Finite Group Databasebeta

← #28384052

Subgroups — #28384052

Order 14272.

338 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 151 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×D4×D223C211yesmaximal, characteristic
2C4×D892C211yesmaximal, characteristic, complemented
3D4×Dic223C211yesmaximal, characteristic, complemented
4#20297266C211yesmaximal, characteristic, complemented
5C4×C22⋊D223C212yesmaximal, complemented
6(C22⋊D223)⋊C4C212yesmaximal, complemented
7#20297262C211yesmaximal, characteristic, complemented
8#20297261C211yesmaximal, characteristic, complemented
9C2×(C2×D223)⋊C4C212yesmaximal, complemented
10C22⋊C4×D223C212yesmaximal, complemented
11C4×D4×C223C211yesmaximal, characteristic, complemented
12C23×D223C2212yes—
13C2×D892C2211yescharacteristic
14C2×C22⋊D223C2212yes—
15D4×D223C418yescomplemented
16C22×Dic223C2212yescomplemented
17C2×C4×D223C2211yescharacteristic
18C2×C4×D223C2212yescomplemented
19C2×C4×D223C2212yescomplemented
20C4⋊Dic223C2211yescharacteristic, complemented
21Dic223⋊C4C2211yescharacteristic, complemented
22Dic223⋊C4C2211yescharacteristic, complemented
23C22⋊Dic223C2212yescomplemented
24(C2×D223)⋊C4C2211yescharacteristic
25(C2×D223)⋊C4C2212yes—
26(C2×D223)⋊C4C2211yescharacteristic
27(C2×D223)⋊C4C2212yescomplemented
28(C2×D223)⋊C4C2212yescomplemented
29C4×Dic223C2211yescharacteristic, complemented
30C4×Dic223C2212yescomplemented
31C2×D4×C223C2211yescharacteristic
32C22×C4×C223C2212yescomplemented
33C4⋊C4×C223C2211yescharacteristic, complemented
34C22⋊C4×C223C2212yescomplemented
35C42×C223C2211yescharacteristic, complemented
36C22×D223—42no—
37C22×D223—42no—
38C22×D223C2×C418yescomplemented
39C22×D223C2311yescharacteristic
40C22×D223C2312yes—
41D892C2×C414yescomplemented
42C22⋊D223C2×C418yescomplemented
43C2×Dic223—42no—
44C2×Dic223C2311yescharacteristic
45C2×Dic223C2312yes—
46C2×Dic223C2311yescharacteristic
47C2×Dic223C2311yescharacteristic
48C4×D223—42no—
49C4×D223—42no—
50C4×D223C2×C414yescomplemented
51C23×C223C2312yes—
52D4×C223C2×C414yescomplemented
53C2×C4×C223—42no—
54C2×C4×C223C2311yescharacteristic
55C2×C4×C223C2311yescharacteristic
56C2×C4×C223C2312yes—
57C2×C4×C223C2311yescharacteristic
58C22×C223—22no—
59C22×C223—22no—
60C22×C223C22×C414yes—
61C22×C223C2411yescharacteristic
62C2×D223—22no—
63C2×D223—22no—
64C2×D223—21no—
65C2×D223—21no—
66C2×D223—48no—
67C2×D223C22×C412yes—
68C2×D223C22×C414yes—
69C4×C223—21no—
70C4×C223—22no—
71C4×C223—21no—
72C4×C223C22×C412yes—
73Dic223—21no—
74Dic223—22no—
75Dic223—21no—
76Dic223C22×C412yes—
77C2×C223—24no—
78C2×C223C23×C411yescharacteristic
79C2×C223H(5,𝔽2)11yescharacteristic
80C2×C223H(5,𝔽2)11yescharacteristic
81D223—22no—
82D223—24no—
83C223#51811yescharacteristic, complemented, Sylow
84#518—2231nomaximal, Sylow
85C22×D4—2231no—
86C4×D4—2231no—
87C4×D4—2232no—
88C4×D4—2231no—
89C4×D4—2232no—
90C4×D4—2231no—
91C4×D4D22311yescharacteristic, complemented
92#117—2231no—
