Finite Group Databasebeta

← #36378591

Subgroups — #36378591

Order 22592.

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#26015038C211yesmaximal, characteristic, complemented
2#26015037C211yesmaximal, characteristic, complemented
3#26015020C211yesmaximal, characteristic, complemented
4#26015019C212yesmaximal, complemented
5#26015016C212yesmaximal, complemented
6#26015013C212yesmaximal, complemented
7#26015012C211yesmaximal, characteristic, complemented
8C2×#19150722C211yesmaximal, characteristic, complemented
9D8×D353C212yesmaximal, complemented
10#26015000C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C353C211yesmaximal, characteristic, complemented
12#19150738C2212yescomplemented
13#19150737—21no—
14#19150737C2211yescharacteristic
15#19150737C2211yescharacteristic, complemented
16#19150736—21no—
17#19150736C2212yescomplemented
18C2×D1412—21no—
19C2×D1412C2211yescharacteristic, complemented
20C2×C22⋊D353—22no—
21D4×D353—21no—
22D4×D353—22no—
23D4×D353C2212yescomplemented
24D4×D353C2212yescomplemented
25#19150727C2211yescharacteristic, complemented
26#19150726C2212yescomplemented
27#19150725C2211yescharacteristic, complemented
28#19150724C2211yescharacteristic, complemented
29D2824C2212yescomplemented
30#19150722C2211yescharacteristic, complemented
31#19150722C2212yescomplemented
32#19150722C2212yescomplemented
33#19150720C2212yescomplemented
34C2×#15667725C2211yescharacteristic, complemented
35C8×D353C2212yescomplemented
36#41×C353C2211yescharacteristic, complemented
37C2×D4×C353C2211yescharacteristic, complemented
38SD16×C353C2212yescomplemented
39D8×C353C2212yescomplemented
40M16×C353C2211yescharacteristic, complemented
41C22×D353—21no—
42C22×D353—22no—
43C22×D353—22no—
44Dic706D411yescharacteristic
45Dic706C2311yescharacteristic
46D1412—22no—
47D1412—21no—
48D1412D411yescharacteristic
49D1412C2312yes—
50D1412C2311yescharacteristic
51C22⋊D353—21no—
52C22⋊D353—22no—
53C22⋊D353—22no—
54C22⋊D353—22no—
55C22⋊D353D412yes—
56C2×Dic353—22no—
57C4×D353—21no—
58C4×D353—22no—
59C4×D353C2312yes—
60#15667725C2311yescharacteristic
61#15667725C2311yescharacteristic
62C23×C353—21no—
63Q8×C353C2311yescharacteristic
64D4×C353—21no—
65D4×C353—21no—
66D4×C353C2311yescharacteristic
67D4×C353C2312yes—
68C2×C4×C353—21no—
69C2×C4×C353C2311yescharacteristic
70C8×C353C2312yes—
71C22×C353—21no—
72C22×C353—22no—
73C22×C353—81no—
74C22×C353C2×D411yescharacteristic
75C2×D353—21no—
76C2×D353—22no—
77C2×D353—82no—
78C2×D353—82no—
79C2×D353—81no—
80C2×D353C2×D412yes—
81C4×C353—21no—
82C4×C353C2×D411yescharacteristic
83C4×C353C2411yescharacteristic
84Dic353—21no—
85Dic353C2×D412yes—
86C2×C353—21no—
87C2×C353—41no—
88C2×C353—42no—
89C2×C353C22×D411yescharacteristic
90D353—22no—
91D353—41no—
92D353—42no—
93C353#57611yescharacteristic, complemented, Sylow
94#576—3531nomaximal, Sylow
95H(5,𝔽2)—3531no—
96H(5,𝔽2)—3531no—
97AGL(1,ℤ/8)—3532no—
98AGL(1,ℤ/8)—3532no—
99AGL(1,ℤ/8)—3531no—
100AGL(1,ℤ/8)D35311yescharacteristic, complemented
101#135—3531no—
102#135—3532no—
103C2×D8—3531no—
