Finite Group Databasebeta

← #44549332

Subgroups — #44549332

Order 32128.

282 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 51 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#30632264C211yesmaximal, characteristic
2#30631826C211yesmaximal, characteristic, complemented
3#30631606C211yesmaximal, characteristic, complemented
4#30631422C211yesmaximal, characteristic, complemented
5#30631421C211yesmaximal, characteristic, complemented
6#30631412C211yesmaximal, characteristic, complemented
7#448×C251C211yesmaximal, characteristic, complemented
8C2×#16860809—21no—
9C2×#16860797—21no—
10#22548693—21no—
11#22548692—21no—
12#22548689C2211yescharacteristic
13#22548688—21no—
14#22548687—21no—
15#22548687—21no—
16C2×Dic251⋊C4—21no—
17C2×Dic251⋊C4C2211yescharacteristic
18#22548639—21no—
19C22⋊(C251⋊C8)C2211yescharacteristic
20C251⋊#95—21no—
21C251⋊#95C2211yescharacteristic
22C22×Q8×C251C2211yescharacteristic
23C2×SD16×C251—21no—
24#121×C251C2211yescharacteristic
25#103×C251—21no—
26C22⋊C8×C251C2211yescharacteristic
27#16860809—42no—
28#16860809—42no—
29C2×Dic502—21no—
30C2×D1004D411yescharacteristic
31C2×C22⋊D251—21no—
32C2×C22⋊D251—21no—
33C2×C22⋊D251—21no—
34C22×Dic251—21no—
35C22×Dic251—21no—
36C22×Dic251D411yescharacteristic
37C22×Dic251D411yescharacteristic
38C2×C4×D251—21no—
39#16860797—42no—
40C2×C251⋊C8—21no—
41C4⋊Dic251D411yescharacteristic
42Dic251⋊C4—21no—
43Dic251⋊C4—21no—
44Dic251⋊C4—21no—
45Dic251⋊C4—42no—
46C22⋊Dic251—21no—
47(C2×D251)⋊C4—21no—
48(C2×D251)⋊C4—21no—
49C2×Q8×C251—21no—
50C2×Q8×C251—21no—
51C2×Q8×C251—42no—
52C2×D4×C251—21no—
53C2×D4×C251D411yescharacteristic
54C22×C4×C251—21no—
55C22×C4×C251C2311yescharacteristic
56SD16×C251—42no—
57C2×C8×C251—21no—
58C4⋊C4×C251D411yescharacteristic
59C22⋊C4×C251—21no—
60C22×D251—21no—
61Dic502—42no—
62D1004—41no—
63C22⋊D251—42no—
64C22⋊D251—41no—
65C22⋊D251—41no—
66C22⋊D251—81no—
67C2×Dic251—21no—
68C2×Dic251—21no—
69C2×Dic251—22no—
70C2×Dic251—21no—
71C2×Dic251—21no—
72C2×Dic251—42no—
73C2×Dic251—41no—
74C2×Dic251—41no—
75C2×Dic251—41no—
76C2×Dic251SD1612yescomplemented
77C4×D251—42no—
78C251⋊C8—41no—
79C23×C251—21no—
80C23×C251C2×D411yescharacteristic
81Q8×C251—22no—
82Q8×C251—22no—
83Q8×C251—41no—
84Q8×C251—41no—
85D4×C251—41no—
86D4×C251—41no—
87D4×C251—41no—
88C2×C4×C251—22no—
89C2×C4×C251—21no—
90C2×C4×C251—41no—
91C2×C4×C251—41no—
92C2×C4×C251—41no—
93C2×C4×C251C2×D411yescharacteristic
94C2×C4×C251C2×D411yescharacteristic
95C8×C251—41no—
96C22×C251—21no—
97C22×C251—21no—
98C22×C251—41no—
99C22×C251—41no—
100C22×C251—41no—
101C22×C251#12011yescharacteristic
102C22×C251C2×SD1611yescharacteristic
103C22×C251AGL(1,ℤ/8)11yescharacteristic
104C2×D251—41no—
105C2×D251—41no—
106C2×D251—41no—
107C4×C251—21no—
108C4×C251—21no—
109C4×C251—41no—
110C4×C251—42no—
111Dic251—22no—
112Dic251—42no—
113Dic251—41no—
114Dic251—41no—
115C2×C251—21no—
116C2×C251—21no—
117C2×C251—81no—
118C2×C251#45011yescharacteristic
