Finite Group Databasebeta

← #44560846

Subgroups — #44560846

Order 32128.

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×#22548741C212yesmaximal, complemented
2C2×#22548737C211yesmaximal, characteristic, complemented
3#30632254C212yesmaximal, complemented
4#30632075C214yesmaximal, complemented
5#30632070C212yesmaximal, complemented
6#30632062C212yesmaximal, complemented
7#30631642C211yesmaximal, characteristic, complemented
8C22×D8×C251C211yesmaximal, characteristic, complemented
9C2×#16860810C2212yes—
10#22548741—28no—
11#22548737—24no—
12C2×Dic1004C2211yescharacteristic, complemented
13C2×#16860799C2212yescomplemented
14C2×#16860798C2214yescomplemented
15C2×D2008C2211yescharacteristic, complemented
16C2×#16860795C2214yescomplemented
17C2×#16860794C2211yescharacteristic
18#22548711C2214yescomplemented
19#22548702C2214yescomplemented
20#22548686—24no—
21#22548684—24no—
22#22548674C2211yescharacteristic
23#22548625C2212yescomplemented
24C8×Dic251C2212yescomplemented
25C22×D4×C251C2212yes—
26C2×D8×C251—24no—
27C2×D8×C251C2212yes—
28C2×D8×C251C2212yes—
29C22×C8×C251C2211yescharacteristic
30#16860810—24no—
31#16860810—24no—
32C2×Dic502C2312yes—
33C2×D1004C2312yes—
34C2×C22⋊D251—28no—
35C2×C22⋊D251—22no—
36C2×C4×D251—22no—
37Dic1004—22no—
38#16860799—24no—
39#16860798—28no—
40D2008—22no—
41#16860795—28no—
42#16860794D414yescomplemented
43C2×C251⋊C8C2312yes—
44C4⋊Dic251D412yes—
45C22⋊Dic251—28no—
46C22⋊Dic251D412yes—
47C4×Dic251C2312yes—
48C24×C251—22no—
49C2×D4×C251—24no—
50C2×D4×C251—24no—
51C2×D4×C251—22no—
52C2×D4×C251C2314yes—
53C22×C4×C251C2311yescharacteristic
54D8×C251—24no—
55D8×C251—24no—
56C2×C8×C251D414yescomplemented
57C2×C8×C251C2311yescharacteristic
58C2×C8×C251C2311yescharacteristic
59C22×D251—22no—
60Dic502—22no—
61Dic502—22no—
62D1004—22no—
63D1004—22no—
64C22⋊D251—44no—
65C22⋊D251—88no—
66C2×Dic251—22no—
67C2×Dic251C2×D414yes—
68C4×D251—44no—
69C251⋊C8C2×D414yes—
70C23×C251—24no—
71C23×C251—44no—
72C23×C251—84no—
73C23×C251C2×D411yescharacteristic
74D4×C251—24no—
75D4×C251—24no—
76D4×C251—44no—
77C2×C4×C251C2×D411yescharacteristic
78C2×C4×C251C2×D412yes—
79C2×C4×C251C2×D412yes—
80C2×C4×C251C2411yescharacteristic
81C8×C251C2×D412yes—
82C8×C251C2×D412yes—
83C22×C251—22no—
84C22×C251—44no—
85C22×C251—48no—
86C22×C251—88no—
87C22×C251#12712yes—
88C22×C251C22×D411yescharacteristic
89C2×D251—42no—
90C2×D251—44no—
91C4×C251#12712yes—
92C4×C251C22×D411yescharacteristic
93C4×C251C22×D411yescharacteristic
94Dic251—24no—
95Dic251—42no—
96C2×C251—22no—
97C2×C251—84no—
98C2×C251C2×#12711yescharacteristic
99C2×C251#57612yes—
100D251—82no—
101C251#315511yescharacteristic, complemented, Sylow
102#3155—2511nomaximal, Sylow
103C2×AGL(1,ℤ/8)—2512no—
104C2×#135—2511no—
105C22×D8D25111yescharacteristic, complemented
106#535—2512no—
107#496—2514no—
108#495—2512no—
109#493—2512no—
110#405—2511no—
111C2×#41—2512no—
112C22×D4C2×D25112yes—
113AGL(1,ℤ/8)—5028no—
114#135—5024no—
115C2×Q16—2511no—
116C2×SD16—2514no—
117C2×SD16—2512no—
118C2×D8—24no—
119C2×D8—2514no—
120C2×D8—2511no—
121C2×D8C2×D25112yes—
122C2×D8C2×D25112yes—
123C2×M16—2511no—
124C22×C8C2×D25111yescharacteristic
125#127—2514no—
126#124—2514no—
127#121—5024no—
128#120—5024no—
129#117—2511no—
130#97—2512no—
131C4×C8—2512no—
132C24—22no—
133#41—5024no—
134#41—5024no—
135C2×Q8—2512no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—5022no—
140C2×D4—5028no—
141C2×D4—2512no—
142C2×D4C22×D25114yes—
143C22×C4—5022no—
144C22×C4C22×D25111yescharacteristic
145Q16—5022no—
146SD16—5028no—
147SD16—5024no—
148D8—24no—
149D8—24no—
150D8—5028no—
151D8—5022no—
152M16—2514no—
153C2×C8—2512no—
154C2×C8C22⋊D25114yescomplemented
155C2×C8C22×D25111yescharacteristic
156C2×C8C22×D25111yescharacteristic
157C4⋊C4—2512no—
158C22⋊C4—5028no—
159C22⋊C4—2512no—
160C42—2512no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—5022no—
165C23D4×D25111yescharacteristic
166Q8—5022no—
167Q8—5022no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—20088no—
172D4—10044no—
173D4—5022no—
174D4—5022no—
175C2×C4—10044no—
176C2×C4—5022no—
177C2×C4—2514no—
178C2×C4D4×D25112yes—
179C2×C4D4×D25111yescharacteristic
180C2×C4C2×C22⋊D25112yes—
181C2×C4C23×D25111yescharacteristic
182C8—2514no—
183C8C2×C22⋊D25112yes—
184C8C2×C22⋊D25112yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—10042no—
190C22—10044no—
191C22#2254871112yes—
192C22C2×D4×D25111yescharacteristic
193C4—10042no—
194C4—5024no—
195C4#2254871112yes—
196C4C2×D4×D25111yescharacteristic
197C4C22×C22⋊D25111yescharacteristic
198C2—22no—
199C2—84no—
200C2—20082no—
201C2C2×#2254871111yescharacteristic
202C2#3063248812yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×D8×C251C2
socleC22×C251C22×D4
hypercenterC22×D8D251
hypocenterC251#3155
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. C2×#22548737C2semidirect
3. #30632254C2semidirect
4. #30632075C2semidirect
5. #30632070C2semidirect
6. #30632062C2semidirect
7. #30631642C2semidirect
8. C22×D8×C251C2semidirect
12. C2×Dic1004C22semidirect
13. C2×#16860799C22semidirect
14. C2×#16860798C22semidirect
15. C2×D2008C22semidirect
16. C2×#16860795C22semidirect
18. #22548711C22semidirect
19. #22548702C22semidirect
23. #22548625C22semidirect
24. C8×Dic251C22semidirect
42. #16860794D4semidirect
56. C2×C8×C251D4semidirect
101. C251#3155semidirect
105. C22×D8D251semidirect
154. C2×C8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
C2×#225487372
#306322542
#306320752
#306320702
#306320622
#306316422
C22×D8×C2512
#3155251