C2×Q8—4391no—
125C2×Q8C2×D43911yescharacteristic, complemented
126C2×D4—8781no—
127C2×D4—8781no—
128C2×D4—4391no—
129C2×D4—4391no—
130C22×C4—8781no—
131C22×C4—8781no—
132C22×C4—4391no—
133Q16—4392no—
134Q16C2×D43912yescomplemented
135SD16—4392no—
136SD16—4392no—
137SD16—4392no—
138SD16C2×D43912yescomplemented
139D8—4392no—
140D8—4392no—
141M16—4391no—
142M16—4392no—
143M16C2×D43911yescharacteristic, complemented
144C2×C8—4392no—
145C23—8781no—
146C23—8781no—
147C23—4391no—
148Q8—21no—
149Q8—8781no—
150Q8—4391no—
151Q8C22×D43911yescharacteristic
152Q8C22×D43912yes—
153D4—21no—
154D4—8781no—
155D4—8782no—
156D4—8782no—
157D4—8781no—
158D4—8781no—
159D4—8781no—
160D4—8781no—
161D4—4392no—
162D4—4391no—
163D4C22×D43911yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—8781no—
167C2×C4—8781no—
168C2×C4—8782no—
169C2×C4—8782no—
170C2×C4—8781no—
171C2×C4—8781no—
172C2×C4—4391no—
173C2×C4—4391no—
174C2×C4—4391no—
175C2×C4—4391no—
176C2×C4—4391no—
177C2×C4C22×D43911yescharacteristic
178C8—4392no—
179C8C22×D43912yes—
180C22—21no—
181C22—8782no—
182C22—8781no—
183C22—4391no—
184C22—4391no—
185C22—17561no—
186C22—17562no—
187C22—17562no—
188C22D4×D43911yescharacteristic
189C4—21no—
190C4—22no—
191C4—8781no—
192C4—4391no—
193C4—4391no—
194C4D4×D43911yescharacteristic
195C4C23×D43911yescharacteristic
196C2—21no—
197C2—41no—
198C2—8781no—
199C2—8781no—
200C2—17562no—
201C2—17561no—
202C2C2×D4×D43911yescharacteristic
subgroupG/H
centerC2C2×D4×D439
commutator, cosocleC4×C439C24
FrattiniC4C23×D439
Fitting#137×C439C2
socleC2×C439C22×D4
hypercenter#137D439
hypocenterC439#575
LCSG ▷ C4×C439 ▷ C2×C439 ▷ C439
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#20296330C2semidirect
2. #41×D439C2semidirect
3. #28380775C2semidirect
4. #28380774C2semidirect
5. #28380771C2semidirect
6. #28380770C2semidirect
7. #28380768C2semidirect
8. #28380766C2semidirect
9. #28380764C2semidirect
10. M16×D439C2semidirect
11. #137×C439C2semidirect
15. #20296330C22semidirect
16. #20296330C22semidirect
20. #20296328C22semidirect
21. Q8×D439C22semidirect
23. C2×D1756C22semidirect
25. D4×D439C22semidirect
29. #20296319C22semidirect
30. #20296318C22semidirect
31. #20296316C22semidirect
32. #20296316C22semidirect
33. D3512C22semidirect
34. #20296314C22semidirect
35. #20296313C22semidirect
36. #20296312C22semidirect
37. C8×D439C22semidirect
39. C2×Q8×C439C22semidirect
40. Q16×C439C22semidirect
41. SD16×C439C22semidirect
42. M16×C439C22semidirect
101. C439#575semidirect
107. #137D439semidirect
125. C2×Q8C2×D439semidirect
134. Q16C2×D439semidirect
138. SD16C2×D439semidirect
143. M16C2×D439semidirect

Maximal subgroups

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

subgroupindex
C2×#202963302
#41×D4392
#283807752
#283807742
#283807712
#283807702
#283807682
#283807662
#283807642
M16×D4392
#137×C4392
#575439