Finite Group Databasebeta

← #28417079

Subgroups — #28417079

Order 14912.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16836593C211yesmaximal, characteristic, complemented
2#21418522C211yesmaximal, characteristic, complemented
3#21418516C212yesmaximal, complemented
4#21418514C211yesmaximal, characteristic, complemented
5#21418507C212yesmaximal, complemented
6#21418506C211yesmaximal, characteristic, complemented
7#21418504C212yesmaximal, complemented
8D4×Dic233C211yesmaximal, characteristic, complemented
9C4×C22⋊D233C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic233C212yesmaximal, complemented
11C22×D4×C233C211yesmaximal, characteristic, complemented
12#16836593—24no—
13C2×Dic466C2211yescharacteristic, complemented
14C2×C22⋊D233C2211yescharacteristic
15C2×C22⋊D233C2212yescomplemented
16C2×C22⋊D233C2212yescomplemented
17C22×Dic233C2212yescomplemented
18C22×Dic233C2212yescomplemented
19C2×C4×D233C2211yescharacteristic, complemented
20C4⋊Dic233C2211yescharacteristic, complemented
21#16836573C2211yescharacteristic, complemented
22#16836573C2212yescomplemented
23C22⋊Dic233C2211yescharacteristic
24C22⋊Dic233C2212yes—
25C22⋊Dic233C2212yescomplemented
26C22⋊Dic233C2212yescomplemented
27C22⋊Dic233C2212yescomplemented
28C22⋊Dic233C2212yescomplemented
29#16836571C2211yescharacteristic, complemented
30C4×Dic233C2211yescharacteristic, complemented
31C24×C233C2212yes—
32C2×D4×C233—24no—
33C2×D4×C233C2211yescharacteristic
34C2×D4×C233C2212yes—
35C2×D4×C233C2211yescharacteristic
36C22×C4×C233C2211yescharacteristic
37C22×D233C2311yescharacteristic
38Dic466—22no—
39C22⋊D233—24no—
40C22⋊D233—22no—
41C22⋊D233—42no—
42C2×Dic233—24no—
43C2×Dic233—42no—
44C2×Dic233C2311yescharacteristic
45C2×Dic233C2311yescharacteristic
46C2×Dic233C2312yes—
47C2×Dic233C2311yescharacteristic
48C2×Dic233C2312yes—
49C4×D233—22no—
50C23×C233—24no—
51C23×C233—24no—
52C23×C233—42no—
53C23×C233C2312yes—
54C23×C233C2312yes—
55C23×C233C2311yescharacteristic
56D4×C233—24no—
57D4×C233—22no—
58D4×C233D414yescomplemented
59C2×C4×C233—22no—
60C2×C4×C233C2311yescharacteristic
61C2×C4×C233C2311yescharacteristic
62C22×C233—21no—
63C22×C233—22no—
64C22×C233—21no—
65C22×C233—22no—
66C22×C233—22no—
67C22×C233—22no—
68C22×C233—22no—
69C22×C233—48no—
70C22×C233C2×D414yes—
71C22×C233#4112yescomplemented
72C22×C233C2411yescharacteristic
73C2×D233—21no—
74C2×D233—21no—
75C2×D233—21no—
76C4×C233—21no—
77C4×C233C2×D412yes—
78Dic233—21no—
79Dic233—22no—
80Dic233—21no—
81Dic233—21no—
82Dic233—22no—
83C2×C233—22no—
84C2×C233—24no—
85C2×C233—42no—
86C2×C233C22×D411yescharacteristic
87C2×C233C2×#4111yescharacteristic
88C2×C233H(5,𝔽2)11yescharacteristic
89D233—41no—
90C233#54611yescharacteristic, complemented, Sylow
91#546—2331nomaximal, Sylow
92C2×#41—2331no—
93C22×D4D23311yescharacteristic, complemented
94#124—2331no—
