Finite Group Databasebeta

← #44549679

Subgroups — #44549679

Order 32128.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#30632431C211yesmaximal, characteristic
2#30632093C211yesmaximal, characteristic, complemented
3#30632049C211yesmaximal, characteristic, complemented
4#30631428C211yesmaximal, characteristic, complemented
5#30631427C211yesmaximal, characteristic, complemented
6#30631426C211yesmaximal, characteristic, complemented
7#497×C251C211yesmaximal, characteristic, complemented
8#22548707C2211yescharacteristic
9#22548703C2211yescharacteristic
10#22548699—21no—
11#22548697—21no—
12#22548692—21no—
13#22548691—21no—
14#22548688—21no—
15#22548687—21no—
16#22548648—21no—
17#22548646—21no—
18#22548638—21no—
19#22548635—21no—
20#22548626C2211yescharacteristic
21#22548626C2211yescharacteristic
22C2×Q16×C251—21no—
23C2×SD16×C251—21no—
24C4⋊Q8×C251C2211yescharacteristic
25#124×C251C2211yescharacteristic
26#97×C251C2211yescharacteristic
27C2×Dic502D411yescharacteristic
28C2×D1004D411yescharacteristic
29C2×C22⋊D251—21no—
30C2×C22⋊D251—21no—
31C22×Dic251—21no—
32C2×C4×D251—21no—
33C2×C251⋊C8—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251D411yescharacteristic
37C4⋊Dic251D411yescharacteristic
38Dic251⋊C4—21no—
39Dic251⋊C4—21no—
40Dic251⋊C4—21no—
41C22⋊Dic251—21no—
42(C2×D251)⋊C4—21no—
43(C2×D251)⋊C4—21no—
44(C2×D251)⋊C4—21no—
45C2×Q8×C251—21no—
46C2×Q8×C251D411yescharacteristic
47C2×D4×C251D411yescharacteristic
48Q16×C251—42no—
49SD16×C251—42no—
50C2×C8×C251—21no—
51C4⋊C4×C251—21no—
52C4⋊C4×C251—21no—
53C22⋊C4×C251—21no—
54C22⋊C4×C251—21no—
55C42×C251C2311yescharacteristic
56C22×D251—21no—
57Dic502—41no—
58D1004—41no—
59C22⋊D251—81no—
60C22⋊D251—81no—
61C2×Dic251—21no—
62C2×Dic251—21no—
63C2×Dic251—21no—
64C2×Dic251—81no—
65C4×D251—81no—
66C251⋊C8—22no—
67C251⋊C8—41no—
68C23×C251—21no—
69Q8×C251—41no—
70Q8×C251—41no—
71Q8×C251—41no—
72D4×C251—41no—
73C2×C4×C251—21no—
74C2×C4×C251—41no—
75C2×C4×C251C2×D411yescharacteristic
76C2×C4×C251C2×D411yescharacteristic
77C2×C4×C251C2×D411yescharacteristic
78C8×C251—22no—
79C22×C251—41no—
80C22×C251—41no—
81C22×C251—41no—
82C22×C251#12011yescharacteristic
83C2×D251—41no—
84C2×D251—41no—
85C2×D251—41no—
86C4×C251—21no—
87C4×C251—21no—
88C4×C251—21no—
89C4×C251—41no—
90C4×C251—81no—
91Dic251—41no—
92Dic251—41no—
93Dic251—41no—
94C2×C251—81no—
95C2×C251#45311yescharacteristic
96C2×C251#45511yescharacteristic
97C2×C251#45611yescharacteristic
98D251—81no—
99C251#166111yescharacteristic, complemented, Sylow
100#1661—2511nomaximal, Sylow
101#562—2511no—
102#501—2511no—
103#497D25111yescharacteristic, complemented
104#489—2511no—
105#330—2511no—
106#329—2511no—
107#329—2511no—
108C2×Q16—21no—
109C2×SD16—21no—
110C4⋊Q8C2×D25111yescharacteristic
111#125—2511no—
112#124—2511no—
113#124C2×D25111yescharacteristic
114#123—5021no—
115#123—5021no—
116C22⋊Q8—5021no—
117C22⋊Q8—5021no—
118#121—5021no—
119#121—5021no—
120#107—5021no—
121#106—5021no—
122#103—5021no—
123#102—5021no—
124#97—2511no—
125#97—2511no—
126#97C2×D25111yescharacteristic
127C2×Q8—21no—
128C2×Q8—2511no—
129C2×Q8C22⋊D25111yescharacteristic
130C2×D4—5021no—
131C2×D4—5021no—
132C2×D4—2511no—
133C2×D4C22⋊D25111yescharacteristic
134C22×C4—5021no—
135C22×C4—5021no—
136Q16—42no—
137SD16—42no—
138C2×C8—21no—
139C2×C8—5021no—
140C2×C8—5021no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—5021no—
144C4⋊C4—5021no—
145C4⋊C4—5021no—
146C4⋊C4—5021no—
147C4⋊C4—2511no—
148C4⋊C4—2511no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—5021no—
152C22⋊C4—5021no—
153C22⋊C4—5021no—
154C22⋊C4—5021no—
155C42C22×D25111yescharacteristic
156C23—21no—
157C23—5021no—
158Q8—41no—
159Q8—41no—
160Q8—41no—
161Q8—10041no—
162D4—41no—
163D4—20081no—
164D4—20081no—
165D4—10041no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—20081no—
169C2×C4—20081no—
170C2×C4—5021no—
171C2×C4—5021no—
172C2×C4—5021no—
173C2×C4D4×D25111yescharacteristic
174C2×C4D4×D25111yescharacteristic
175C2×C4C2×C22⋊D25111yescharacteristic
176C8—22no—
177C8—10041no—
178C8—5022no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—10041no—
183C22—10041no—
184C22—10041no—
185C22#2254868311yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—81no—
191C4—10041no—
192C4—10041no—
193C4—10041no—
194C2—81no—
195C2—20081no—
196C2#3063185411yescharacteristic
197C2#3063186311yescharacteristic
198C2#3063186811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#497×C251C2
socleC22×C251#120
prime radical#30632431C2
hypercenter#497D251
hypocenterC251#1661
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #30632093C2semidirect
3. #30632049C2semidirect
4. #30631428C2semidirect
5. #30631427C2semidirect
6. #30631426C2semidirect
7. #497×C251C2semidirect
99. C251#1661semidirect
103. #497D251semidirect

Maximal subgroups

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

subgroupindex
#306324312
#306320932
#306320492
#306314282
#306314272
#306314262
#497×C2512
#1661251