Finite Group Databasebeta

← #44559841

Subgroups — #44559841

Order 32128.

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

normal subgroups only

This group has 584 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
2C4×#16860810C211yesmaximal, characteristic, complemented
3#30631884C214yesmaximal, complemented
4#30631879C214yesmaximal, complemented
5#30631711C211yesmaximal, characteristic, complemented
6#30631692C212yesmaximal, complemented
7C2×#127×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9#22548741—28no—
10C2×#16860798C2214yescomplemented
11C2×#16860795C2214yescomplemented
12C2×#16860794C2212yes—
13C4×Dic502C2212yescomplemented
14C4×D1004C2212yescomplemented
15C4×C22⋊D251—22no—
16#22548674C2211yescharacteristic
17#22548672—21no—
18C42×D251—21no—
19C4⋊(C251⋊C8)C2214yescomplemented
20#22548636C2218yescomplemented
21C22×D4×C251—22no—
22C22×D4×C251C2212yes—
23#127×C251—21no—
24#127×C251—21no—
25#127×C251C2214yes—
26C2×C42×C251C2211yescharacteristic
27#16860810—22no—
28#16860810D414yescomplemented
29C2×Dic502C2311yescharacteristic
30C2×D1004C2311yescharacteristic
31C2×C22⋊D251—21no—
32C2×C4×D251—21no—
33C2×C4×D251—22no—
34#16860798—28no—
35#16860795—28no—
36#16860794—24no—
37C2×C251⋊C8C2314yes—
38C4⋊Dic251C2312yes—
39Dic251⋊C4—22no—
40C22⋊Dic251—21no—
41(C2×D251)⋊C4—22no—
42C4×Dic251—21no—
43C4×Dic251—22no—
44C24×C251—22no—
45C2×D4×C251—24no—
46C2×D4×C251—22no—
47C2×D4×C251—24no—
48C2×D4×C251—24no—
49C2×D4×C251—24no—
50C2×D4×C251—48no—
51C2×D4×C251C2314yes—
52C22×C4×C251D412yes—
53C22×C4×C251C2311yescharacteristic
54C42×C251D411yescharacteristic
55C42×C251D411yescharacteristic
56C42×C251C2312yes—
57C22×D251—21no—
58Dic502—21no—
59Dic502C2×D412yescomplemented
60D1004—21no—
61D1004C2×D412yescomplemented
62C22⋊D251—24no—
63C2×Dic251—21no—
64C2×Dic251—22no—
65C4×D251—24no—
66C4×D251—44no—
67C251⋊C8—24no—
68C23×C251—24no—
69C23×C251—48no—
70C23×C251—44no—
71C23×C251C2×D411yescharacteristic
72D4×C251—24no—
73D4×C251—24no—
74D4×C251—48no—
75D4×C251—44no—
76D4×C251—48no—
77C2×C4×C251—24no—
78C2×C4×C251C2×D412yes—
79C2×C4×C251C2×D412yes—
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D412yes—
82C2×C4×C251#4112yes—
83C2×C4×C251C2411yescharacteristic
84C22×C251—22no—
85C22×C251—44no—
86C22×C251—48no—
87C22×C251—88no—
88C22×C251#12112yes—
89C22×C251C22×D411yescharacteristic
90C2×D251—22no—
91C2×D251—42no—
92C4×C251—22no—
93C4×C251#12112yes—
94C4×C251AGL(1,ℤ/8)14yescomplemented
95C4×C251C22×D411yescharacteristic
96C4×C251C2×#4111yescharacteristic
97Dic251—22no—
98Dic251—42no—
99C2×C251—22no—
100C2×C251—84no—
101C2×C251C2×#12111yescharacteristic
102C2×C251C2×AGL(1,ℤ/8)12yes—
103D251—42no—
104C251#303811yescharacteristic, complemented, Sylow
105#3038—2511nomaximal, Sylow
