Finite Group Databasebeta

← #27164526

Subgroups — #27164526

Order 12736.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 107 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×#15691885C211yesmaximal, characteristic, complemented
2C22×C4×D199C211yesmaximal, characteristic, complemented
3#20263938C211yesmaximal, characteristic, complemented
4#20263933C212yesmaximal, complemented
5#20263928C212yesmaximal, complemented
6#20263925C211yesmaximal, characteristic, complemented
7#20263923C211yesmaximal, characteristic, complemented
8#20263917C212yesmaximal, complemented
9(C22⋊D199)⋊C4C212yesmaximal, complemented
10#20263911C211yesmaximal, characteristic, complemented
11#123×C199C211yesmaximal, characteristic, complemented
12C23×D199C2211yescharacteristic
13#15691885—24no—
14C2×Dic398C2211yescharacteristic, complemented
15C2×D796C2212yes—
16C2×C22⋊D199C2211yescharacteristic
17C2×C22⋊D199C2212yes—
18C2×C22⋊D199C2211yescharacteristic, complemented
19C22×Dic199C2211yescharacteristic
20C22×Dic199C2211yescharacteristic, complemented
21C2×C4×D199—24no—
22C2×C4×D199C2212yes—
23C2×C4×D199C2212yes—
24C2×C4×D199C2211yescharacteristic, complemented
25Dic199⋊C4C2212yescomplemented
26Dic199⋊C4C2212yescomplemented
27C22⋊Dic199C2211yescharacteristic
28(C2×D199)⋊C4C2212yes—
29(C2×D199)⋊C4C2212yes—
30(C2×D199)⋊C4C2212yescomplemented
31C4×Dic199C2212yescomplemented
32C2×D4×C199C2211yescharacteristic
33C22×C4×C199C2211yescharacteristic
34C4⋊C4×C199C2212yescomplemented
35C22⋊C4×C199C2211yescharacteristic
36C22⋊C4×C199C2212yescomplemented
37C22×D199—22no—
38C22×D199—24no—
39C22×D199C2311yescharacteristic
40C22×D199C2312yes—
41Dic398—22no—
42D796—24no—
43C22⋊D199—22no—
44C22⋊D199—24no—
45C22⋊D199—22no—
46C2×Dic199—22no—
47C2×Dic199D414yescomplemented
48C2×Dic199C2312yes—
49C2×Dic199C2311yescharacteristic
50C2×Dic199C2311yescharacteristic
51C2×Dic199C2311yescharacteristic
52C4×D199—24no—
53C4×D199—24no—
54C4×D199—22no—
55C23×C199C2311yescharacteristic
56C23×C199C2311yescharacteristic
57D4×C199—22no—
58C2×C4×C199—22no—
59C2×C4×C199C2312yes—
60C2×C4×C199C2312yes—
61C2×C4×C199C2311yescharacteristic
62C22×C199—22no—
63C22×C199—21no—
64C22×C199—22no—
65C22×C199C2×D412yes—
66C22×C199C2411yescharacteristic
67C2×D199—28no—
68C2×D199—22no—
69C2×D199—22no—
70C2×D199—21no—
71C2×D199—22no—
72C2×D199#4114yescomplemented
73C4×C199—22no—
74C4×C199—21no—
75C4×C199—22no—
76Dic199—21no—
77Dic199—22no—
78Dic199C2×D412yes—
79Dic199C2×D412yes—
80C2×C199—22no—
81C2×C199—41no—
82C2×C199C22×D411yescharacteristic
83C2×C199C2×#4112yes—
84D199—24no—
85D199—41no—
86C199#52511yescharacteristic, complemented, Sylow
87#525—1991nomaximal, Sylow
88C2×#41—1991no—
89C23×C4—1991no—
90#123—1991no—
91#123D19911yescharacteristic, complemented
92C22⋊Q8—1992no—
93#121—1992no—
94#120—1991no—
95#120—1991no—
96C4×D4—1992no—
97C4×D4—1992no—
98#117—1991no—
99C24—1991no—
100#41—3984no—
101C2×Q8—1991no—
102C2×D4—1992no—
103C2×D4—1991no—
104C2×D4—1991no—
105C2×D4—1992no—
106C2×D4C2×D19911yescharacteristic
107C22×C4—3984no—
108C22×C4—1991no—
109C22×C4—1992no—
110C22×C4—1992no—
111C22×C4—1991no—
112C22×C4—1991no—
113C22×C4C2×D19911yescharacteristic
114C4⋊C4—1992no—
115C4⋊C4—1992no—
116C4⋊C4C2×D19912yescomplemented
117C22⋊C4—1991no—
118C22⋊C4—1992no—
119C22⋊C4—1992no—
120C22⋊C4—1992no—
121C22⋊C4C2×D19911yescharacteristic
122C22⋊C4C2×D19912yescomplemented
123C42—1992no—
124C23—3984no—
125C23—3982no—
126C23—1991no—
127C23—1992no—
128C23C22×D19911yescharacteristic
129C23C22×D19911yescharacteristic
130Q8—3982no—
131D4—22no—
132D4—3984no—
133D4—3984no—
134D4—3982no—
135D4—3982no—
136C2×C4—22no—
137C2×C4—3984no—
138C2×C4—3984no—
139C2×C4—3982no—
140C2×C4—3982no—
141C2×C4—1992no—
142C2×C4—1991no—
143C2×C4—1994no—
144C2×C4—1991no—
145C2×C4—1991no—
146C2×C4C22×D19912yes—
147C2×C4C22×D19911yescharacteristic
148C2×C4C22×D19912yes—
149C22—22no—
150C22—21no—
151C22—22no—
152C22—3988no—
153C22—3981no—
154C22—3982no—
155C22—3982no—
156C22—3982no—
157C22—1994no—
158C22D4×D19912yes—
159C22C23×D19911yescharacteristic
160C4—22no—
161C4—21no—
162C4—22no—
163C4—3981no—
164C4—3982no—
165C4—1992no—
166C4—1992no—
167C2—22no—
168C2—41no—
169C2—7961no—
170C2—3984no—
171C2C2×D4×D19911yescharacteristic
172C2#41×D19912yes—
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#123×C199C2
hypercenter#123D199
hypocenterC199#525
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#15691885C2semidirect
2. C22×C4×D199C2semidirect
3. #20263938C2semidirect
4. #20263933C2semidirect
5. #20263928C2semidirect
6. #20263925C2semidirect
7. #20263923C2semidirect
8. #20263917C2semidirect
9. (C22⋊D199)⋊C4C2semidirect
10. #20263911C2semidirect
11. #123×C199C2semidirect
14. C2×Dic398C22semidirect
18. C2×C22⋊D199C22semidirect
20. C22×Dic199C22semidirect
24. C2×C4×D199C22semidirect
25. Dic199⋊C4C22semidirect
26. Dic199⋊C4C22semidirect
30. (C2×D199)⋊C4C22semidirect
31. C4×Dic199C22semidirect
34. C4⋊C4×C199C22semidirect
36. C22⋊C4×C199C22semidirect
47. C2×Dic199D4semidirect
72. C2×D199#41semidirect
86. C199#525semidirect
91. #123D199semidirect
116. C4⋊C4C2×D199semidirect
122. C22⋊C4C2×D199semidirect

Maximal subgroups

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

subgroupindex
C2×#156918852
C22×C4×D1992
#202639382
#202639332
#202639282
#202639252
#202639232
#202639172
(C22⋊D199)⋊C42
#202639112
#123×C1992
#525199