Finite Group Databasebeta

← #30657378

Subgroups — #30657378

Order 16448.

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×#16861619C211yesmaximal, characteristic, complemented
2C22×D1028C211yesmaximal, characteristic, complemented
3#22551581C211yesmaximal, characteristic, complemented
4#22551577C212yesmaximal, complemented
5#22551567C211yesmaximal, characteristic, complemented
6#22551567C212yesmaximal, complemented
7#22551562C212yesmaximal, complemented
8#22551554C211yesmaximal, characteristic, complemented
9C4×C22⋊D257C211yesmaximal, characteristic, complemented
10C22⋊C4×D257C212yesmaximal, complemented
11C22⋊Q8×C257C211yesmaximal, characteristic, complemented
12C23×D257C2212yes—
13#16861619—24no—
14C2×D1028—24no—
15C2×D1028C2212yes—
16C2×D1028C2211yescharacteristic
17C2×D1028C2211yescharacteristic
18C2×D1028C2212yescomplemented
19C2×D1028C2212yescomplemented
20C2×C22⋊D257C2211yescharacteristic
21C2×C4×D257C2211yescharacteristic, complemented
22C2×C4×D257C2212yescomplemented
23C2×C4×D257C2212yescomplemented
24#16861597C2211yescharacteristic, complemented
25C22⋊Dic257C2211yescharacteristic
26#16861595C2211yescharacteristic, complemented
27#16861595C2212yescomplemented
28#16861595C2212yescomplemented
29#16861595C2212yescomplemented
30#16861595C2212yescomplemented
31C4×Dic257C2211yescharacteristic, complemented
32C2×Q8×C257C2211yescharacteristic, complemented
33C22×C4×C257C2211yescharacteristic
34C4⋊C4×C257C2211yescharacteristic, complemented
35C4⋊C4×C257C2212yescomplemented
36C22⋊C4×C257C2212yes—
37C22×D257—24no—
38C22×D257—24no—
39C22×D257—42no—
40C22×D257C2312yes—
41C22×D257C2311yescharacteristic
42C22×D257C2312yes—
43D1028—24no—
44D1028—24no—
45D1028—22no—
46D1028—42no—
47D1028D414yescomplemented
48C22⋊D257—22no—
49C2×Dic257C2311yescharacteristic
50C2×Dic257C2311yescharacteristic
51C2×Dic257C2311yescharacteristic
52C4×D257—24no—
53C4×D257—22no—
54C4×D257—42no—
55C23×C257C2311yescharacteristic
56Q8×C257—22no—
57C2×C4×C257—22no—
58C2×C4×C257C2312yes—
59C2×C4×C257C2312yes—
60C2×C4×C257C2311yescharacteristic
61C2×C4×C257C2311yescharacteristic
62C22×C257—21no—
63C22×C257—21no—
64C22×C257#4112yescomplemented
65C22×C257C2411yescharacteristic
66C2×D257—22no—
67C2×D257—22no—
68C2×D257—21no—
69C2×D257—22no—
70C2×D257—21no—
71C2×D257—22no—
72C2×D257—21no—
73C2×D257—22no—
74C2×D257—48no—
75C2×D257C2×D414yes—
76C4×C257—21no—
77C4×C257—22no—
78C4×C257—22no—
79C4×C257C2×D412yes—
80Dic257—21no—
81Dic257—21no—
82Dic257—21no—
83C2×C257—22no—
84C2×C257C22×D411yescharacteristic
85C2×C257C2×#4111yescharacteristic
86C2×C257H(5,𝔽2)11yescharacteristic
87D257—24no—
88D257—41no—
89D257—42no—
90C257#54611yescharacteristic, complemented, Sylow
91#546—2571nomaximal, Sylow
92C2×#41—2571no—
93C22×D4—2571no—
94#124—2571no—
95#123—2572no—
96C22⋊Q8D25711yescharacteristic, complemented
97#121—2571no—
