Finite Group Databasebeta

← #28412280

Subgroups — #28412280

Order 14656.

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×#16814667C211yesmaximal, characteristic, complemented
2#41×D229C211yesmaximal, characteristic, complemented
3#20301020C212yesmaximal, complemented
4#20301019C211yesmaximal, characteristic, complemented
5#20301016C211yesmaximal, characteristic, complemented
6#20301015C212yesmaximal, complemented
7#20301013C211yesmaximal, characteristic, complemented
8#20301011C212yesmaximal, complemented
9#20301009C212yesmaximal, complemented
10M16×D229C211yesmaximal, characteristic, complemented
11#137×C229C211yesmaximal, characteristic, complemented
12#16814667—21no—
13#16814667—21no—
14#16814667C2212yes—
15#16814667C2212yescomplemented
16#16814667C2211yescharacteristic, complemented
17#16814666—21no—
18#16814666C2211yescharacteristic
19#16814665—21no—
20#16814665C2211yescharacteristic, complemented
21Q8×D229C2211yescharacteristic, complemented
22C2×D916—21no—
23C2×D916C2211yescharacteristic, complemented
24D4×D229—21no—
25D4×D229C2211yescharacteristic, complemented
26C2×C4×D229—21no—
27C2×C4×D229—21no—
28C2×C4×D229C2211yescharacteristic
29#16814656C2212yescomplemented
30#16814655C2212yescomplemented
31#16814653C2212yescomplemented
32#16814653C2212yescomplemented
33D1832C2212yescomplemented
34#16814651C2212yescomplemented
35#16814650C2211yescharacteristic, complemented
36#16814649C2212yescomplemented
37C8×D229C2212yescomplemented
38#41×C229C2211yescharacteristic
39C2×Q8×C229C2211yescharacteristic, complemented
40Q16×C229C2212yescomplemented
41SD16×C229C2212yescomplemented
42M16×C229C2211yescharacteristic, complemented
43C22×D229—21no—
44C22×D229—21no—
45C22×D229D411yescharacteristic
46Dic458—21no—
47Dic458C2311yescharacteristic
48D916—21no—
49D916—22no—
50D916—22no—
51D916—21no—
52D916C2312yes—
53D916C2311yescharacteristic
54C22⋊D229—21no—
55C22⋊D229—21no—
56C22⋊D229—21no—
57C2×Dic229—21no—
58C2×Dic229D411yescharacteristic
59C4×D229—21no—
60C4×D229—21no—
61C4×D229—21no—
62C4×D229—22no—
63C4×D229—22no—
64C4×D229D411yescharacteristic
65C4×D229D411yescharacteristic
66C4×D229C2311yescharacteristic
67C4×D229C2311yescharacteristic
68#14270080C2312yes—
69Q8×C229—21no—
70Q8×C229C2312yes—
71Q8×C229C2311yescharacteristic
72D4×C229—21no—
73D4×C229C2311yescharacteristic
74C2×C4×C229—21no—
75C2×C4×C229—21no—
76C2×C4×C229C2311yescharacteristic
77C8×C229C2312yes—
78C22×C229—21no—
79C22×C229C2×D411yescharacteristic
80C2×D229—21no—
81C2×D229—22no—
82C2×D229—41no—
83C2×D229—42no—
84C2×D229—42no—
85C2×D229C2×D411yescharacteristic
86C2×D229C2×D411yescharacteristic
87C4×C229—21no—
88C4×C229—22no—
89C4×C229C2×D411yescharacteristic
90C4×C229C2411yescharacteristic
91Dic229—21no—
92Dic229C2×D411yescharacteristic
93Dic229C2×D411yescharacteristic
94C2×C229—21no—
95C2×C229—41no—
96C2×C229C22×D411yescharacteristic
97D229—21no—
98D229—21no—
99D229—41no—
100D229—42no—
101C229#57511yescharacteristic, complemented, Sylow
102#575—2291nomaximal, Sylow
103C2×#41—2291no—
104C2×#41—2291no—
105#137—2291no—
106#137—2292no—
107#137D22911yescharacteristic, complemented
