Finite Group Databasebeta

← #44549342

Subgroups — #44549342

Order 32128.

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

normal subgroups only

This group has 260 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#30632274C211yesmaximal, characteristic
2#30631843C211yesmaximal, characteristic, complemented
3#30631598C211yesmaximal, characteristic, complemented
4#30631425C211yesmaximal, characteristic, complemented
5#30631421C211yesmaximal, characteristic, complemented
6#30631410C211yesmaximal, characteristic, complemented
7#448×C251C211yesmaximal, characteristic, complemented
8C2×#16860800—21no—
9#22548710—21no—
10#22548701—21no—
11#22548699—21no—
12#22548694C2211yescharacteristic
13#22548687—21no—
14#22548683—21no—
15D4×Dic251—21no—
16C2×(C2×D251)⋊C4—21no—
17C2×(C2×D251)⋊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
27C23×D251D411yescharacteristic
28C2×Dic502D411yescharacteristic
29C2×C22⋊D251—21no—
30C2×C22⋊D251—21no—
31C22×Dic251—21no—
32C22×Dic251—21no—
33C22×Dic251D411yescharacteristic
34#16860800—42no—
35C2×C251⋊C8—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251D411yescharacteristic
38Dic251⋊C4—21no—
39C22⋊Dic251—21no—
40C22⋊Dic251—21no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43(C2×D251)⋊C4—21no—
44(C2×D251)⋊C4—21no—
45(C2×D251)⋊C4—42no—
46C4×Dic251—21no—
47C2×Q8×C251—21no—
48C2×Q8×C251—21no—
49C2×Q8×C251—42no—
50C2×D4×C251—21no—
51C2×D4×C251D411yescharacteristic
52C22×C4×C251—21no—
53C22×C4×C251C2311yescharacteristic
54SD16×C251—42no—
55C2×C8×C251—21no—
56C4⋊C4×C251D411yescharacteristic
57C22⋊C4×C251—21no—
58C22×D251—21no—
59C22×D251—21no—
60C22×D251—21no—
61C22×D251—41no—
62C22×D251—41no—
63Dic502—41no—
64C22⋊D251—81no—
65C22⋊D251—81no—
66C2×Dic251—21no—
67C2×Dic251—21no—
68C2×Dic251—21no—
69C2×Dic251—41no—
70C2×Dic251—41no—
71C2×Dic251—41no—
72C2×Dic251—41no—
73C2×Dic251—81no—
74C251⋊C8—41no—
75C23×C251—21no—
76C23×C251C2×D411yescharacteristic
77Q8×C251—22no—
78Q8×C251—22no—
79Q8×C251—41no—
80Q8×C251—41no—
81D4×C251—41no—
82D4×C251—41no—
83D4×C251—41no—
84C2×C4×C251—22no—
85C2×C4×C251—21no—
86C2×C4×C251—41no—
87C2×C4×C251—41no—
88C2×C4×C251—41no—
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251C2×D411yescharacteristic
91C8×C251—41no—
92C22×C251—21no—
93C22×C251—21no—
94C22×C251—41no—
95C22×C251—41no—
96C22×C251—41no—
97C22×C251#12011yescharacteristic
98C22×C251#13511yescharacteristic
99C22×C251#13711yescharacteristic
100C2×D251—42no—
101C2×D251—41no—
102C2×D251—41no—
103C2×D251—41no—
104C2×D251—81no—
105C4×C251—21no—
106C4×C251—21no—
107C4×C251—41no—
108C4×C251—42no—
109Dic251—41no—
110Dic251—41no—
111Dic251—41no—
112Dic251—81no—
113C2×C251—21no—
114C2×C251—21no—
115C2×C251—81no—
116C2×C251#45211yescharacteristic
117C2×C251#45511yescharacteristic
118C2×C251B(4,𝔽2)11yescharacteristic
119D251—81no—
120C251#160611yescharacteristic, complemented, Sylow
121#1606—2511nomaximal, Sylow
122#538—2511no—
123#451—2511no—
124#448D25111yescharacteristic, complemented
125#393—2511no—
126#328—2511no—
127#327—2511no—
128#323—2511no—
129C22×Q8C2×D25111yescharacteristic
130C2×Q16—5021no—
131C2×SD16—21no—
132#126—5021no—
133#124—5021no—
134#123—5021no—
135C22⋊Q8—2511no—
136#121—5021no—
137#121C2×D25111yescharacteristic
138#120—5021no—
139C4×D4—5021no—
140C2×C22⋊C4—5021no—
141C2×C22⋊C4—2511no—
142#103—21no—
143#103—5021no—
144C22⋊C8—2511no—
145C22⋊C8C2×D25111yescharacteristic
146#95—5021no—
147#95—2511no—
148C24—2511no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—2511no—
153C2×D4—21no—
154C2×D4—5021no—
155C2×D4—5021no—
156C2×D4C22⋊D25111yescharacteristic
157C22×C4—21no—
158C22×C4—5021no—
159C22×C4—5021no—
160C22×C4—2511no—
161C22×C4C22×D25111yescharacteristic
162Q16—10042no—
163SD16—42no—
164C2×C8—21no—
165C2×C8—5021no—
166C4⋊C4—5021no—
167C4⋊C4—5021no—
168C4⋊C4—2511no—
169C4⋊C4C22⋊D25111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—10042no—
172C22⋊C4—5021no—
173C22⋊C4—5021no—
174C22⋊C4—5021no—
175C22⋊C4—5021no—
176C22⋊C4—5021no—
177C22⋊C4—5021no—
178C42—5021no—
179C23—21no—
180C23—10041no—
181C23—10041no—
182C23—5021no—
183C23—5021no—
184C23—5021no—
185C23D4×D25111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—10041no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—20081no—
195D4—20081no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—20081no—
202C2×C4—10041no—
203C2×C4—10041no—
204C2×C4—10041no—
205C2×C4—10041no—
206C2×C4—5021no—
207C2×C4—5021no—
208C2×C4—5021no—
209C2×C4D4×D25111yescharacteristic
210C2×C4C2×C22⋊D25111yescharacteristic
211C8—41no—
212C8—10041no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—20081no—
219C22—10042no—
220C22—10041no—
221C22—10041no—
222C22—10041no—
223C22#2254868311yescharacteristic
224C22#2254873611yescharacteristic
225C22#2254874411yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—20081no—
231C4—10041no—
232C4—10041no—
233C4—10041no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—20081no—
238C2#3063184611yescharacteristic
239C2#3063186311yescharacteristic
240C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#448×C251C2
socleC22×C251#120
prime radical#30632274C2
hypercenter#448D251
hypocenterC251#1606
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. #30631843C2semidirect
3. #30631598C2semidirect
4. #30631425C2semidirect
5. #30631421C2semidirect
6. #30631410C2semidirect
7. #448×C251C2semidirect
120. C251#1606semidirect
124. #448D251semidirect

Maximal subgroups

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

subgroupindex
#306322742
#306318432
#306315982
#306314252
#306314212
#306314102
#448×C2512
#1606251