Finite Group Databasebeta

← #28417076

Subgroups — #28417076

Order 14912.

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

normal subgroups only

This group has 334 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×#16836595C211yesmaximal, characteristic, complemented
2C22×D932C211yesmaximal, characteristic, complemented
3#21418524C211yesmaximal, characteristic, complemented
4#21418520C212yesmaximal, complemented
5#21418510C211yesmaximal, characteristic, complemented
6#21418510C212yesmaximal, complemented
7#21418505C212yesmaximal, complemented
8#21418497C211yesmaximal, characteristic, complemented
9C4×C22⋊D233C211yesmaximal, characteristic, complemented
10C22⋊C4×D233C212yesmaximal, complemented
11C22⋊Q8×C233C211yesmaximal, characteristic, complemented
12C23×D233C2212yes—
13#16836595—24no—
14C2×D932—24no—
15C2×D932C2212yes—
16C2×D932C2211yescharacteristic
17C2×D932C2211yescharacteristic
18C2×D932C2212yescomplemented
19C2×D932C2212yescomplemented
20C2×C22⋊D233C2211yescharacteristic
21C2×C4×D233C2211yescharacteristic, complemented
22C2×C4×D233C2212yescomplemented
23C2×C4×D233C2212yescomplemented
24#16836573C2211yescharacteristic, complemented
25C22⋊Dic233C2211yescharacteristic
26#16836571C2211yescharacteristic, complemented
27#16836571C2212yescomplemented
28#16836571C2212yescomplemented
29#16836571C2212yescomplemented
30#16836571C2212yescomplemented
31C4×Dic233C2211yescharacteristic, complemented
32C2×Q8×C233C2211yescharacteristic, complemented
33C22×C4×C233C2211yescharacteristic
34C4⋊C4×C233C2211yescharacteristic, complemented
35C4⋊C4×C233C2212yescomplemented
36C22⋊C4×C233C2212yes—
37C22×D233—24no—
38C22×D233—24no—
39C22×D233—42no—
40C22×D233C2312yes—
41C22×D233C2311yescharacteristic
42C22×D233C2312yes—
43D932—24no—
44D932—24no—
45D932—22no—
46D932—42no—
47D932D414yescomplemented
48C22⋊D233—22no—
49C2×Dic233C2311yescharacteristic
50C2×Dic233C2311yescharacteristic
51C2×Dic233C2311yescharacteristic
52C4×D233—24no—
53C4×D233—22no—
54C4×D233—42no—
55C23×C233C2311yescharacteristic
56Q8×C233—22no—
57C2×C4×C233—22no—
58C2×C4×C233C2312yes—
59C2×C4×C233C2312yes—
60C2×C4×C233C2311yescharacteristic
61C2×C4×C233C2311yescharacteristic
62C22×C233—21no—
63C22×C233—21no—
64C22×C233#4112yescomplemented
65C22×C233C2411yescharacteristic
66C2×D233—22no—
67C2×D233—22no—
68C2×D233—21no—
69C2×D233—22no—
70C2×D233—21no—
71C2×D233—22no—
72C2×D233—21no—
73C2×D233—22no—
74C2×D233—48no—
75C2×D233C2×D414yes—
76C4×C233—21no—
77C4×C233—22no—
78C4×C233—22no—
79C4×C233C2×D412yes—
80Dic233—21no—
81Dic233—21no—
82Dic233—21no—
83C2×C233—22no—
84C2×C233C22×D411yescharacteristic
85C2×C233C2×#4111yescharacteristic
86C2×C233H(5,𝔽2)11yescharacteristic
87D233—24no—
88D233—41no—
89D233—42no—
90C233#54611yescharacteristic, complemented, Sylow
91#546—2331nomaximal, Sylow
92C2×#41—2331no—
93C22×D4—2331no—
94#124—2331no—
95#123—2332no—
96C22⋊Q8D23311yescharacteristic, complemented
