Finite Group Databasebeta

← #36342109

Subgroups — #36342109

Order 21568.

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

normal subgroups only

This group has 268 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
1#24925005C211yesmaximal, characteristic, complemented
2#24925004C211yesmaximal, characteristic, complemented
3#24924987C211yesmaximal, characteristic, complemented
4#24924986C212yesmaximal, complemented
5#24924983C212yesmaximal, complemented
6#24924980C212yesmaximal, complemented
7#24924979C211yesmaximal, characteristic, complemented
8C2×#19129170C211yesmaximal, characteristic, complemented
9D8×D337C212yesmaximal, complemented
10#24924967C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C337C211yesmaximal, characteristic, complemented
12#19129186C2212yescomplemented
13#19129185—21no—
14#19129185C2211yescharacteristic
15#19129185C2211yescharacteristic, complemented
16#19129184—21no—
17#19129184C2212yescomplemented
18C2×D1348—21no—
19C2×D1348C2211yescharacteristic, complemented
20C2×C22⋊D337—22no—
21D4×D337—21no—
22D4×D337—22no—
23D4×D337C2212yescomplemented
24D4×D337C2212yescomplemented
25#19129175C2211yescharacteristic, complemented
26#19129174C2212yescomplemented
27#19129173C2211yescharacteristic, complemented
28#19129172C2211yescharacteristic, complemented
29D2696C2212yescomplemented
30#19129170C2211yescharacteristic, complemented
31#19129170C2212yescomplemented
32#19129170C2212yescomplemented
33#19129168C2212yescomplemented
34C2×#14582106C2211yescharacteristic, complemented
35C8×D337C2212yescomplemented
36#41×C337C2211yescharacteristic, complemented
37C2×D4×C337C2211yescharacteristic, complemented
38SD16×C337C2212yescomplemented
39D8×C337C2212yescomplemented
40M16×C337C2211yescharacteristic, complemented
41C22×D337—21no—
42C22×D337—22no—
43C22×D337—22no—
44Dic674D411yescharacteristic
45Dic674C2311yescharacteristic
46D1348—22no—
47D1348—21no—
48D1348D411yescharacteristic
49D1348C2312yes—
50D1348C2311yescharacteristic
51C22⋊D337—21no—
52C22⋊D337—22no—
53C22⋊D337—22no—
54C22⋊D337—22no—
55C22⋊D337D412yes—
56C2×Dic337—22no—
57C4×D337—21no—
58C4×D337—22no—
59C4×D337C2312yes—
60#14582106C2311yescharacteristic
61#14582106C2311yescharacteristic
62C23×C337—21no—
63Q8×C337C2311yescharacteristic
64D4×C337—21no—
65D4×C337—21no—
66D4×C337C2311yescharacteristic
67D4×C337C2312yes—
68C2×C4×C337—21no—
69C2×C4×C337C2311yescharacteristic
70C8×C337C2312yes—
71C22×C337—21no—
72C22×C337—22no—
73C22×C337—81no—
74C22×C337C2×D411yescharacteristic
75C2×D337—21no—
76C2×D337—22no—
77C2×D337—82no—
78C2×D337—82no—
79C2×D337—81no—
80C2×D337C2×D412yes—
81C4×C337—21no—
82C4×C337C2×D411yescharacteristic
83C4×C337C2411yescharacteristic
84Dic337—21no—
85Dic337C2×D412yes—
86C2×C337—21no—
87C2×C337—41no—
88C2×C337—42no—
89C2×C337C22×D411yescharacteristic
90D337—22no—
91D337—41no—
92D337—42no—
93C337#57611yescharacteristic, complemented, Sylow
94#576—3371nomaximal, Sylow
95H(5,𝔽2)—3371no—
96H(5,𝔽2)—3371no—
97AGL(1,ℤ/8)—3372no—
98AGL(1,ℤ/8)—3372no—
99AGL(1,ℤ/8)—3371no—
100AGL(1,ℤ/8)D33711yescharacteristic, complemented
