Finite Group Databasebeta

← #28384390

Subgroups — #28384390

Order 14272.

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

normal subgroups only

This group has 258 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#20297351C211yesmaximal, characteristic, complemented
2#20297349C211yesmaximal, characteristic, complemented
3#20297337C211yesmaximal, characteristic, complemented
4#20297333C212yesmaximal, complemented
5#20297328C212yesmaximal, complemented
6#20297324C211yesmaximal, characteristic, complemented
7#20297323C212yesmaximal, complemented
8C2×#16813158C211yesmaximal, characteristic, complemented
9SD16×D223C212yesmaximal, complemented
10#20297312C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C223C211yesmaximal, characteristic, complemented
12#16813170—21no—
13#16813170C2211yescharacteristic
14#16813170C2211yescharacteristic
15#16813169—21no—
16#16813169—22no—
17#16813169C2212yes—
18#16813169C2212yescomplemented
19C2×Dic446—21no—
20C2×Dic446C2211yescharacteristic, complemented
21Q8×D223C2212yescomplemented
22C2×C22⋊D223—22no—
23D4×D223—21no—
24D4×D223C2212yes—
25Dic892C2212yescomplemented
26#16813160C2211yescharacteristic, complemented
27#16813159C2212yescomplemented
28#16813158C2212yescomplemented
29#16813158C2212yescomplemented
30#16813158C2211yescharacteristic, complemented
31#16813157C2211yescharacteristic, complemented
32#16813155C2211yescharacteristic, complemented
33#16813153C2212yescomplemented
34C2×C223⋊C8C2211yescharacteristic, complemented
35C8×D223C2212yescomplemented
36#41×C223C2211yescharacteristic, complemented
37C2×D4×C223C2211yescharacteristic
38SD16×C223C2212yescomplemented
39D8×C223C2212yescomplemented
40M16×C223C2211yescharacteristic, complemented
41C22×D223—22no—
42Dic446—21no—
43Dic446—22no—
44Dic446D411yescharacteristic
45Dic446C2312yes—
46Dic446C2311yescharacteristic
47D892D411yescharacteristic
48D892C2311yescharacteristic
49C22⋊D223—21no—
50C22⋊D223—22no—
51C22⋊D223—22no—
52C22⋊D223—22no—
53C22⋊D223D412yes—
54C2×Dic223—21no—
55C2×Dic223—22no—
56C2×Dic223—22no—
57C4×D223—22no—
58C4×D223—21no—
59C4×D223C2312yes—
60C223⋊C8C2311yescharacteristic
61C223⋊C8C2311yescharacteristic
62C23×C223—21no—
63Q8×C223C2311yescharacteristic
64D4×C223—21no—
65D4×C223—21no—
66D4×C223C2311yescharacteristic
67D4×C223C2312yes—
68C2×C4×C223—21no—
69C2×C4×C223C2311yescharacteristic
70C8×C223C2312yes—
71C22×C223—21no—
72C22×C223—22no—
73C22×C223—81no—
74C22×C223C2×D411yescharacteristic
75C2×D223—21no—
76C2×D223—82no—
77C2×D223C2×D412yes—
78C4×C223—21no—
79C4×C223C2×D411yescharacteristic
80C4×C223C2411yescharacteristic
81Dic223—21no—
82Dic223—22no—
83Dic223C2×D412yes—
84C2×C223—21no—
85C2×C223—42no—
86C2×C223—41no—
87C2×C223C22×D411yescharacteristic
88D223—22no—
89D223—41no—
90C223#57711yescharacteristic, complemented, Sylow
91#577—2231nomaximal, Sylow
922−1+4—2231no—
93H(5,𝔽2)—2231no—
94#137—2232no—
95#137—2231no—
96AGL(1,ℤ/8)—2232no—
