Finite Group Databasebeta

← #44549473

Subgroups — #44549473

Order 32128.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#30632407C211yesmaximal, characteristic, complemented
2#30632033C211yesmaximal, characteristic, complemented
3#30631884C211yesmaximal, characteristic, complemented
4#30631434C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
7#459×C251C211yesmaximal, characteristic
8C2×#16860798—21no—
9#22548715C2211yescharacteristic
10#22548708—22no—
11#22548702—21no—
12#22548694—21no—
13C4×Dic502C2211yescharacteristic
14#22548680—21no—
15#22548674—21no—
16C4⋊(C251⋊C8)C2211yescharacteristic
17#22548636—22no—
18#22548636—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×D8×C251—21no—
21#127×C251C2211yescharacteristic
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#102×C251—21no—
25C2×Dic502—21no—
26C2×Dic502D411yescharacteristic
27#16860798—42no—
28C2×C251⋊C8—21no—
29C2×C251⋊C8—21no—
30C4⋊Dic251—21no—
31C4⋊Dic251D411yescharacteristic
32C4⋊Dic251D412yes—
33Dic251⋊C4—22no—
34Dic251⋊C4—21no—
35C22⋊Dic251—22no—
36C22⋊Dic251—22no—
37C4×Dic251—22no—
38C4×Dic251—21no—
39C2×D4×C251—21no—
40C2×D4×C251—21no—
41C2×D4×C251—21no—
42C2×D4×C251D411yescharacteristic
43C22×C4×C251—21no—
44D8×C251—42no—
45C2×C8×C251—21no—
46C4⋊C4×C251D411yescharacteristic
47C22⋊C4×C251—21no—
48C42×C251C2311yescharacteristic
49Dic502—21no—
50Dic502—21no—
51Dic502—41no—
52Dic502—41no—
53C2×Dic251—21no—
54C2×Dic251—21no—
55C2×Dic251—22no—
56C251⋊C8—41no—
57C251⋊C8—41no—
58C23×C251—21no—
59C23×C251—41no—
60D4×C251—21no—
61D4×C251—21no—
62D4×C251—42no—
63D4×C251—41no—
64D4×C251—41no—
65D4×C251—41no—
66D4×C251—41no—
67C2×C4×C251—21no—
68C2×C4×C251—41no—
69C2×C4×C251—41no—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C8×C251—41no—
74C22×C251—41no—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251—81no—
78C22×C251—81no—
79C22×C251—81no—
80C22×C251#12011yescharacteristic
81C4×C251—21no—
82C4×C251—41no—
83C4×C251#13511yescharacteristic
84C4×C251#13511yescharacteristic
85C4×C251AGL(1,ℤ/8)11yescharacteristic
86C4×C251AGL(1,ℤ/8)11yescharacteristic
87Dic251—41no—
88Dic251—42no—
89Dic251—41no—
90C2×C251—81no—
91C2×C251—161no—
92C2×C251#44911yescharacteristic
93C2×C251#44911yescharacteristic
94C2×C251#45411yescharacteristic
95C251#162811yescharacteristic, complemented, Sylow
96#1628—2511nomaximal, Sylow
97#556—2511no—
98#486—2511no—
99#460—2511no—
100#459D25111yescharacteristic
101#332—2511no—
102Q8⋊C8—2511no—
103D4⋊C8—2511no—
104C2×SD16—5021no—
105C2×D8—21no—
106C4⋊Q8—2511no—
107#127C2×D25111yescharacteristic
108#126—5022no—
109#124—5021no—
110C22⋊Q8—5021no—
111C4×Q8—5021no—
112C4×Q8—2511no—
113C4×D4C2×D25111yescharacteristic
114#117—5021no—
115C4⋊C8—2511no—
116C4⋊C8C2×D25111yescharacteristic
117#102—21no—
118#102—5021no—
119#102—5022no—
120C4×C8—2511no—
121C2×Q8—5021no—
122C2×Q8—2511no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4C22⋊D25111yescharacteristic
127C22×C4—21no—
128SD16—10042no—
129D8—42no—
130C2×C8—21no—
131C2×C8—5021no—
132C2×C8—5021no—
133C4⋊C4—5022no—
134C4⋊C4—5021no—
135C4⋊C4—5021no—
136C4⋊C4—2511no—
137C4⋊C4—2512no—
138C4⋊C4C22⋊D25111yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—5022no—
141C22⋊C4—5022no—
142C42—5022no—
143C42—5021no—
144C42C22×D25111yescharacteristic
145C23—21no—
146C23—41no—
147Q8—10041no—
148Q8—10041no—
149Q8—5021no—
150Q8—5021no—
151D4—21no—
152D4—21no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—5021no—
162C2×C4—5021no—
163C2×C4—5022no—
164C2×C4D4×D25111yescharacteristic
165C2×C4D4×D25111yescharacteristic
166C2×C4C2×C22⋊D25111yescharacteristic
167C8—41no—
168C8—10041no—
169C8—10041no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#2254868311yescharacteristic
177C4—21no—
178C4—41no—
179C4—10041no—
180C4—10042no—
181C4—10041no—
182C4#2254873411yescharacteristic
183C4#2254873511yescharacteristic
184C4#2254873911yescharacteristic
185C4#2254874111yescharacteristic
186C2—81no—
187C2—161no—
188C2#3063182911yescharacteristic
189C2#3063183011yescharacteristic
190C2#3063185811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting, prime radical#459×C251C2
socleC22×C251#120
hypercenter#459D251
hypocenterC251#1628
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #30632407C2semidirect
2. #30632033C2semidirect
3. #30631884C2semidirect
4. #30631434C2semidirect
5. #30631417C2semidirect
6. D4⋊(C251⋊C8)C2semidirect
95. C251#1628semidirect

Maximal subgroups

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

subgroupindex
#306324072
#306320332
#306318842
#306314342
#306314172
D4⋊(C251⋊C8)2
#459×C2512
#1628251