101#135—3371no—
102#135—3372no—
103C2×D8—3371no—
104C2×D8—3372no—
105#131—3371no—
106#41—3371no—
107#41—3372no—
108#41—3372no—
109#41—3371no—
110#41—6741no—
111#41—6741no—
112#41C2×D33711yescharacteristic, complemented
113C2×D4—3372no—
114C2×D4—3372no—
115C2×D4—3371no—
116C2×D4—6741no—
117C2×D4—6742no—
118C2×D4—6741no—
119C2×D4—6742no—
120C2×D4C2×D33711yescharacteristic, complemented
121Q16—3371no—
122SD16—3371no—
123SD16—3371no—
124SD16—3372no—
125SD16C2×D33712yescomplemented
126D8—3372no—
127D8—3371no—
128D8—3372no—
129D8—3372no—
130D8C2×D33712yescomplemented
131M16—3372no—
132M16C2×D33711yescharacteristic, complemented
133C2×C8—3371no—
134C2×C8—3372no—
135C23—21no—
136C23—6742no—
137C23—6742no—
138C23—6741no—
139Q8—3371no—
140Q8—3371no—
141Q8C22×D33711yescharacteristic
142D4—21no—
143D4—21no—
144D4—3372no—
145D4—3371no—
146D4—3372no—
147D4—3371no—
148D4—6742no—
149D4—6742no—
150D4—6742no—
151D4—6742no—
152D4—6741no—
153D4—6741no—
154D4C22×D33712yes—
155D4C22×D33711yescharacteristic
156C2×C4—21no—
157C2×C4—3372no—
158C2×C4—6742no—
159C2×C4—6742no—
160C2×C4—6741no—
161C2×C4C22×D33711yescharacteristic
162C8—3371no—
163C8—3371no—
164C8C22×D33712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—3372no—
169C22—26961no—
170C22—26962no—
171C22—26962no—
172C22—6742no—
173C22—6741no—
174C22D4×D33711yescharacteristic
175C4—21no—
176C4—3372no—
177C4—6741no—
178C4D4×D33711yescharacteristic
179C4C23×D33711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—13482no—
184C2—13481no—
185C2—6742no—
186C2C2×D4×D33711yescharacteristic
subgroupG/H
centerC2C2×D4×D337
commutator, cosocleC4×C337C24
FrattiniC4C23×D337
FittingAGL(1,ℤ/8)×C337C2
socleC2×C337C22×D4
hypercenterAGL(1,ℤ/8)D337
hypocenterC337#576
LCSG ▷ C4×C337 ▷ C2×C337 ▷ C337
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #24925005C2semidirect
2. #24925004C2semidirect
3. #24924987C2semidirect
4. #24924986C2semidirect
5. #24924983C2semidirect
6. #24924980C2semidirect
7. #24924979C2semidirect
8. C2×#19129170C2semidirect
9. D8×D337C2semidirect
10. #24924967C2semidirect
11. AGL(1,ℤ/8)×C337C2semidirect
12. #19129186C22semidirect
15. #19129185C22semidirect
17. #19129184C22semidirect
19. C2×D1348C22semidirect
23. D4×D337C22semidirect
24. D4×D337C22semidirect
25. #19129175C22semidirect
26. #19129174C22semidirect
27. #19129173C22semidirect
28. #19129172C22semidirect
29. D2696C22semidirect
30. #19129170C22semidirect
31. #19129170C22semidirect
32. #19129170C22semidirect
33. #19129168C22semidirect
34. C2×#14582106C22semidirect
35. C8×D337C22semidirect
36. #41×C337C22semidirect
37. C2×D4×C337C22semidirect
38. SD16×C337C22semidirect
39. D8×C337C22semidirect
40. M16×C337C22semidirect
93. C337#576semidirect
100. AGL(1,ℤ/8)D337semidirect
112. #41C2×D337semidirect
120. C2×D4C2×D337semidirect
125. SD16C2×D337semidirect
130. D8C2×D337semidirect
132. M16C2×D337semidirect

Maximal subgroups

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

subgroupindex
#249250052
#249250042
#249249872
#249249862
#249249832
#249249802
#249249792
C2×#191291702
D8×D3372
#249249672
AGL(1,ℤ/8)×C3372
#576337