106C2×AGL(1,ℤ/8)—2512no—
107C2×#127D25111yescharacteristic, complemented
108C4×#41—2511no—
109#460—2514no—
110#459—2514no—
111#423—2511no—
112#418—2512no—
113C2×#41—2511no—
114C22×D4—22no—
115C22×D4C2×D25112yes—
116AGL(1,ℤ/8)—5028no—
117C2×SD16—2514no—
118C2×D8—2514no—
119C2×M16—2512no—
120#127—21no—
121#127—21no—
122#127C2×D25114yes—
123C4×Q8—2512no—
124C4×D4—5022no—
125C4×D4—2512no—
126#117—5021no—
127#117—2511no—
128C2×C42—5021no—
129C2×C42C2×D25111yescharacteristic
130C4⋊C8—2514no—
131#102—2518no—
132C24—22no—
133#41—5022no—
134#41—2514no—
135C2×Q8—2511no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—48no—
142C2×D4—5021no—
143C2×D4—2511no—
144C2×D4C22×D25114yes—
145C22×C4—5022no—
146C22×C4—5021no—
147C22×C4C22⋊D25112yes—
148C22×C4C22×D25111yescharacteristic
149SD16—5028no—
150D8—5028no—
151M16—5024no—
152C2×C8—2514no—
153C4⋊C4—5022no—
154C4⋊C4—2512no—
155C22⋊C4—5021no—
156C22⋊C4—5022no—
157C42—5021no—
158C42—5022no—
159C42C22⋊D25111yescharacteristic
160C42C22⋊D25111yescharacteristic
161C42C22×D25112yes—
162C23—24no—
163C23—48no—
164C23—44no—
165C23—5021no—
166C23C2×C22⋊D25111yescharacteristic
167Q8—5021no—
168Q8—2512no—
169D4—24no—
170D4—24no—
171D4—48no—
172D4—48no—
173D4—44no—
174D4—5024no—
175D4—5021no—
176D4—2512no—
177C2×C4—24no—
178C2×C4—10044no—
179C2×C4—5022no—
180C2×C4—5021no—
181C2×C4—5024no—
182C2×C4D4×D25112yes—
183C2×C4C2×C22⋊D25112yes—
184C2×C4C2×C22⋊D25112yes—
185C2×C4C2×C22⋊D25111yescharacteristic
186C2×C4#1686080912yes—
187C2×C4C23×D25111yescharacteristic
188C8—5024no—
189C22—22no—
190C22—48no—
191C22—44no—
192C22—88no—
193C22—10042no—
194C22—5022no—
195C22#2254868612yes—
196C22C22×C22⋊D25111yescharacteristic
197C4—22no—
198C4—10042no—
199C4—5022no—
200C4#2254868612yes—
201C4#2254874114yescomplemented
202C4C2×D4×D25111yescharacteristic
203C4C2×#1686080911yescharacteristic
204C2—22no—
205C2—84no—
206C2—10042no—
207C2C2×#2254868611yescharacteristic
208C2C2×#2254874112yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#127×C251C2
socleC22×C251C22×D4
hypercenterC2×#127D251
hypocenterC251#3038
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. C4×#16860810C2semidirect
3. #30631884C2semidirect
4. #30631879C2semidirect
5. #30631711C2semidirect
6. #30631692C2semidirect
7. C2×#127×C251C2semidirect
10. C2×#16860798C22semidirect
11. C2×#16860795C22semidirect
13. C4×Dic502C22semidirect
14. C4×D1004C22semidirect
19. C4⋊(C251⋊C8)C22semidirect
20. #22548636C22semidirect
28. #16860810D4semidirect
59. Dic502C2×D4semidirect
61. D1004C2×D4semidirect
94. C4×C251AGL(1,ℤ/8)semidirect
104. C251#3038semidirect
107. C2×#127D251semidirect
201. C4#22548741semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
C4×#168608102
#306318842
#306318792
#306317112
#306316922
C2×#127×C2512
#3038251