Finite Group Databasebeta

← #36374616

Subgroups — #36374616

Order 22336.

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×D349C211yesmaximal, characteristic
2C4×D1396C211yesmaximal, characteristic, complemented
3D4×Dic349C211yesmaximal, characteristic, complemented
4#26013601C211yesmaximal, characteristic, complemented
5C4×C22⋊D349C212yesmaximal, complemented
6#26013599C212yesmaximal, complemented
7#26013597C211yesmaximal, characteristic, complemented
8#26013596C211yesmaximal, characteristic, complemented
9C2×#19150210C212yesmaximal, complemented
10C22⋊C4×D349C212yesmaximal, complemented
11C4×D4×C349C211yesmaximal, characteristic, complemented
12C23×D349C2212yes—
13C2×D1396C2211yescharacteristic
14C2×C22⋊D349C2212yes—
15D4×D349C418yescomplemented
16C22×Dic349C2212yescomplemented
17C2×C4×D349C2211yescharacteristic
18C2×C4×D349C2212yescomplemented
19C2×C4×D349C2212yescomplemented
20C4⋊Dic349C2211yescharacteristic, complemented
21#19150212C2211yescharacteristic, complemented
22#19150212C2211yescharacteristic, complemented
23C22⋊Dic349C2212yescomplemented
24#19150210C2211yescharacteristic
25#19150210C2212yes—
26#19150210C2211yescharacteristic
27#19150210C2212yescomplemented
28#19150210C2212yescomplemented
29C4×Dic349C2211yescharacteristic, complemented
30C4×Dic349C2212yescomplemented
31C2×D4×C349C2211yescharacteristic
32C22×C4×C349C2212yescomplemented
33C4⋊C4×C349C2211yescharacteristic, complemented
34C22⋊C4×C349C2212yescomplemented
35C42×C349C2211yescharacteristic, complemented
36C22×D349—42no—
37C22×D349—42no—
38C22×D349C2×C418yescomplemented
39C22×D349C2311yescharacteristic
40C22×D349C2312yes—
41D1396C2×C414yescomplemented
42C22⋊D349C2×C418yescomplemented
43C2×Dic349—42no—
44C2×Dic349C2311yescharacteristic
45C2×Dic349C2312yes—
46C2×Dic349C2311yescharacteristic
47C2×Dic349C2311yescharacteristic
48C4×D349—42no—
49C4×D349—42no—
50C4×D349C2×C414yescomplemented
51C23×C349C2312yes—
52D4×C349C2×C414yescomplemented
53C2×C4×C349—42no—
54C2×C4×C349C2311yescharacteristic
55C2×C4×C349C2311yescharacteristic
56C2×C4×C349C2312yes—
57C2×C4×C349C2311yescharacteristic
58C22×C349—22no—
59C22×C349—22no—
60C22×C349C22×C414yes—
61C22×C349C2411yescharacteristic
62C2×D349—22no—
63C2×D349—22no—
64C2×D349—21no—
65C2×D349—21no—
66C2×D349—48no—
67C2×D349C22×C412yes—
68C2×D349C22×C414yes—
69C4×C349—21no—
70C4×C349—22no—
71C4×C349—21no—
72C4×C349C22×C412yes—
73Dic349—21no—
74Dic349—22no—
75Dic349—21no—
76Dic349C22×C412yes—
77C2×C349—24no—
78C2×C349C23×C411yescharacteristic
79C2×C349H(5,𝔽2)11yescharacteristic
80C2×C349H(5,𝔽2)11yescharacteristic
81D349—22no—
82D349—24no—
83C349#51811yescharacteristic, complemented, Sylow
84#518—3491nomaximal, Sylow
85C22×D4—3491no—
86C4×D4—3491no—
87C4×D4—3492no—
88C4×D4—3491no—
89C4×D4—3492no—
90C4×D4—3491no—
91C4×D4D34911yescharacteristic, complemented
92#117—3491no—
93#117—3491no—
94C2×C22⋊C4—3492no—