119C2×C251#45611yescharacteristic
120C2×C251B(4,𝔽2)11yescharacteristic
121D251—81no—
122C251#160411yescharacteristic, complemented, Sylow
123#1604—2511nomaximal, Sylow
124#537—2511no—
125#448—2511no—
126#448D25111yescharacteristic, complemented
127#395—2511no—
128#327—2511no—
129#327—2511no—
130#324—2511no—
131C2×#41—5021no—
132C22×Q8C2×D25111yescharacteristic
133C2×SD16—21no—
134C2×SD16—5021no—
135C22⋊Q8—5021no—
136C22⋊Q8—5021no—
137#121—5021no—
138#121—5021no—
139#121—5021no—
140#121—2511no—
141#121C2×D25111yescharacteristic
142C2×C4⋊C4—5021no—
143C2×C4⋊C4—2511no—
144#103—21no—
145#103—5021no—
146C22⋊C8—2511no—
147C22⋊C8C2×D25111yescharacteristic
148#95—5021no—
149#95—2511no—
150#41—10042no—
151#41—10042no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—42no—
155C2×Q8—5021no—
156C2×D4—21no—
157C2×D4—5021no—
158C2×D4—5021no—
159C2×D4—5021no—
160C2×D4—2511no—
161C2×D4C22⋊D25111yescharacteristic
162C22×C4—21no—
163C22×C4—5021no—
164C22×C4—5021no—
165C22×C4—5021no—
166C22×C4—2511no—
167C22×C4—2511no—
168C22×C4C22×D25111yescharacteristic
169SD16—42no—
170SD16—10042no—
171C2×C8—21no—
172C2×C8—5021no—
173C4⋊C4—10042no—
174C4⋊C4—5021no—
175C4⋊C4—5021no—
176C4⋊C4—5021no—
177C4⋊C4—2511no—
178C4⋊C4C22⋊D25111yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—5021no—
181C22⋊C4—5021no—
182C22⋊C4—5021no—
183C23—21no—
184C23—5021no—
185C23D4×D25111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—10042no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—20081no—
195D4—10041no—
196D4—10041no—
197D4—10041no—
198D4—10042no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—10042no—
205C2×C4—10041no—
206C2×C4—10041no—
207C2×C4—10042no—
208C2×C4—10041no—
209C2×C4—5022no—
210C2×C4—5021no—
211C2×C4—5021no—
212C2×C4—5021no—
213C2×C4—5021no—
214C2×C4—2512no—
215C2×C4D4×D25111yescharacteristic
216C2×C4C2×C22⋊D25111yescharacteristic
217C8—41no—
218C8—10041no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—10041no—
225C22—10041no—
226C22—10041no—
227C22#2254868311yescharacteristic
228C22SD16×D25111yescharacteristic
229C22#2254874211yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—42no—
234C4—10042no—
235C4—10041no—
236C4—10041no—
237C4—5022no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—20081no—
242C2#3063183611yescharacteristic
243C2#3063186811yescharacteristic
244C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#448×C251C2
socleC22×C251#120
prime radical#30632264C2
hypercenter#448D251
hypocenterC251#1604
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #30631826C2semidirect
3. #30631606C2semidirect
4. #30631422C2semidirect
5. #30631421C2semidirect
6. #30631412C2semidirect
7. #448×C251C2semidirect
76. C2×Dic251SD16semidirect
122. C251#1604semidirect
126. #448D251semidirect

Maximal subgroups

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

subgroupindex
#306322642
#306318262
#306316062
#306314222
#306314212
#306314122
#448×C2512
#1604251