Finite Group Databasebeta

← #28384384

Subgroups — #28384384

Order 14272.

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#20297350C211yesmaximal, characteristic, complemented
2#20297349C211yesmaximal, characteristic, complemented
3#20297332C211yesmaximal, characteristic, complemented
4#20297331C212yesmaximal, complemented
5#20297328C212yesmaximal, complemented
6#20297325C212yesmaximal, complemented
7#20297324C211yesmaximal, characteristic, complemented
8C2×#16813155C211yesmaximal, characteristic, complemented
9D8×D223C212yesmaximal, complemented
10#20297312C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C223C211yesmaximal, characteristic, complemented
12#16813171C2212yescomplemented
13#16813170—21no—
14#16813170C2211yescharacteristic
15#16813170C2211yescharacteristic, complemented
16#16813169—21no—
17#16813169C2212yescomplemented
18C2×D892—21no—
19C2×D892C2211yescharacteristic, complemented
20C2×C22⋊D223—22no—
21D4×D223—21no—
22D4×D223—22no—
23D4×D223C2212yescomplemented
24D4×D223C2212yescomplemented
25#16813160C2211yescharacteristic, complemented
26#16813159C2212yescomplemented
27#16813158C2211yescharacteristic, complemented
28#16813157C2211yescharacteristic, complemented
29D1784C2212yescomplemented
30#16813155C2211yescharacteristic, complemented
31#16813155C2212yescomplemented
32#16813155C2212yescomplemented
33#16813153C2212yescomplemented
34C2×C223⋊C8C2211yescharacteristic, complemented
35C8×D223C2212yescomplemented
36#41×C223C2211yescharacteristic, complemented
37C2×D4×C223C2211yescharacteristic, complemented
38SD16×C223C2212yescomplemented
39D8×C223C2212yescomplemented
40M16×C223C2211yescharacteristic, complemented
41C22×D223—21no—
42C22×D223—22no—
43C22×D223—22no—
44Dic446D411yescharacteristic
45Dic446C2311yescharacteristic
46D892—22no—
47D892—21no—
48D892D411yescharacteristic
49D892C2312yes—
50D892C2311yescharacteristic
51C22⋊D223—21no—
52C22⋊D223—22no—
53C22⋊D223—22no—
54C22⋊D223—22no—
55C22⋊D223D412yes—
56C2×Dic223—22no—
57C4×D223—21no—
58C4×D223—22no—
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—22no—
77C2×D223—82no—
78C2×D223—82no—
79C2×D223—81no—
80C2×D223C2×D412yes—
81C4×C223—21no—
82C4×C223C2×D411yescharacteristic
83C4×C223C2411yescharacteristic
84Dic223—21no—
85Dic223C2×D412yes—
86C2×C223—21no—
87C2×C223—41no—
88C2×C223—42no—
89C2×C223C22×D411yescharacteristic
90D223—22no—
91D223—41no—
92D223—42no—
93C223#57611yescharacteristic, complemented, Sylow
94#576—2231nomaximal, Sylow
95H(5,𝔽2)—2231no—
96H(5,𝔽2)—2231no—
97AGL(1,ℤ/8)—2232no—
98AGL(1,ℤ/8)—2232no—
99AGL(1,ℤ/8)—2231no—
100AGL(1,ℤ/8)D22311yescharacteristic, complemented
101#135—2231no—
102#135—2232no—
103C2×D8—2231no—
104C2×D8—2232no—
105#131—2231no—
106#41—4461no—
107#41—4461no—
108#41—2232no—
109#41—2232no—
110#41—2231no—
111#41—2231no—
112#41C2×D22311yescharacteristic, complemented
113C2×D4—4461no—
114C2×D4—4462no—
115C2×D4—4461no—
116C2×D4—4462no—
117C2×D4—2232no—
118C2×D4—2232no—
119C2×D4—2231no—
120C2×D4C2×D22311yescharacteristic, complemented
121Q16—2231no—
122SD16—2231no—
123SD16—2231no—
124SD16—2232no—
125SD16C2×D22312yescomplemented
126D8—2232no—
127D8—2231no—
128D8—2232no—
129D8—2232no—
130D8C2×D22312yescomplemented
131M16—2232no—
132M16C2×D22311yescharacteristic, complemented
133C2×C8—2231no—
134C2×C8—2232no—
135C23—21no—
136C23—4462no—
137C23—4462no—
138C23—4461no—
139Q8—2231no—
140Q8—2231no—
141Q8C22×D22311yescharacteristic
142D4—21no—
143D4—21no—
144D4—4462no—
145D4—4462no—
146D4—4461no—
147D4—4462no—
148D4—4462no—
149D4—4461no—
150D4—2232no—
151D4—2231no—
152D4—2232no—
153D4—2231no—
154D4C22×D22312yes—
155D4C22×D22311yescharacteristic
156C2×C4—21no—
157C2×C4—4462no—
158C2×C4—4462no—
159C2×C4—4461no—
160C2×C4—2232no—
161C2×C4C22×D22311yescharacteristic
162C8—2231no—
163C8—2231no—
164C8C22×D22312yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—4462no—
169C22—4461no—
170C22—2232no—
171C22—17841no—
172C22—17842no—
173C22—17842no—
174C22D4×D22311yescharacteristic
175C4—21no—
176C4—4461no—
177C4—2232no—
178C4D4×D22311yescharacteristic
179C4C23×D22311yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—8922no—
184C2—8921no—
185C2—4462no—
186C2C2×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#576
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. #20297350C2semidirect
2. #20297349C2semidirect
3. #20297332C2semidirect
4. #20297331C2semidirect
5. #20297328C2semidirect
6. #20297325C2semidirect
7. #20297324C2semidirect
8. C2×#16813155C2semidirect
9. D8×D223C2semidirect
10. #20297312C2semidirect
11. AGL(1,ℤ/8)×C223C2semidirect
12. #16813171C22semidirect
15. #16813170C22semidirect
17. #16813169C22semidirect
19. C2×D892C22semidirect
23. D4×D223C22semidirect
24. D4×D223C22semidirect
25. #16813160C22semidirect
26. #16813159C22semidirect
27. #16813158C22semidirect
28. #16813157C22semidirect
29. D1784C22semidirect
30. #16813155C22semidirect
31. #16813155C22semidirect
32. #16813155C22semidirect
33. #16813153C22semidirect
34. C2×C223⋊C8C22semidirect
35. C8×D223C22semidirect
36. #41×C223C22semidirect
37. C2×D4×C223C22semidirect
38. SD16×C223C22semidirect
39. D8×C223C22semidirect
40. M16×C223C22semidirect
93. C223#576semidirect
100. AGL(1,ℤ/8)D223semidirect
112. #41C2×D223semidirect
120. C2×D4C2×D223semidirect
125. SD16C2×D223semidirect
130. D8C2×D223semidirect
132. M16C2×D223semidirect

Maximal subgroups

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

subgroupindex
#202973502
#202973492
#202973322
#202973312
#202973282
#202973252
#202973242
C2×#168131552
D8×D2232
#202973122
AGL(1,ℤ/8)×C2232
#576223