Finite Group Databasebeta

← #27164814

Subgroups — #27164814

Order 12736.

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×#15691885C211yesmaximal, characteristic, complemented
2#41×D199C211yesmaximal, characteristic, complemented
3#20263988C211yesmaximal, characteristic, complemented
4#20263985C211yesmaximal, characteristic, complemented
5#20263984C212yesmaximal, complemented
6#20263981C211yesmaximal, characteristic, complemented
7#20263979C212yesmaximal, complemented
8#20263976C212yesmaximal, complemented
9#20263974C212yesmaximal, complemented
10M16×D199C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C199C211yesmaximal, characteristic, complemented
12#15691887C2211yescharacteristic, complemented
13#15691886—21no—
14#15691886C2211yescharacteristic
15#15691885—21no—
16#15691885—21no—
17#15691885—21no—
18#15691885C2212yes—
19#15691885C2211yescharacteristic, complemented
20#15691885C2212yescomplemented
21C2×Dic398C2211yescharacteristic, complemented
22Q8×D199C2211yescharacteristic, complemented
23C2×C22⋊D199—21no—
24D4×D199—21no—
25D4×D199C2211yescharacteristic, complemented
26C22×Dic199—21no—
27C2×C4×D199—21no—
28C2×C4×D199C2211yescharacteristic
29Dic796C2212yescomplemented
30#15691876C2212yescomplemented
31#15691875C2212yescomplemented
32#15691874C2212yescomplemented
33#15691874C2212yescomplemented
34#15691871C2212yescomplemented
35#15691870C2211yescharacteristic, complemented
36#15691869C2212yescomplemented
37C8×D199C2212yescomplemented
38#41×C199C2211yescharacteristic
39C2×D4×C199C2211yescharacteristic, complemented
40SD16×C199C2212yescomplemented
41D8×C199C2212yescomplemented
42M16×C199C2211yescharacteristic, complemented
43C22×D199—21no—
44C22×D199D411yescharacteristic
45Dic398—21no—
46Dic398—21no—
47Dic398C2311yescharacteristic
48Dic398C2312yes—
49D796—21no—
50D796C2311yescharacteristic
51C22⋊D199—21no—
52C22⋊D199—21no—
53C22⋊D199—22no—
54C22⋊D199—22no—
55C22⋊D199—21no—
56C2×Dic199—21no—
57C2×Dic199—21no—
58C2×Dic199—22no—
59C2×Dic199—22no—
60C2×Dic199D411yescharacteristic
61C4×D199—21no—
62C4×D199—21no—
63C4×D199—21no—
64C4×D199D411yescharacteristic
65C4×D199D411yescharacteristic
66C4×D199C2311yescharacteristic
67C4×D199C2311yescharacteristic
68C199⋊C8C2312yes—
69C23×C199—21no—
70Q8×C199C2311yescharacteristic
71D4×C199—21no—
72D4×C199—21no—
73D4×C199C2311yescharacteristic
74D4×C199C2312yes—
75C2×C4×C199—21no—
76C2×C4×C199C2311yescharacteristic
77C8×C199C2312yes—
78C22×C199—21no—
79C22×C199—22no—
80C22×C199—42no—
81C22×C199C2×D411yescharacteristic
82C2×D199—21no—
83C2×D199—41no—
84C2×D199—42no—
85C2×D199C2×D411yescharacteristic
86C2×D199C2×D411yescharacteristic
87C4×C199—21no—
88C4×C199C2×D411yescharacteristic
89C4×C199C2411yescharacteristic
90Dic199—21no—
91Dic199—22no—
92Dic199C2×D411yescharacteristic
93Dic199C2×D411yescharacteristic
94C2×C199—21no—
95C2×C199—41no—
96C2×C199—42no—
97C2×C199C22×D411yescharacteristic
98D199—21no—
99D199—21no—
100D199—41no—
101C199#57511yescharacteristic, complemented, Sylow
102#575—1991nomaximal, Sylow
103C2×#41—1991no—
104C2×#41—1991no—
105#137—1991no—
106#137—1992no—
107#137—1991no—
108AGL(1,ℤ/8)—1991no—