98#121—2572no—
99#120—2572no—
100C4×D4—2571no—
101C4×D4—2571no—
102C2×C22⋊C4—2572no—
103C24—2572no—
104#41—5144no—
105C2×Q8C2×D25711yescharacteristic, complemented
106C2×D4—2571no—
107C2×D4—2571no—
108C2×D4—2572no—
109C2×D4—2572no—
110C2×D4—2571no—
111C2×D4—2572no—
112C2×D4—5144no—
113C22×C4—2572no—
114C22×C4—2571no—
115C22×C4—2572no—
116C22×C4C2×D25711yescharacteristic
117C4⋊C4—2571no—
118C4⋊C4C2×D25711yescharacteristic, complemented
119C4⋊C4C2×D25712yescomplemented
120C22⋊C4—2571no—
121C22⋊C4—2572no—
122C22⋊C4—2572no—
123C22⋊C4—2572no—
124C22⋊C4—2571no—
125C22⋊C4—2572no—
126C22⋊C4C2×D25712yes—
127C42—2571no—
128C23—2572no—
129C23—2572no—
130C23—2571no—
131C23—5144no—
132C23—5144no—
133C23—10282no—
134C23C22×D25711yescharacteristic
135Q8—22no—
136D4—2574no—
137D4—5144no—
138D4—5144no—
139D4—5142no—
140D4—5142no—
141D4—10282no—
142C2×C4—2571no—
143C2×C4—2571no—
144C2×C4—2571no—
145C2×C4—22no—
146C2×C4—5144no—
147C2×C4—5142no—
148C2×C4—10282no—
149C2×C4C22×D25712yes—
150C2×C4C22×D25712yes—
151C2×C4C22×D25711yescharacteristic
152C2×C4C22×D25711yescharacteristic
153C22—2574no—
154C22—21no—
155C22—21no—
156C22—5142no—
157C22—5142no—
158C22—5141no—
159C22—5142no—
160C22—5141no—
161C22—5142no—
162C22—5141no—
163C22—5142no—
164C22—10288no—
165C22#1686161912yescomplemented
166C22C23×D25711yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—5141no—
171C4—5141no—
172C4—5141no—
173C4D4×D25712yes—
174C2—22no—
175C2—5144no—
176C2—10282no—
177C2—10281no—
178C2C2×D4×D25711yescharacteristic
179C2C2×#1686161911yescharacteristic
180C2#2255163811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
FittingC22⋊Q8×C257C2
hypercenterC22⋊Q8D257
hypocenterC257#546
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861619C2semidirect
2. C22×D1028C2semidirect
3. #22551581C2semidirect
4. #22551577C2semidirect
5. #22551567C2semidirect
6. #22551567C2semidirect
7. #22551562C2semidirect
8. #22551554C2semidirect
9. C4×C22⋊D257C2semidirect
10. C22⋊C4×D257C2semidirect
11. C22⋊Q8×C257C2semidirect
18. C2×D1028C22semidirect
19. C2×D1028C22semidirect
21. C2×C4×D257C22semidirect
22. C2×C4×D257C22semidirect
23. C2×C4×D257C22semidirect
24. #16861597C22semidirect
26. #16861595C22semidirect
27. #16861595C22semidirect
28. #16861595C22semidirect
29. #16861595C22semidirect
30. #16861595C22semidirect
31. C4×Dic257C22semidirect
32. C2×Q8×C257C22semidirect
34. C4⋊C4×C257C22semidirect
35. C4⋊C4×C257C22semidirect
47. D1028D4semidirect
64. C22×C257#41semidirect
90. C257#546semidirect
96. C22⋊Q8D257semidirect
105. C2×Q8C2×D257semidirect
118. C4⋊C4C2×D257semidirect
119. C4⋊C4C2×D257semidirect
165. C22#16861619semidirect

Maximal subgroups

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

subgroupindex
C2×#168616192
C22×D10282
#225515812
#225515772
#225515672
#225515622
#225515542
C4×C22⋊D2572
C22⋊C4×D2572
C22⋊Q8×C2572
#546257