Finite Group Databasebeta

← #43248513

Subgroups — #43248513

Order 28352.

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×#20296786C211yesmaximal, characteristic, complemented
2#41×D443C211yesmaximal, characteristic, complemented
3#28381919C211yesmaximal, characteristic, complemented
4#28381916C211yesmaximal, characteristic, complemented
5#28381915C212yesmaximal, complemented
6#28381912C211yesmaximal, characteristic, complemented
7#28381910C212yesmaximal, complemented
8#28381907C212yesmaximal, complemented
9#28381905C212yesmaximal, complemented
10M16×D443C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C443C211yesmaximal, characteristic, complemented
12#20296788C2211yescharacteristic, complemented
13#20296787—21no—
14#20296787C2211yescharacteristic
15#20296786—21no—
16#20296786—21no—
17#20296786—21no—
18#20296786C2212yes—
19#20296786C2211yescharacteristic, complemented
20#20296786C2212yescomplemented
21C2×Dic886C2211yescharacteristic, complemented
22Q8×D443C2211yescharacteristic, complemented
23C2×C22⋊D443—21no—
24D4×D443—21no—
25D4×D443C2211yescharacteristic, complemented
26C22×Dic443—21no—
27C2×C4×D443—21no—
28C2×C4×D443C2211yescharacteristic
29Dic1772C2212yescomplemented
30#20296777C2212yescomplemented
31#20296776C2212yescomplemented
32#20296775C2212yescomplemented
33#20296775C2212yescomplemented
34#20296772C2212yescomplemented
35#20296771C2211yescharacteristic, complemented
36#20296770C2212yescomplemented
37C8×D443C2212yescomplemented
38#41×C443C2211yescharacteristic
39C2×D4×C443C2211yescharacteristic, complemented
40SD16×C443C2212yescomplemented
41D8×C443C2212yescomplemented
42M16×C443C2211yescharacteristic, complemented
43C22×D443—21no—
44C22×D443D411yescharacteristic
45Dic886—21no—
46Dic886—21no—
47Dic886C2311yescharacteristic
48Dic886C2312yes—
49D1772—21no—
50D1772C2311yescharacteristic
51C22⋊D443—21no—
52C22⋊D443—21no—
53C22⋊D443—22no—
54C22⋊D443—22no—
55C22⋊D443—21no—
56C2×Dic443—21no—
57C2×Dic443—21no—
58C2×Dic443—22no—
59C2×Dic443—22no—
60C2×Dic443D411yescharacteristic
61C4×D443—21no—
62C4×D443—21no—
63C4×D443—21no—
64C4×D443D411yescharacteristic
65C4×D443D411yescharacteristic
66C4×D443C2311yescharacteristic
67C4×D443C2311yescharacteristic
68C443⋊C8C2312yes—
69C23×C443—21no—
70Q8×C443C2311yescharacteristic
71D4×C443—21no—
72D4×C443—21no—
73D4×C443C2311yescharacteristic
74D4×C443C2312yes—
75C2×C4×C443—21no—
76C2×C4×C443C2311yescharacteristic
77C8×C443C2312yes—
78C22×C443—21no—
79C22×C443—22no—
80C22×C443—42no—
81C22×C443C2×D411yescharacteristic
82C2×D443—21no—
83C2×D443—41no—
84C2×D443—42no—
85C2×D443C2×D411yescharacteristic
86C2×D443C2×D411yescharacteristic
87C4×C443—21no—
88C4×C443C2×D411yescharacteristic
89C4×C443C2411yescharacteristic
90Dic443—21no—
91Dic443—22no—
92Dic443C2×D411yescharacteristic
93Dic443C2×D411yescharacteristic
94C2×C443—21no—
95C2×C443—41no—
96C2×C443—42no—
97C2×C443C22×D411yescharacteristic
98D443—21no—
99D443—21no—
100D443—41no—
101C443#57511yescharacteristic, complemented, Sylow
102#575—4431nomaximal, Sylow
103C2×#41—4431no—
104C2×#41—4431no—
105#137—4431no—
106#137—4432no—
107#137—4431no—
108AGL(1,ℤ/8)—4431no—