93#117—2231no—
94C2×C22⋊C4—2232no—
95C2×C22⋊C4—2232no—
96C24—2232no—
97C2×D4—2232no—
98C2×D4—2238no—
99C2×D4—2231no—
100C2×D4C2×D22311yescharacteristic
101C22×C4—2232no—
102C22×C4—2232no—
103C22×C4—2232no—
104C22×C4—2231no—
105C22×C4C2×D22312yescomplemented
106C4⋊C4—2231no—
107C4⋊C4—2231no—
108C4⋊C4—2231no—
109C4⋊C4C2×D22311yescharacteristic, complemented
110C22⋊C4—2232no—
111C22⋊C4—2232no—
112C22⋊C4—2231no—
113C22⋊C4—2231no—
114C22⋊C4—2232no—
115C22⋊C4—2232no—
116C22⋊C4C2×D22312yescomplemented
117C42—2231no—
118C42—2232no—
119C42C2×D22311yescharacteristic, complemented
120C23—8922no—
121C23—8922no—
122C23—2238no—
123C23—2231no—
124C23—2232no—
125C23C22×D22312yes—
126D4—2238no—
127D4—2234no—
128D4C4×D22314yescomplemented
129C2×C4—42no—
130C2×C4—8922no—
131C2×C4—8922no—
132C2×C4—8922no—
133C2×C4—2231no—
134C2×C4—2232no—
135C2×C4—2231no—
136C2×C4—2231no—
137C2×C4—2234no—
138C2×C4C22×D22312yes—
139C2×C4C22×D22311yescharacteristic
140C2×C4C22×D22311yescharacteristic
141C2×C4C22×D22311yescharacteristic
142C22—22no—
143C22—22no—
144C22—8928no—
145C22—4462no—
146C22—4461no—
147C22—4462no—
148C22—4461no—
149C22—2232no—
150C22—2234no—
151C22C2×C4×D22314yes—
152C22C23×D22311yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—4461no—
157C4—4462no—
158C4—4461no—
159C4—2232no—
160C4C2×C4×D22312yes—
161C2—24no—
162C2—4462no—
163C2—4464no—
164C2C22×C4×D22311yescharacteristic
165C2#2029734911yescharacteristic
166C2#2029735011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D223
commutatorC2×C223C23×C4
FittingC4×D4×C223C2
socle, cosocleC22×C223C24
prime radicalC2×D4×D223C2
hypercenterC4×D4D223
hypocenterC223#518
LCSG ▷ C2×C223 ▷ C223
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×D892C2semidirect
3. D4×Dic223C2semidirect
4. #20297266C2semidirect
5. C4×C22⋊D223C2semidirect
6. (C22⋊D223)⋊C4C2semidirect
7. #20297262C2semidirect
8. #20297261C2semidirect
9. C2×(C2×D223)⋊C4C2semidirect
10. C22⋊C4×D223C2semidirect
11. C4×D4×C223C2semidirect
15. D4×D223C4semidirect
16. C22×Dic223C22semidirect
18. C2×C4×D223C22semidirect
19. C2×C4×D223C22semidirect
20. C4⋊Dic223C22semidirect
21. Dic223⋊C4C22semidirect
22. Dic223⋊C4C22semidirect
23. C22⋊Dic223C22semidirect
27. (C2×D223)⋊C4C22semidirect
28. (C2×D223)⋊C4C22semidirect
29. C4×Dic223C22semidirect
30. C4×Dic223C22semidirect
32. C22×C4×C223C22semidirect
33. C4⋊C4×C223C22semidirect
34. C22⋊C4×C223C22semidirect
35. C42×C223C22semidirect
38. C22×D223C2×C4semidirect
41. D892C2×C4semidirect
42. C22⋊D223C2×C4semidirect
50. C4×D223C2×C4semidirect
52. D4×C223C2×C4semidirect
83. C223#518semidirect
91. C4×D4D223semidirect
105. C22×C4C2×D223semidirect
109. C4⋊C4C2×D223semidirect
116. C22⋊C4C2×D223semidirect
119. C42C2×D223semidirect
128. D4C4×D223semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2232
C4×D8922
D4×Dic2232
#202972662
C4×C22⋊D2232
(C22⋊D223)⋊C42
#202972622
#202972612
C2×(C2×D223)⋊C42
C22⋊C4×D2232
C4×D4×C2232
#518223