Finite Group Databasebeta

← #49131742

Subgroups — #49131742

Order 36224.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22583636C212yesmaximal, complemented
2C2×#22583632C211yesmaximal, characteristic, complemented
3#31827032C212yesmaximal, complemented
4#31826853C214yesmaximal, complemented
5#31826848C212yesmaximal, complemented
6#31826840C212yesmaximal, complemented
7#31826420C211yesmaximal, characteristic, complemented
8C22×D8×C283C211yesmaximal, characteristic, complemented
9C2×#17985392C2212yes—
10#22583636—28no—
11#22583632—24no—
12C2×Dic1132C2211yescharacteristic, complemented
13C2×#17985381C2212yescomplemented
14C2×#17985380C2214yescomplemented
15C2×D2264C2211yescharacteristic, complemented
16C2×#17985377C2214yescomplemented
17C2×#17985376C2211yescharacteristic
18#22583606C2214yescomplemented
19#22583597C2214yescomplemented
20#22583581—24no—
21#22583579—24no—
22#22583569C2211yescharacteristic
23#22583520C2212yescomplemented
24C8×Dic283C2212yescomplemented
25C22×D4×C283C2212yes—
26C2×D8×C283—24no—
27C2×D8×C283C2212yes—
28C2×D8×C283C2212yes—
29C22×C8×C283C2211yescharacteristic
30#17985392—24no—
31#17985392—24no—
32C2×Dic566C2312yes—
33C2×D1132C2312yes—
34C2×C22⋊D283—28no—
35C2×C22⋊D283—22no—
36C2×C4×D283—22no—
37Dic1132—22no—
38#17985381—24no—
39#17985380—28no—
40D2264—22no—
41#17985377—28no—
42#17985376D414yescomplemented
43C2×C283⋊C8C2312yes—
44C4⋊Dic283D412yes—
45C22⋊Dic283—28no—
46C22⋊Dic283D412yes—
47C4×Dic283C2312yes—
48C24×C283—22no—
49C2×D4×C283—24no—
50C2×D4×C283—24no—
51C2×D4×C283—22no—
52C2×D4×C283C2314yes—
53C22×C4×C283C2311yescharacteristic
54D8×C283—24no—
55D8×C283—24no—
56C2×C8×C283D414yescomplemented
57C2×C8×C283C2311yescharacteristic
58C2×C8×C283C2311yescharacteristic
59C22×D283—22no—
60Dic566—22no—
61Dic566—22no—
62D1132—22no—
63D1132—22no—
64C22⋊D283—44no—
65C22⋊D283—88no—
66C2×Dic283—22no—
67C2×Dic283C2×D414yes—
68C4×D283—44no—
69C283⋊C8C2×D414yes—
70C23×C283—24no—
71C23×C283—44no—
72C23×C283—84no—
73C23×C283C2×D411yescharacteristic
74D4×C283—24no—
75D4×C283—24no—
76D4×C283—44no—
77C2×C4×C283C2×D411yescharacteristic
78C2×C4×C283C2×D412yes—
79C2×C4×C283C2×D412yes—
80C2×C4×C283C2411yescharacteristic
81C8×C283C2×D412yes—
82C8×C283C2×D412yes—
83C22×C283—22no—
84C22×C283—44no—
85C22×C283—48no—
86C22×C283—88no—
87C22×C283#12712yes—
88C22×C283C22×D411yescharacteristic
89C2×D283—42no—
90C2×D283—44no—
91C4×C283#12712yes—
92C4×C283C22×D411yescharacteristic
93C4×C283C22×D411yescharacteristic
94Dic283—24no—
95Dic283—42no—
96C2×C283—22no—
97C2×C283—84no—
98C2×C283C2×#12711yescharacteristic
99C2×C283#57612yes—
100D283—82no—
101C283#315511yescharacteristic, complemented, Sylow
102#3155—2831nomaximal, Sylow
103C2×AGL(1,ℤ/8)—2832no—
104C2×#135—2831no—
105C22×D8D28311yescharacteristic, complemented
106#535—2832no—
107#496—2834no—
108#495—2832no—
109#493—2832no—
110#405—2831no—
111C2×#41—2832no—
112C22×D4C2×D28312yes—
113AGL(1,ℤ/8)—5668no—
114#135—5664no—
115C2×Q16—2831no—
116C2×SD16—2834no—
117C2×SD16—2832no—
118C2×D8—24no—
119C2×D8—2834no—
120C2×D8—2831no—
121C2×D8C2×D28312yes—
122C2×D8C2×D28312yes—
123C2×M16—2831no—
124C22×C8C2×D28311yescharacteristic
125#127—2834no—
126#124—2834no—
127#121—5664no—
128#120—5664no—
129#117—2831no—
130#97—2832no—
131C4×C8—2832no—
132C24—22no—
133#41—5664no—
134#41—5664no—
135C2×Q8—2832no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—2832no—
140C2×D4—5668no—
141C2×D4—5662no—
142C2×D4C22×D28314yes—
143C22×C4—5662no—
144C22×C4C22×D28311yescharacteristic
145Q16—5662no—
146SD16—5668no—
147SD16—5664no—
148D8—24no—
149D8—24no—
150D8—5668no—
151D8—5662no—
152M16—2834no—
153C2×C8—2832no—
154C2×C8C22⋊D28314yescomplemented
155C2×C8C22×D28311yescharacteristic
156C2×C8C22×D28311yescharacteristic
157C4⋊C4—2832no—
158C22⋊C4—2832no—
159C22⋊C4—5668no—
160C42—2832no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—5662no—
165C23D4×D28311yescharacteristic
166Q8—5662no—
167Q8—5662no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—5662no—
172D4—5662no—
173D4—11324no—
174D4—22648no—
175C2×C4—2834no—
176C2×C4—5662no—
177C2×C4—11324no—
178C2×C4D4×D28311yescharacteristic
179C2×C4D4×D28312yes—
180C2×C4C2×C22⋊D28312yes—
181C2×C4C23×D28311yescharacteristic
182C8—2834no—
183C8C2×C22⋊D28312yes—
184C8C2×C22⋊D28312yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—11322no—
190C22—11324no—
191C22#2258360612yes—
192C22C2×D4×D28311yescharacteristic
193C4—5664no—
194C4—11322no—
195C4#2258360612yes—
196C4C2×D4×D28311yescharacteristic
197C4C22×C22⋊D28311yescharacteristic
198C2—22no—
199C2—84no—
200C2—22642no—
201C2C2×#2258360611yescharacteristic
202C2#3182726612yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC22×D8×C283C2
socleC22×C283C22×D4
hypercenterC22×D8D283
hypocenterC283#3155
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#22583636C2semidirect
2. C2×#22583632C2semidirect
3. #31827032C2semidirect
4. #31826853C2semidirect
5. #31826848C2semidirect
6. #31826840C2semidirect
7. #31826420C2semidirect
8. C22×D8×C283C2semidirect
12. C2×Dic1132C22semidirect
13. C2×#17985381C22semidirect
14. C2×#17985380C22semidirect
15. C2×D2264C22semidirect
16. C2×#17985377C22semidirect
18. #22583606C22semidirect
19. #22583597C22semidirect
23. #22583520C22semidirect
24. C8×Dic283C22semidirect
42. #17985376D4semidirect
56. C2×C8×C283D4semidirect
101. C283#3155semidirect
105. C22×D8D283semidirect
154. C2×C8C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836362
C2×#225836322
#318270322
#318268532
#318268482
#318268402
#318264202
C22×D8×C2832
#3155283