95#123—2332no—
96C22⋊Q8—2331no—
97#121—2331no—
98#121—2332no—
99#120—2332no—
100C4×D4—2331no—
101C4×D4—2331no—
102C2×C22⋊C4—2332no—
103C24C2×D23312yes—
104#41—4664no—
105C2×Q8—2331no—
106C2×D4—24no—
107C2×D4—2332no—
108C2×D4—2331no—
109C2×D4—2332no—
110C2×D4C2×D23312yes—
111C2×D4C2×D23311yescharacteristic
112C2×D4C2×D23311yescharacteristic
113C22×C4—2332no—
114C22×C4—2332no—
115C22×C4—2331no—
116C22×C4C2×D23311yescharacteristic
117C4⋊C4—2331no—
118C4⋊C4—2331no—
119C4⋊C4—2332no—
120C22⋊C4—2331no—
121C22⋊C4—2332no—
122C22⋊C4—2332no—
123C22⋊C4—2332no—
124C22⋊C4—2332no—
125C22⋊C4—2332no—
126C22⋊C4—2331no—
127C42—2331no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—2331no—
132C23C22×D23312yes—
133C23C22×D23312yes—
134C23C22×D23311yescharacteristic
135Q8—4662no—
136D4—24no—
137D4—22no—
138D4—9322no—
139D4—4664no—
140D4—4662no—
141D4C22⋊D23314yescomplemented
142C2×C4—22no—
143C2×C4—9322no—
144C2×C4—4664no—
145C2×C4—4662no—
146C2×C4—2331no—
147C2×C4—2331no—
148C2×C4—2332no—
149C2×C4—2331no—
150C2×C4—2332no—
151C2×C4C22×D23311yescharacteristic
152C2×C4C22×D23311yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—4661no—
162C22—4661no—
163C22—4661no—
164C22C2×C22⋊D23314yes—
165C22#1683659312yescomplemented
166C22C23×D23311yescharacteristic
167C4—21no—
168C4—4661no—
169C4—4662no—
170C4—4661no—
171C4—4661no—
172C4—4662no—
173C4C2×C22⋊D23312yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—9321no—
178C2C22×C22⋊D23311yescharacteristic
179C2C2×#1683659311yescharacteristic
180C2#2141858011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D233
commutator, socle, cosocleC22×C233C24
FittingC22×D4×C233C2
hypercenterC22×D4D233
hypocenterC233#546
LCSG ▷ C22×C233 ▷ C233
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16836593C2semidirect
2. #21418522C2semidirect
3. #21418516C2semidirect
4. #21418514C2semidirect
5. #21418507C2semidirect
6. #21418506C2semidirect
7. #21418504C2semidirect
8. D4×Dic233C2semidirect
9. C4×C22⋊D233C2semidirect
10. C2×C22⋊Dic233C2semidirect
11. C22×D4×C233C2semidirect
13. C2×Dic466C22semidirect
15. C2×C22⋊D233C22semidirect
16. C2×C22⋊D233C22semidirect
17. C22×Dic233C22semidirect
18. C22×Dic233C22semidirect
19. C2×C4×D233C22semidirect
20. C4⋊Dic233C22semidirect
21. #16836573C22semidirect
22. #16836573C22semidirect
25. C22⋊Dic233C22semidirect
26. C22⋊Dic233C22semidirect
27. C22⋊Dic233C22semidirect
28. C22⋊Dic233C22semidirect
29. #16836571C22semidirect
30. C4×Dic233C22semidirect
58. D4×C233D4semidirect
71. C22×C233#41semidirect
90. C233#546semidirect
93. C22×D4D233semidirect
141. D4C22⋊D233semidirect
165. C22#16836593semidirect

Maximal subgroups

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

subgroupindex
C2×#168365932
#214185222
#214185162
#214185142
#214185072
#214185062
#214185042
D4×Dic2332
C4×C22⋊D2332
C2×C22⋊Dic2332
C22×D4×C2332
#546233