108AGL(1,ℤ/8)—2291no—
109AGL(1,ℤ/8)—2292no—
110AGL(1,ℤ/8)—2291no—
111#135—2292no—
112#135—2292no—
113C2×M16—2291no—
114#41—4581no—
115#41—4581no—
116#41—4581no—
117#41—4581no—
118#41—2292no—
119#41—2291no—
120#41—2292no—
121#41—2291no—
122#41—2291no—
123#41C2×D22911yescharacteristic
124C2×Q8—2291no—
125C2×Q8C2×D22911yescharacteristic, complemented
126C2×D4—4581no—
127C2×D4—4581no—
128C2×D4—2291no—
129C2×D4—2291no—
130C22×C4—4581no—
131C22×C4—4581no—
132C22×C4—2291no—
133Q16—2292no—
134Q16C2×D22912yescomplemented
135SD16—2292no—
136SD16—2292no—
137SD16—2292no—
138SD16C2×D22912yescomplemented
139D8—2292no—
140D8—2292no—
141M16—2291no—
142M16—2292no—
143M16C2×D22911yescharacteristic, complemented
144C2×C8—2292no—
145C23—4581no—
146C23—4581no—
147C23—2291no—
148Q8—21no—
149Q8—4581no—
150Q8—2291no—
151Q8C22×D22911yescharacteristic
152Q8C22×D22912yes—
153D4—21no—
154D4—4581no—
155D4—4582no—
156D4—4582no—
157D4—4581no—
158D4—4581no—
159D4—4581no—
160D4—4581no—
161D4—2292no—
162D4—2291no—
163D4C22×D22911yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—4581no—
167C2×C4—4581no—
168C2×C4—4582no—
169C2×C4—4582no—
170C2×C4—4581no—
171C2×C4—4581no—
172C2×C4—2291no—
173C2×C4—2291no—
174C2×C4—2291no—
175C2×C4—2291no—
176C2×C4—2291no—
177C2×C4C22×D22911yescharacteristic
178C8—2292no—
179C8C22×D22912yes—
180C22—21no—
181C22—9162no—
182C22—9161no—
183C22—9162no—
184C22—4582no—
185C22—4581no—
186C22—2291no—
187C22—2291no—
188C22D4×D22911yescharacteristic
189C4—21no—
190C4—22no—
191C4—4581no—
192C4—2291no—
193C4—2291no—
194C4D4×D22911yescharacteristic
195C4C23×D22911yescharacteristic
196C2—21no—
197C2—41no—
198C2—9162no—
199C2—9161no—
200C2—4581no—
201C2—4581no—
202C2C2×D4×D22911yescharacteristic
subgroupG/H
centerC2C2×D4×D229
commutator, cosocleC4×C229C24
FrattiniC4C23×D229
Fitting#137×C229C2
socleC2×C229C22×D4
hypercenter#137D229
hypocenterC229#575
LCSG ▷ C4×C229 ▷ C2×C229 ▷ C229
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#16814667C2semidirect
2. #41×D229C2semidirect
3. #20301020C2semidirect
4. #20301019C2semidirect
5. #20301016C2semidirect
6. #20301015C2semidirect
7. #20301013C2semidirect
8. #20301011C2semidirect
9. #20301009C2semidirect
10. M16×D229C2semidirect
11. #137×C229C2semidirect
15. #16814667C22semidirect
16. #16814667C22semidirect
20. #16814665C22semidirect
21. Q8×D229C22semidirect
23. C2×D916C22semidirect
25. D4×D229C22semidirect
29. #16814656C22semidirect
30. #16814655C22semidirect
31. #16814653C22semidirect
32. #16814653C22semidirect
33. D1832C22semidirect
34. #16814651C22semidirect
35. #16814650C22semidirect
36. #16814649C22semidirect
37. C8×D229C22semidirect
39. C2×Q8×C229C22semidirect
40. Q16×C229C22semidirect
41. SD16×C229C22semidirect
42. M16×C229C22semidirect
101. C229#575semidirect
107. #137D229semidirect
125. C2×Q8C2×D229semidirect
134. Q16C2×D229semidirect
138. SD16C2×D229semidirect
143. M16C2×D229semidirect

Maximal subgroups

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

subgroupindex
C2×#168146672
#41×D2292
#203010202
#203010192
#203010162
#203010152
#203010132
#203010112
#203010092
M16×D2292
#137×C2292
#575229