109AGL(1,ℤ/8)—4432no—
110AGL(1,ℤ/8)D44311yescharacteristic, complemented
111#135—4432no—
112#135—4432no—
113C2×M16—4431no—
114#41—8861no—
115#41—8861no—
116#41—8861no—
117#41—8861no—
118#41—4431no—
119#41—4432no—
120#41—4432no—
121#41—4431no—
122#41—4431no—
123#41C2×D44311yescharacteristic
124C2×Q8—4431no—
125C2×Q8—4431no—
126C2×D4—8861no—
127C2×D4—8861no—
128C2×D4—4431no—
129C2×D4C2×D44311yescharacteristic, complemented
130C22×C4—8861no—
131C22×C4—8861no—
132C22×C4—4431no—
133Q16—4432no—
134Q16—4432no—
135SD16—4432no—
136SD16—4432no—
137SD16—4432no—
138SD16C2×D44312yescomplemented
139D8—4432no—
140D8C2×D44312yescomplemented
141M16—4431no—
142M16—4432no—
143M16C2×D44311yescharacteristic, complemented
144C2×C8—4432no—
145C23—21no—
146C23—8861no—
147C23—4431no—
148Q8—8861no—
149Q8—8861no—
150Q8—4431no—
151Q8—4432no—
152Q8C22×D44311yescharacteristic
153D4—21no—
154D4—21no—
155D4—8861no—
156D4—8861no—
157D4—8862no—
158D4—8862no—
159D4—8861no—
160D4—8861no—
161D4—4431no—
162D4C22×D44312yes—
163D4C22×D44311yescharacteristic
164C2×C4—21no—
165C2×C4—8861no—
166C2×C4—8861no—
167C2×C4—8861no—
168C2×C4—8861no—
169C2×C4—8862no—
170C2×C4—8862no—
171C2×C4—8861no—
172C2×C4—4431no—
173C2×C4—4431no—
174C2×C4—4431no—
175C2×C4—4431no—
176C2×C4—4431no—
177C2×C4C22×D44311yescharacteristic
178C8—4432no—
179C8C22×D44312yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—8861no—
184C22—4431no—
185C22—4431no—
186C22—17721no—
187C22—17722no—
188C22D4×D44311yescharacteristic
189C4—21no—
190C4—8861no—
191C4—8862no—
192C4—4431no—
193C4—4431no—
194C4D4×D44311yescharacteristic
195C4C23×D44311yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—8861no—
200C2—8861no—
201C2—17721no—
202C2C2×D4×D44311yescharacteristic
subgroupG/H
centerC2C2×D4×D443
commutator, cosocleC4×C443C24
FrattiniC4C23×D443
FittingAGL(1,ℤ/8)×C443C2
socleC2×C443C22×D4
hypercenterAGL(1,ℤ/8)D443
hypocenterC443#575
LCSG ▷ C4×C443 ▷ C2×C443 ▷ C443
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20296786C2semidirect
2. #41×D443C2semidirect
3. #28381919C2semidirect
4. #28381916C2semidirect
5. #28381915C2semidirect
6. #28381912C2semidirect
7. #28381910C2semidirect
8. #28381907C2semidirect
9. #28381905C2semidirect
10. M16×D443C2semidirect
11. AGL(1,ℤ/8)×C443C2semidirect
12. #20296788C22semidirect
19. #20296786C22semidirect
20. #20296786C22semidirect
21. C2×Dic886C22semidirect
22. Q8×D443C22semidirect
25. D4×D443C22semidirect
29. Dic1772C22semidirect
30. #20296777C22semidirect
31. #20296776C22semidirect
32. #20296775C22semidirect
33. #20296775C22semidirect
34. #20296772C22semidirect
35. #20296771C22semidirect
36. #20296770C22semidirect
37. C8×D443C22semidirect
39. C2×D4×C443C22semidirect
40. SD16×C443C22semidirect
41. D8×C443C22semidirect
42. M16×C443C22semidirect
101. C443#575semidirect
110. AGL(1,ℤ/8)D443semidirect
129. C2×D4C2×D443semidirect
138. SD16C2×D443semidirect
140. D8C2×D443semidirect
143. M16C2×D443semidirect

Maximal subgroups

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

subgroupindex
C2×#202967862
#41×D4432
#283819192
#283819162
#283819152
#283819122
#283819102
#283819072
#283819052
M16×D4432
AGL(1,ℤ/8)×C4432
#575443