Finite Group Databasebeta

← #42122607

Subgroups — #42122607

Order 27712.

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×#20294096C211yesmaximal, characteristic, complemented
2#41×D433C211yesmaximal, characteristic, complemented
3#28357253C211yesmaximal, characteristic, complemented
4#28357250C211yesmaximal, characteristic, complemented
5#28357249C212yesmaximal, complemented
6#28357246C211yesmaximal, characteristic, complemented
7#28357244C212yesmaximal, complemented
8#28357241C212yesmaximal, complemented
9#28357239C212yesmaximal, complemented
10M16×D433C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C433C211yesmaximal, characteristic, complemented
12#20294098C2211yescharacteristic, complemented
13#20294097—21no—
14#20294097C2211yescharacteristic
15#20294096—21no—
16#20294096—21no—
17#20294096—21no—
18#20294096C2212yes—
19#20294096C2211yescharacteristic, complemented
20#20294096C2212yescomplemented
21C2×Dic866C2211yescharacteristic, complemented
22Q8×D433C2211yescharacteristic, complemented
23C2×C22⋊D433—21no—
24D4×D433—21no—
25D4×D433C2211yescharacteristic, complemented
26C22×Dic433—21no—
27C2×C4×D433—21no—
28C2×C4×D433C2211yescharacteristic
29Dic1732C2212yescomplemented
30#20294087C2212yescomplemented
31#20294086C2212yescomplemented
32#20294085C2212yescomplemented
33#20294085C2212yescomplemented
34#20294082C2212yescomplemented
35#20294081C2211yescharacteristic, complemented
36#20294080C2212yescomplemented
37C8×D433C2212yescomplemented
38#41×C433C2211yescharacteristic
39C2×D4×C433C2211yescharacteristic, complemented
40SD16×C433C2212yescomplemented
41D8×C433C2212yescomplemented
42M16×C433C2211yescharacteristic, complemented
43C22×D433—21no—
44C22×D433D411yescharacteristic
45Dic866—21no—
46Dic866—21no—
47Dic866C2311yescharacteristic
48Dic866C2312yes—
49D1732—21no—
50D1732C2311yescharacteristic
51C22⋊D433—21no—
52C22⋊D433—21no—
53C22⋊D433—22no—
54C22⋊D433—22no—
55C22⋊D433—21no—
56C2×Dic433—21no—
57C2×Dic433—21no—
58C2×Dic433—22no—
59C2×Dic433—22no—
60C2×Dic433D411yescharacteristic
61C4×D433—21no—
62C4×D433—21no—
63C4×D433—21no—
64C4×D433D411yescharacteristic
65C4×D433D411yescharacteristic
66C4×D433C2311yescharacteristic
67C4×D433C2311yescharacteristic
68#16812358C2312yes—
69C23×C433—21no—
70Q8×C433C2311yescharacteristic
71D4×C433—21no—
72D4×C433—21no—
73D4×C433C2311yescharacteristic
74D4×C433C2312yes—
75C2×C4×C433—21no—
76C2×C4×C433C2311yescharacteristic
77C8×C433C2312yes—
78C22×C433—21no—
79C22×C433—22no—
80C22×C433—42no—
81C22×C433C2×D411yescharacteristic
82C2×D433—21no—
83C2×D433—41no—
84C2×D433—42no—
85C2×D433C2×D411yescharacteristic
86C2×D433C2×D411yescharacteristic
87C4×C433—21no—
88C4×C433C2×D411yescharacteristic
89C4×C433C2411yescharacteristic
90Dic433—21no—
91Dic433—22no—
92Dic433C2×D411yescharacteristic
93Dic433C2×D411yescharacteristic
94C2×C433—21no—
95C2×C433—41no—
96C2×C433—42no—
97C2×C433C22×D411yescharacteristic
98D433—21no—
99D433—21no—
100D433—41no—
101C433#57511yescharacteristic, complemented, Sylow
102#575—4331nomaximal, Sylow
103C2×#41—4331no—
104C2×#41—4331no—
105#137—4331no—
106#137—4332no—
107#137—4331no—
108AGL(1,ℤ/8)—4331no—
