Finite Group Databasebeta

← #37554990

Subgroups — #37554990

Order 23488.

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#26044307C211yesmaximal, characteristic, complemented
2#26044306C211yesmaximal, characteristic, complemented
3#26044289C211yesmaximal, characteristic, complemented
4#26044288C212yesmaximal, complemented
5#26044285C212yesmaximal, complemented
6#26044282C212yesmaximal, complemented
7#26044281C211yesmaximal, characteristic, complemented
8C2×#19154562C211yesmaximal, characteristic, complemented
9D8×D367C212yesmaximal, complemented
10#26044269C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C367C211yesmaximal, characteristic, complemented
12#19154578C2212yescomplemented
13#19154577—21no—
14#19154577C2211yescharacteristic
15#19154577C2211yescharacteristic, complemented
16#19154576—21no—
17#19154576C2212yescomplemented
18C2×D1468—21no—
19C2×D1468C2211yescharacteristic, complemented
20C2×C22⋊D367—22no—
21D4×D367—21no—
22D4×D367—22no—
23D4×D367C2212yescomplemented
24D4×D367C2212yescomplemented
25#19154567C2211yescharacteristic, complemented
26#19154566C2212yescomplemented
27#19154565C2211yescharacteristic, complemented
28#19154564C2211yescharacteristic, complemented
29D2936C2212yescomplemented
30#19154562C2211yescharacteristic, complemented
31#19154562C2212yescomplemented
32#19154562C2212yescomplemented
33#19154560C2212yescomplemented
34C2×C367⋊C8C2211yescharacteristic, complemented
35C8×D367C2212yescomplemented
36#41×C367C2211yescharacteristic, complemented
37C2×D4×C367C2211yescharacteristic, complemented
38SD16×C367C2212yescomplemented
39D8×C367C2212yescomplemented
40M16×C367C2211yescharacteristic, complemented
41C22×D367—21no—
42C22×D367—22no—
43C22×D367—22no—
44Dic734D411yescharacteristic
45Dic734C2311yescharacteristic
46D1468—22no—
47D1468—21no—
48D1468D411yescharacteristic
49D1468C2312yes—
50D1468C2311yescharacteristic
51C22⋊D367—21no—
52C22⋊D367—22no—
53C22⋊D367—22no—
54C22⋊D367—22no—
55C22⋊D367D412yes—
56C2×Dic367—22no—
57C4×D367—21no—
58C4×D367—22no—
59C4×D367C2312yes—
60C367⋊C8C2311yescharacteristic
61C367⋊C8C2311yescharacteristic
62C23×C367—21no—
63Q8×C367C2311yescharacteristic
64D4×C367—21no—
65D4×C367—21no—
66D4×C367C2311yescharacteristic
67D4×C367C2312yes—
68C2×C4×C367—21no—
69C2×C4×C367C2311yescharacteristic
70C8×C367C2312yes—
71C22×C367—21no—
72C22×C367—22no—
73C22×C367—81no—
74C22×C367C2×D411yescharacteristic
75C2×D367—21no—
76C2×D367—22no—
77C2×D367—82no—
78C2×D367—82no—
79C2×D367—81no—
80C2×D367C2×D412yes—
81C4×C367—21no—
82C4×C367C2×D411yescharacteristic
83C4×C367C2411yescharacteristic
84Dic367—21no—
85Dic367C2×D412yes—
86C2×C367—21no—
87C2×C367—41no—
88C2×C367—42no—
89C2×C367C22×D411yescharacteristic
90D367—22no—
91D367—41no—
92D367—42no—
93C367#57611yescharacteristic, complemented, Sylow
94#576—3671nomaximal, Sylow
95H(5,𝔽2)—3671no—
96H(5,𝔽2)—3671no—
97AGL(1,ℤ/8)—3672no—
98AGL(1,ℤ/8)—3672no—
99AGL(1,ℤ/8)—3671no—
100AGL(1,ℤ/8)D36711yescharacteristic, complemented
101#135—3671no—
102#135—3672no—
103C2×D8—3671no—