97AGL(1,ℤ/8)D22311yescharacteristic, complemented
98#135—2231no—
99#135—2232no—
100C2×SD16—2231no—
101C2×SD16—2232no—
102#131—2231no—
103#41—4461no—
104#41—4461no—
105#41—4462no—
106#41—2232no—
107#41—2232no—
108#41—2231no—
109#41—2231no—
110#41C2×D22311yescharacteristic, complemented
111C2×Q8—4461no—
112C2×Q8—2232no—
113C2×Q8—2231no—
114C2×D4—4461no—
115C2×D4—4462no—
116C2×D4—2232no—
117C2×D4C2×D22311yescharacteristic
118Q16—2231no—
119Q16—2232no—
120SD16—2231no—
121SD16—2232no—
122SD16—2232no—
123SD16—2231no—
124SD16—2232no—
125SD16C2×D22312yescomplemented
126D8—2231no—
127D8C2×D22312yescomplemented
128M16—2232no—
129M16C2×D22311yescharacteristic, complemented
130C2×C8—2231no—
131C2×C8—2232no—
132C23—21no—
133C23—4462no—
134Q8—4462no—
135Q8—4461no—
136Q8—2232no—
137Q8—2231no—
138Q8—2231no—
139Q8C22×D22311yescharacteristic
140D4—21no—
141D4—21no—
142D4—4462no—
143D4—4462no—
144D4—4461no—
145D4—4462no—
146D4—2232no—
147D4—2231no—
148D4—2231no—
149D4C22×D22311yescharacteristic
150D4C22×D22312yes—
151C2×C4—21no—
152C2×C4—4462no—
153C2×C4—4461no—
154C2×C4—4462no—
155C2×C4—4462no—
156C2×C4—4461no—
157C2×C4—2232no—
158C2×C4C22×D22311yescharacteristic
159C8—2231no—
160C8—2231no—
161C8C22×D22312yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—4461no—
166C22—2232no—
167C22—17842no—
168C22D4×D22311yescharacteristic
169C4—21no—
170C4—4461no—
171C4—4462no—
172C4—2232no—
173C4D4×D22311yescharacteristic
174C4C23×D22311yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—8921no—
179C2—4462no—
180C2C2×D4×D22311yescharacteristic
subgroupG/H
centerC2C2×D4×D223
commutator, cosocleC4×C223C24
FrattiniC4C23×D223
FittingAGL(1,ℤ/8)×C223C2
socleC2×C223C22×D4
hypercenterAGL(1,ℤ/8)D223
hypocenterC223#577
LCSG ▷ C4×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20297351C2semidirect
2. #20297349C2semidirect
3. #20297337C2semidirect
4. #20297333C2semidirect
5. #20297328C2semidirect
6. #20297324C2semidirect
7. #20297323C2semidirect
8. C2×#16813158C2semidirect
9. SD16×D223C2semidirect
10. #20297312C2semidirect
11. AGL(1,ℤ/8)×C223C2semidirect
18. #16813169C22semidirect
20. C2×Dic446C22semidirect
21. Q8×D223C22semidirect
25. Dic892C22semidirect
26. #16813160C22semidirect
27. #16813159C22semidirect
28. #16813158C22semidirect
29. #16813158C22semidirect
30. #16813158C22semidirect
31. #16813157C22semidirect
32. #16813155C22semidirect
33. #16813153C22semidirect
34. C2×C223⋊C8C22semidirect
35. C8×D223C22semidirect
36. #41×C223C22semidirect
38. SD16×C223C22semidirect
39. D8×C223C22semidirect
40. M16×C223C22semidirect
90. C223#577semidirect
97. AGL(1,ℤ/8)D223semidirect
110. #41C2×D223semidirect
125. SD16C2×D223semidirect
127. D8C2×D223semidirect
129. M16C2×D223semidirect

Maximal subgroups

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

subgroupindex
#202973512
#202973492
#202973372
#202973332
#202973282
#202973242
#202973232
C2×#168131582
SD16×D2232
#202973122
AGL(1,ℤ/8)×C2232
#577223