95C2×C22⋊C4—3492no—
96C24—3492no—
97C2×D4—3492no—
98C2×D4—3498no—
99C2×D4—3491no—
100C2×D4C2×D34911yescharacteristic
101C22×C4—3491no—
102C22×C4—3492no—
103C22×C4—3492no—
104C22×C4—3492no—
105C22×C4C2×D34912yescomplemented
106C4⋊C4—3491no—
107C4⋊C4—3491no—
108C4⋊C4—3491no—
109C4⋊C4C2×D34911yescharacteristic, complemented
110C22⋊C4—3492no—
111C22⋊C4—3492no—
112C22⋊C4—3491no—
113C22⋊C4—3491no—
114C22⋊C4—3492no—
115C22⋊C4—3492no—
116C22⋊C4C2×D34912yescomplemented
117C42—3492no—
118C42—3491no—
119C42C2×D34911yescharacteristic, complemented
120C23—3498no—
121C23—3491no—
122C23—3492no—
123C23—13962no—
124C23—13962no—
125C23C22×D34912yes—
126D4—3498no—
127D4—3494no—
128D4C4×D34914yescomplemented
129C2×C4—42no—
130C2×C4—3491no—
131C2×C4—3494no—
132C2×C4—3491no—
133C2×C4—3492no—
134C2×C4—3491no—
135C2×C4—13962no—
136C2×C4—13962no—
137C2×C4—13962no—
138C2×C4C22×D34912yes—
139C2×C4C22×D34911yescharacteristic
140C2×C4C22×D34911yescharacteristic
141C2×C4C22×D34911yescharacteristic
142C22—22no—
143C22—22no—
144C22—3492no—
145C22—3494no—
146C22—13968no—
147C22—6982no—
148C22—6981no—
149C22—6982no—
150C22—6981no—
151C22C2×C4×D34914yes—
152C22C23×D34911yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—3492no—
157C4—6981no—
158C4—6982no—
159C4—6981no—
160C4C2×C4×D34912yes—
161C2—24no—
162C2—6982no—
163C2—6984no—
164C2C22×C4×D34911yescharacteristic
165C2#2601368411yescharacteristic
166C2#2601368511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D349
commutatorC2×C349C23×C4
FittingC4×D4×C349C2
socle, cosocleC22×C349C24
prime radicalC2×D4×D349C2
hypercenterC4×D4D349
hypocenterC349#518
LCSG ▷ C2×C349 ▷ C349
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×D1396C2semidirect
3. D4×Dic349C2semidirect
4. #26013601C2semidirect
5. C4×C22⋊D349C2semidirect
6. #26013599C2semidirect
7. #26013597C2semidirect
8. #26013596C2semidirect
9. C2×#19150210C2semidirect
10. C22⋊C4×D349C2semidirect
11. C4×D4×C349C2semidirect
15. D4×D349C4semidirect
16. C22×Dic349C22semidirect
18. C2×C4×D349C22semidirect
19. C2×C4×D349C22semidirect
20. C4⋊Dic349C22semidirect
21. #19150212C22semidirect
22. #19150212C22semidirect
23. C22⋊Dic349C22semidirect
27. #19150210C22semidirect
28. #19150210C22semidirect
29. C4×Dic349C22semidirect
30. C4×Dic349C22semidirect
32. C22×C4×C349C22semidirect
33. C4⋊C4×C349C22semidirect
34. C22⋊C4×C349C22semidirect
35. C42×C349C22semidirect
38. C22×D349C2×C4semidirect
41. D1396C2×C4semidirect
42. C22⋊D349C2×C4semidirect
50. C4×D349C2×C4semidirect
52. D4×C349C2×C4semidirect
83. C349#518semidirect
91. C4×D4D349semidirect
105. C22×C4C2×D349semidirect
109. C4⋊C4C2×D349semidirect
116. C22⋊C4C2×D349semidirect
119. C42C2×D349semidirect
128. D4C4×D349semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3492
C4×D13962
D4×Dic3492
#260136012
C4×C22⋊D3492
#260135992
#260135972
#260135962
C2×#191502102
C22⋊C4×D3492
C4×D4×C3492
#518349