109AGL(1,ℤ/8)—1992no—
110AGL(1,ℤ/8)D19911yescharacteristic, complemented
111#135—1992no—
112#135—1992no—
113C2×M16—1991no—
114#41—3981no—
115#41—3981no—
116#41—3981no—
117#41—3981no—
118#41—1991no—
119#41—1992no—
120#41—1992no—
121#41—1991no—
122#41—1991no—
123#41C2×D19911yescharacteristic
124C2×Q8—1991no—
125C2×Q8—1991no—
126C2×D4—3981no—
127C2×D4—3981no—
128C2×D4—1991no—
129C2×D4C2×D19911yescharacteristic, complemented
130C22×C4—3981no—
131C22×C4—3981no—
132C22×C4—1991no—
133Q16—1992no—
134Q16—1992no—
135SD16—1992no—
136SD16—1992no—
137SD16—1992no—
138SD16C2×D19912yescomplemented
139D8—1992no—
140D8C2×D19912yescomplemented
141M16—1991no—
142M16—1992no—
143M16C2×D19911yescharacteristic, complemented
144C2×C8—1992no—
145C23—21no—
146C23—3981no—
147C23—1991no—
148Q8—3981no—
149Q8—3981no—
150Q8—1991no—
151Q8—1992no—
152Q8C22×D19911yescharacteristic
153D4—21no—
154D4—21no—
155D4—3981no—
156D4—3981no—
157D4—3982no—
158D4—3982no—
159D4—3981no—
160D4—3981no—
161D4—1991no—
162D4C22×D19912yes—
163D4C22×D19911yescharacteristic
164C2×C4—21no—
165C2×C4—3981no—
166C2×C4—3981no—
167C2×C4—3981no—
168C2×C4—3981no—
169C2×C4—3982no—
170C2×C4—3982no—
171C2×C4—3981no—
172C2×C4—1991no—
173C2×C4—1991no—
174C2×C4—1991no—
175C2×C4—1991no—
176C2×C4—1991no—
177C2×C4C22×D19911yescharacteristic
178C8—1992no—
179C8C22×D19912yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—7961no—
184C22—7962no—
185C22—3981no—
186C22—1991no—
187C22—1991no—
188C22D4×D19911yescharacteristic
189C4—21no—
190C4—3981no—
191C4—3982no—
192C4—1991no—
193C4—1991no—
194C4D4×D19911yescharacteristic
195C4C23×D19911yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—7961no—
200C2—3981no—
201C2—3981no—
202C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC2C2×D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
FittingAGL(1,ℤ/8)×C199C2
socleC2×C199C22×D4
hypercenterAGL(1,ℤ/8)D199
hypocenterC199#575
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15691885C2semidirect
2. #41×D199C2semidirect
3. #20263988C2semidirect
4. #20263985C2semidirect
5. #20263984C2semidirect
6. #20263981C2semidirect
7. #20263979C2semidirect
8. #20263976C2semidirect
9. #20263974C2semidirect
10. M16×D199C2semidirect
11. AGL(1,ℤ/8)×C199C2semidirect
12. #15691887C22semidirect
19. #15691885C22semidirect
20. #15691885C22semidirect
21. C2×Dic398C22semidirect
22. Q8×D199C22semidirect
25. D4×D199C22semidirect
29. Dic796C22semidirect
30. #15691876C22semidirect
31. #15691875C22semidirect
32. #15691874C22semidirect
33. #15691874C22semidirect
34. #15691871C22semidirect
35. #15691870C22semidirect
36. #15691869C22semidirect
37. C8×D199C22semidirect
39. C2×D4×C199C22semidirect
40. SD16×C199C22semidirect
41. D8×C199C22semidirect
42. M16×C199C22semidirect
101. C199#575semidirect
110. AGL(1,ℤ/8)D199semidirect
129. C2×D4C2×D199semidirect
138. SD16C2×D199semidirect
140. D8C2×D199semidirect
143. M16C2×D199semidirect

Maximal subgroups

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

subgroupindex
C2×#156918852
#41×D1992
#202639882
#202639852
#202639842
#202639812
#202639792
#202639762
#202639742
M16×D1992
AGL(1,ℤ/8)×C1992
#575199