97#121—2331no—
98#121—2332no—
99#120—2332no—
100C4×D4—2331no—
101C4×D4—2331no—
102C2×C22⋊C4—2332no—
103C24—2332no—
104#41—4664no—
105C2×Q8C2×D23311yescharacteristic, complemented
106C2×D4—4664no—
107C2×D4—2331no—
108C2×D4—2332no—
109C2×D4—2332no—
110C2×D4—2332no—
111C2×D4—2331no—
112C2×D4—2331no—
113C22×C4—2332no—
114C22×C4—2332no—
115C22×C4—2331no—
116C22×C4C2×D23311yescharacteristic
117C4⋊C4—2331no—
118C4⋊C4C2×D23311yescharacteristic, complemented
119C4⋊C4C2×D23312yescomplemented
120C22⋊C4—2331no—
121C22⋊C4—2332no—
122C22⋊C4—2332no—
123C22⋊C4—2332no—
124C22⋊C4—2331no—
125C22⋊C4—2332no—
126C22⋊C4C2×D23312yes—
127C42—2331no—
128C23—9322no—
129C23—4664no—
130C23—4664no—
131C23—2332no—
132C23—2331no—
133C23—2332no—
134C23C22×D23311yescharacteristic
135Q8—22no—
136D4—9322no—
137D4—4664no—
138D4—4664no—
139D4—4662no—
140D4—4662no—
141D4—2334no—
142C2×C4—22no—
143C2×C4—9322no—
144C2×C4—4664no—
145C2×C4—4662no—
146C2×C4—2331no—
147C2×C4—2331no—
148C2×C4—2331no—
149C2×C4C22×D23312yes—
150C2×C4C22×D23312yes—
151C2×C4C22×D23311yescharacteristic
152C2×C4C22×D23311yescharacteristic
153C22—21no—
154C22—21no—
155C22—9328no—
156C22—4661no—
157C22—4662no—
158C22—4661no—
159C22—4662no—
160C22—4661no—
161C22—4662no—
162C22—4662no—
163C22—4662no—
164C22—2334no—
165C22#1683659512yescomplemented
166C22C23×D23311yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—4661no—
171C4—4661no—
172C4—4661no—
173C4D4×D23312yes—
174C2—22no—
175C2—9322no—
176C2—9321no—
177C2—4664no—
178C2C2×D4×D23311yescharacteristic
179C2C2×#1683659511yescharacteristic
180C2#2141858111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D233
commutator, socle, cosocleC22×C233C24
FittingC22⋊Q8×C233C2
hypercenterC22⋊Q8D233
hypocenterC233#546
LCSG ▷ C22×C233 ▷ C233
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16836595C2semidirect
2. C22×D932C2semidirect
3. #21418524C2semidirect
4. #21418520C2semidirect
5. #21418510C2semidirect
6. #21418510C2semidirect
7. #21418505C2semidirect
8. #21418497C2semidirect
9. C4×C22⋊D233C2semidirect
10. C22⋊C4×D233C2semidirect
11. C22⋊Q8×C233C2semidirect
18. C2×D932C22semidirect
19. C2×D932C22semidirect
21. C2×C4×D233C22semidirect
22. C2×C4×D233C22semidirect
23. C2×C4×D233C22semidirect
24. #16836573C22semidirect
26. #16836571C22semidirect
27. #16836571C22semidirect
28. #16836571C22semidirect
29. #16836571C22semidirect
30. #16836571C22semidirect
31. C4×Dic233C22semidirect
32. C2×Q8×C233C22semidirect
34. C4⋊C4×C233C22semidirect
35. C4⋊C4×C233C22semidirect
47. D932D4semidirect
64. C22×C233#41semidirect
90. C233#546semidirect
96. C22⋊Q8D233semidirect
105. C2×Q8C2×D233semidirect
118. C4⋊C4C2×D233semidirect
119. C4⋊C4C2×D233semidirect
165. C22#16836595semidirect

Maximal subgroups

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

subgroupindex
C2×#168365952
C22×D9322
#214185242
#214185202
#214185102
#214185052
#214184972
C4×C22⋊D2332
C22⋊C4×D2332
C22⋊Q8×C2332
#546233