Finite Group Databasebeta

← #44549360

Subgroups — #44549360

Order 32128.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#30632284C211yesmaximal, characteristic
2#30632017C211yesmaximal, characteristic, complemented
3#30632000C211yesmaximal, characteristic
4#30631423C211yesmaximal, characteristic, complemented
5#30631420C211yesmaximal, characteristic
6#30631413C211yesmaximal, characteristic, complemented
7#398×C251C211yesmaximal, characteristic, complemented
8#22548710—21no—
9#22548705—21no—
10#22548700—21no—
11#22548698—21no—
12#22548694C2211yescharacteristic
13#22548693—21no—
14#22548689C2211yescharacteristic
15C4×C22⋊D251—21no—
16#22548648—21no—
17#22548646—21no—
18#22548638—21no—
19#22548635—21no—
20C22⋊(C251⋊C8)C2211yescharacteristic
21C22⋊(C251⋊C8)C2211yescharacteristic
22C2×C4⋊C4×C251—21no—
23C2×C4⋊C4×C251C2211yescharacteristic
24C2×C4⋊C4×C251C2211yescharacteristic
25#95×C251—21no—
26#95×C251C2211yescharacteristic
27C2×Dic502D411yescharacteristic
28C2×D1004D411yescharacteristic
29C2×C22⋊D251—21no—
30C2×C4×D251—21no—
31C2×C251⋊C8—21no—
32C2×C251⋊C8—21no—
33C4⋊Dic251D411yescharacteristic
34C4⋊Dic251D411yescharacteristic
35Dic251⋊C4—21no—
36Dic251⋊C4—21no—
37Dic251⋊C4—21no—
38Dic251⋊C4—21no—
39C22⋊Dic251—21no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43(C2×D251)⋊C4—21no—
44C4×Dic251—21no—
45C22×C4×C251—21no—
46C22×C4×C251—21no—
47C22×C4×C251D411yescharacteristic
48C22×C4×C251D411yescharacteristic
49C22×C4×C251C2311yescharacteristic
50C4⋊C4×C251—21no—
51C4⋊C4×C251—21no—
52C4⋊C4×C251—21no—
53C4⋊C4×C251—21no—
54C4⋊C4×C251—42no—
55C22×D251—21no—
56Dic502—41no—
57D1004—41no—
58C22⋊D251—41no—
59C22⋊D251—41no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—21no—
63C4×D251—81no—
64C251⋊C8—41no—
65C251⋊C8—41no—
66C23×C251C2×D411yescharacteristic
67C2×C4×C251—22no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—21no—
71C2×C4×C251—21no—
72C2×C4×C251—41no—
73C2×C4×C251—41no—
74C2×C4×C251—41no—
75C2×C4×C251—41no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251—41no—
79C2×C4×C251C2×D411yescharacteristic
80C2×C4×C251C2×D411yescharacteristic
81C22×C251—21no—
82C22×C251—21no—
83C22×C251#12011yescharacteristic
84C22×C251#13511yescharacteristic
85C22×C251AGL(1,ℤ/8)11yescharacteristic
86C2×D251—41no—
87C2×D251—41no—
88C2×D251—41no—
89C4×C251—21no—
90C4×C251—21no—
91C4×C251—42no—
92C4×C251—41no—
93C4×C251—41no—
94C4×C251—81no—
95Dic251—41no—
96Dic251—41no—
97Dic251—41no—
98C2×C251—21no—
99C2×C251—21no—
100C2×C251#44911yescharacteristic
101C2×C251#45611yescharacteristic
102C2×C251#45811yescharacteristic
103D251—81no—
104C251#160911yescharacteristic, complemented, Sylow
105#1609—2511nomaximal, Sylow
106#539—2511no—
107#483—2511no—
108#480—2511no—
109#398D25111yescharacteristic, complemented
110#328—2511no—
111#327—2511no—
112#324—2511no—
113#126—5021no—
114#125—5021no—
115#123—5021no—
116#123—5021no—
117C22⋊Q8—5021no—
118C22⋊Q8—2511no—
119#121—2511no—
120C4×D4—5021no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×D25111yescharacteristic
123C2×C4⋊C4C2×D25111yescharacteristic
124#107—5021no—
125#106—5021no—
126#103—5021no—
127#102—5021no—
128C22⋊C8—2511no—
129C22⋊C8—2511no—
130#95—21no—
131#95C2×D25111yescharacteristic
132C2×Q8—2511no—
133C2×D4—5021no—
134C2×D4—2511no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—5021no—
138C22×C4C22⋊D25111yescharacteristic
139C22×C4C22⋊D25111yescharacteristic
140C22×C4C22×D25111yescharacteristic
141C2×C8—5021no—
142C2×C8—5021no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—5021no—
149C4⋊C4—5021no—
150C4⋊C4—5021no—
151C4⋊C4—5021no—
152C4⋊C4—2511no—
153C4⋊C4—2511no—
154C22⋊C4—5021no—
155C22⋊C4—5021no—
156C22⋊C4—5021no—
157C22⋊C4—5021no—
158C22⋊C4—5021no—
159C42—5021no—
160C23—5021no—
161C23C2×C22⋊D25111yescharacteristic
162Q8—10041no—
163D4—10041no—
164D4—10041no—
165D4—10041no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—20081no—
179C2×C4—5021no—
180C2×C4—5021no—
181C2×C4—5021no—
182C2×C4D4×D25111yescharacteristic
183C2×C4D4×D25111yescharacteristic
184C8—10041no—
185C8—10041no—
186C22—21no—
187C22—21no—
188C22—10041no—
189C22—10041no—
190C22—10041no—
191C22#2254868311yescharacteristic
192C22#2254873511yescharacteristic
193C22#2254874111yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4—10041no—
201C4—10041no—
202C4—10041no—
203C2—21no—
204C2—21no—
205C2—20081no—
206C2#3063183011yescharacteristic
207C2#3063186911yescharacteristic
208C2#3063187611yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#398×C251C2
socleC22×C251#120
prime radical#22548689C22
hypercenter#398D251
hypocenterC251#1609
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #30632017C2semidirect
4. #30631423C2semidirect
6. #30631413C2semidirect
7. #398×C251C2semidirect
104. C251#1609semidirect
109. #398D251semidirect

Maximal subgroups

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

subgroupindex
#306322842
#306320172
#306320002
#306314232
#306314202
#306314132
#398×C2512
#1609251