109AGL(1,ℤ/8)—4332no—
110AGL(1,ℤ/8)D43311yescharacteristic, complemented
111#135—4332no—
112#135—4332no—
113C2×M16—4331no—
114#41—8661no—
115#41—8661no—
116#41—8661no—
117#41—8661no—
118#41—4331no—
119#41—4332no—
120#41—4331no—
121#41—4332no—
122#41—4331no—
123#41C2×D43311yescharacteristic
124C2×Q8—4331no—
125C2×Q8—4331no—
126C2×D4—8661no—
127C2×D4—8661no—
128C2×D4—4331no—
129C2×D4C2×D43311yescharacteristic, complemented
130C22×C4—8661no—
131C22×C4—8661no—
132C22×C4—4331no—
133Q16—4332no—
134Q16—4332no—
135SD16—4332no—
136SD16—4332no—
137SD16—4332no—
138SD16C2×D43312yescomplemented
139D8—4332no—
140D8C2×D43312yescomplemented
141M16—4331no—
142M16—4332no—
143M16C2×D43311yescharacteristic, complemented
144C2×C8—4332no—
145C23—21no—
146C23—8661no—
147C23—4331no—
148Q8—8661no—
149Q8—8661no—
150Q8—4331no—
151Q8—4332no—
152Q8C22×D43311yescharacteristic
153D4—21no—
154D4—21no—
155D4—8661no—
156D4—8661no—
157D4—8662no—
158D4—8662no—
159D4—8661no—
160D4—8661no—
161D4—4331no—
162D4C22×D43312yes—
163D4C22×D43311yescharacteristic
164C2×C4—21no—
165C2×C4—8661no—
166C2×C4—8661no—
167C2×C4—8661no—
168C2×C4—8661no—
169C2×C4—8662no—
170C2×C4—8662no—
171C2×C4—8661no—
172C2×C4—4331no—
173C2×C4—4331no—
174C2×C4—4331no—
175C2×C4—4331no—
176C2×C4—4331no—
177C2×C4C22×D43311yescharacteristic
178C8—4332no—
179C8C22×D43312yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—8661no—
184C22—4331no—
185C22—4331no—
186C22—17321no—
187C22—17322no—
188C22D4×D43311yescharacteristic
189C4—21no—
190C4—8661no—
191C4—8662no—
192C4—4331no—
193C4—4331no—
194C4D4×D43311yescharacteristic
195C4C23×D43311yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—8661no—
200C2—8661no—
201C2—17321no—
202C2C2×D4×D43311yescharacteristic
subgroupG/H
centerC2C2×D4×D433
commutator, cosocleC4×C433C24
FrattiniC4C23×D433
FittingAGL(1,ℤ/8)×C433C2
socleC2×C433C22×D4
hypercenterAGL(1,ℤ/8)D433
hypocenterC433#575
LCSG ▷ C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20294096C2semidirect
2. #41×D433C2semidirect
3. #28357253C2semidirect
4. #28357250C2semidirect
5. #28357249C2semidirect
6. #28357246C2semidirect
7. #28357244C2semidirect
8. #28357241C2semidirect
9. #28357239C2semidirect
10. M16×D433C2semidirect
11. AGL(1,ℤ/8)×C433C2semidirect
12. #20294098C22semidirect
19. #20294096C22semidirect
20. #20294096C22semidirect
21. C2×Dic866C22semidirect
22. Q8×D433C22semidirect
25. D4×D433C22semidirect
29. Dic1732C22semidirect
30. #20294087C22semidirect
31. #20294086C22semidirect
32. #20294085C22semidirect
33. #20294085C22semidirect
34. #20294082C22semidirect
35. #20294081C22semidirect
36. #20294080C22semidirect
37. C8×D433C22semidirect
39. C2×D4×C433C22semidirect
40. SD16×C433C22semidirect
41. D8×C433C22semidirect
42. M16×C433C22semidirect
101. C433#575semidirect
110. AGL(1,ℤ/8)D433semidirect
129. C2×D4C2×D433semidirect
138. SD16C2×D433semidirect
140. D8C2×D433semidirect
143. M16C2×D433semidirect

Maximal subgroups

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

subgroupindex
C2×#202940962
#41×D4332
#283572532
#283572502
#283572492
#283572462
#283572442
#283572412
#283572392
M16×D4332
AGL(1,ℤ/8)×C4332
#575433