104C2×D8—3672no—
105#131—3671no—
106#41—3671no—
107#41—3672no—
108#41—3672no—
109#41—3671no—
110#41—7341no—
111#41—7341no—
112#41C2×D36711yescharacteristic, complemented
113C2×D4—3672no—
114C2×D4—3672no—
115C2×D4—3671no—
116C2×D4—7341no—
117C2×D4—7342no—
118C2×D4—7341no—
119C2×D4—7342no—
120C2×D4C2×D36711yescharacteristic, complemented
121Q16—3671no—
122SD16—3671no—
123SD16—3671no—
124SD16—3672no—
125SD16C2×D36712yescomplemented
126D8—3672no—
127D8—3671no—
128D8—3672no—
129D8—3672no—
130D8C2×D36712yescomplemented
131M16—3672no—
132M16C2×D36711yescharacteristic, complemented
133C2×C8—3671no—
134C2×C8—3672no—
135C23—21no—
136C23—7342no—
137C23—7342no—
138C23—7341no—
139Q8—3671no—
140Q8—3671no—
141Q8C22×D36711yescharacteristic
142D4—21no—
143D4—21no—
144D4—3672no—
145D4—3671no—
146D4—3672no—
147D4—3671no—
148D4—7342no—
149D4—7342no—
150D4—7342no—
151D4—7342no—
152D4—7341no—
153D4—7341no—
154D4C22×D36712yes—
155D4C22×D36711yescharacteristic
156C2×C4—21no—
157C2×C4—3672no—
158C2×C4—7342no—
159C2×C4—7342no—
160C2×C4—7341no—
161C2×C4C22×D36711yescharacteristic
162C8—3671no—
163C8—3671no—
164C8C22×D36712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—3672no—
169C22—29361no—
170C22—29362no—
171C22—29362no—
172C22—7342no—
173C22—7341no—
174C22D4×D36711yescharacteristic
175C4—21no—
176C4—3672no—
177C4—7341no—
178C4D4×D36711yescharacteristic
179C4C23×D36711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—14682no—
184C2—14681no—
185C2—7342no—
186C2C2×D4×D36711yescharacteristic
subgroupG/H
centerC2C2×D4×D367
commutator, cosocleC4×C367C24
FrattiniC4C23×D367
FittingAGL(1,ℤ/8)×C367C2
socleC2×C367C22×D4
hypercenterAGL(1,ℤ/8)D367
hypocenterC367#576
LCSG ▷ C4×C367 ▷ C2×C367 ▷ C367
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26044307C2semidirect
2. #26044306C2semidirect
3. #26044289C2semidirect
4. #26044288C2semidirect
5. #26044285C2semidirect
6. #26044282C2semidirect
7. #26044281C2semidirect
8. C2×#19154562C2semidirect
9. D8×D367C2semidirect
10. #26044269C2semidirect
11. AGL(1,ℤ/8)×C367C2semidirect
12. #19154578C22semidirect
15. #19154577C22semidirect
17. #19154576C22semidirect
19. C2×D1468C22semidirect
23. D4×D367C22semidirect
24. D4×D367C22semidirect
25. #19154567C22semidirect
26. #19154566C22semidirect
27. #19154565C22semidirect
28. #19154564C22semidirect
29. D2936C22semidirect
30. #19154562C22semidirect
31. #19154562C22semidirect
32. #19154562C22semidirect
33. #19154560C22semidirect
34. C2×C367⋊C8C22semidirect
35. C8×D367C22semidirect
36. #41×C367C22semidirect
37. C2×D4×C367C22semidirect
38. SD16×C367C22semidirect
39. D8×C367C22semidirect
40. M16×C367C22semidirect
93. C367#576semidirect
100. AGL(1,ℤ/8)D367semidirect
112. #41C2×D367semidirect
120. C2×D4C2×D367semidirect
125. SD16C2×D367semidirect
130. D8C2×D367semidirect
132. M16C2×D367semidirect

Maximal subgroups

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

subgroupindex
#260443072
#260443062
#260442892
#260442882
#260442852
#260442822
#260442812
C2×#191545622
D8×D3672
#260442692
AGL(1,ℤ/8)×C3672
#576367