104C2×D8—3532no—
105#131—3531no—
106#41—3531no—
107#41—3532no—
108#41—3532no—
109#41—3531no—
110#41—7061no—
111#41—7061no—
112#41C2×D35311yescharacteristic, complemented
113C2×D4—3532no—
114C2×D4—3532no—
115C2×D4—3531no—
116C2×D4—7061no—
117C2×D4—7062no—
118C2×D4—7061no—
119C2×D4—7062no—
120C2×D4C2×D35311yescharacteristic, complemented
121Q16—3531no—
122SD16—3531no—
123SD16—3531no—
124SD16—3532no—
125SD16C2×D35312yescomplemented
126D8—3532no—
127D8—3531no—
128D8—3532no—
129D8—3532no—
130D8C2×D35312yescomplemented
131M16—3532no—
132M16C2×D35311yescharacteristic, complemented
133C2×C8—3531no—
134C2×C8—3532no—
135C23—21no—
136C23—7062no—
137C23—7062no—
138C23—7061no—
139Q8—3531no—
140Q8—3531no—
141Q8C22×D35311yescharacteristic
142D4—21no—
143D4—21no—
144D4—3532no—
145D4—3531no—
146D4—3532no—
147D4—3531no—
148D4—7062no—
149D4—7062no—
150D4—7062no—
151D4—7062no—
152D4—7061no—
153D4—7061no—
154D4C22×D35312yes—
155D4C22×D35311yescharacteristic
156C2×C4—21no—
157C2×C4—3532no—
158C2×C4—7062no—
159C2×C4—7062no—
160C2×C4—7061no—
161C2×C4C22×D35311yescharacteristic
162C8—3531no—
163C8—3531no—
164C8C22×D35312yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—28241no—
169C22—28242no—
170C22—28242no—
171C22—3532no—
172C22—7062no—
173C22—7061no—
174C22D4×D35311yescharacteristic
175C4—21no—
176C4—3532no—
177C4—7061no—
178C4D4×D35311yescharacteristic
179C4C23×D35311yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—14122no—
184C2—14121no—
185C2—7062no—
186C2C2×D4×D35311yescharacteristic
subgroupG/H
centerC2C2×D4×D353
commutator, cosocleC4×C353C24
FrattiniC4C23×D353
FittingAGL(1,ℤ/8)×C353C2
socleC2×C353C22×D4
hypercenterAGL(1,ℤ/8)D353
hypocenterC353#576
LCSG ▷ C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26015038C2semidirect
2. #26015037C2semidirect
3. #26015020C2semidirect
4. #26015019C2semidirect
5. #26015016C2semidirect
6. #26015013C2semidirect
7. #26015012C2semidirect
8. C2×#19150722C2semidirect
9. D8×D353C2semidirect
10. #26015000C2semidirect
11. AGL(1,ℤ/8)×C353C2semidirect
12. #19150738C22semidirect
15. #19150737C22semidirect
17. #19150736C22semidirect
19. C2×D1412C22semidirect
23. D4×D353C22semidirect
24. D4×D353C22semidirect
25. #19150727C22semidirect
26. #19150726C22semidirect
27. #19150725C22semidirect
28. #19150724C22semidirect
29. D2824C22semidirect
30. #19150722C22semidirect
31. #19150722C22semidirect
32. #19150722C22semidirect
33. #19150720C22semidirect
34. C2×#15667725C22semidirect
35. C8×D353C22semidirect
36. #41×C353C22semidirect
37. C2×D4×C353C22semidirect
38. SD16×C353C22semidirect
39. D8×C353C22semidirect
40. M16×C353C22semidirect
93. C353#576semidirect
100. AGL(1,ℤ/8)D353semidirect
112. #41C2×D353semidirect
120. C2×D4C2×D353semidirect
125. SD16C2×D353semidirect
130. D8C2×D353semidirect
132. M16C2×D353semidirect

Maximal subgroups

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

subgroupindex
#260150382
#260150372
#260150202
#260150192
#260150162
#260150132
#260150122
C2×#191507222
D8×D3532
#260150002
AGL(1,ℤ/8)×C3532
#576353