Finite Group Databasebeta

← #44560762

Subgroups — #44560762

Order 32128.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#30632236C211yesmaximal, characteristic
2#125×D251C211yesmaximal, characteristic, complemented
3#30632054C212yesmaximal, complemented
4#30632050C212yesmaximal, complemented
5#30632044C211yesmaximal, characteristic, complemented
6#30632042C211yesmaximal, characteristic, complemented
7#30632041C211yesmaximal, characteristic, complemented
8#30631687C212yesmaximal, complemented
9#30631685C212yesmaximal, complemented
10C4×#16860793C211yesmaximal, characteristic, complemented
11#488×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2211yescharacteristic
13C2×#16860809C2211yescharacteristic
14C2×#16860793C2212yes—
15#22548713C2211yescharacteristic
16#22548711C2211yescharacteristic
17#22548707C2211yescharacteristic
18#22548706C2212yes—
19#22548705—22no—
20#22548703C2211yescharacteristic
21#22548688—22no—
22(C22⋊D251)⋊C4—22no—
23C4⋊C4×D251—22no—
24C4⋊C4×D251C2212yes—
25C42×D251C2211yescharacteristic
26#22548640C2212yes—
27#22548639C2212yes—
28#22548638C2212yes—
29#22548637C2212yes—
30#22548636C2212yes—
31#22548635C2212yes—
32#22548625C2212yes—
33C4×C251⋊C8C2211yescharacteristic
34#125×C251C2211yescharacteristic
35#124×C251C2211yescharacteristic
36#103×C251C2212yes—
37#102×C251C2212yes—
38C4×C8×C251C2211yescharacteristic
39#16860811—24no—
40#16860809—24no—
41C2×Dic502C2311yescharacteristic
42C2×D1004—21no—
43C2×D1004C2311yescharacteristic
44C2×C22⋊D251—21no—
45C2×C22⋊D251—22no—
46C22×Dic251—21no—
47C2×C4×D251—22no—
48C2×C4×D251—21no—
49C2×C4×D251D412yes—
50C2×C4×D251C2311yescharacteristic
51#16860793—24no—
52C2×C251⋊C8C2312yes—
53C4⋊Dic251—22no—
54C4⋊Dic251C2312yes—
55Dic251⋊C4—22no—
56Dic251⋊C4—22no—
57Dic251⋊C4—22no—
58Dic251⋊C4—22no—
59(C2×D251)⋊C4—22no—
60C4×Dic251D412yes—
61C4×Dic251C2311yescharacteristic
62C2×Q8×C251C2311yescharacteristic
63C2×D4×C251C2311yescharacteristic
64C2×C8×C251C2312yes—
65C4⋊C4×C251—22no—
66C4⋊C4×C251C2312yes—
67C22⋊C4×C251—22no—
68C42×C251C2311yescharacteristic
69C22×D251—21no—
70C22×D251C2×D411yescharacteristic
71Dic502—22no—
72D1004—22no—
73D1004—42no—
74C22⋊D251—44no—
75C22⋊D251—42no—
76C2×Dic251—21no—
77C2×Dic251—22no—
78C2×Dic251—42no—
79C2×Dic251C2×D412yes—
80C2×Dic251C2×D411yescharacteristic
81C4×D251—42no—
82C4×D251—44no—
83C4×D251—42no—
84C4×D251#4114yes—
85C251⋊C8—22no—
86C23×C251—21no—
87Q8×C251—22no—
88D4×C251—22no—
89C2×C4×C251—21no—
90C2×C4×C251—22no—
91C2×C4×C251C2×D412yes—
92C2×C4×C251C2411yescharacteristic
93C8×C251—22no—
94C22×C251—41no—
95C22×C251—42no—
96C22×C251C22×D411yescharacteristic
97C2×D251—22no—
98C2×D251—41no—
99C2×D251—42no—
100C2×D251#12412yes—
101C4×C251—22no—
102C4×C251—41no—
103C4×C251—42no—
104C4×C251C2×#4112yes—
105Dic251—22no—
106Dic251—41no—
107Dic251—42no—
108Dic251#12412yes—
109C2×C251—81no—
110C2×C251C2×#12411yescharacteristic
111C2×C251#57512yes—
112D251—22no—
113D251—81no—
114C251#314211yescharacteristic, complemented, Sylow
115#3142—2511nomaximal, Sylow
116#532—2511no—
117C2×#125—2511no—
118#491—2512no—
119#490—2512no—
120#488—2511no—
121#488—2511no—
122#488—2511no—
123#488D25111yescharacteristic, complemented
124#417—2512no—
125#417—2512no—
126C4×M16—2511no—
127C2×#41—2511no—
128C2×#41—2511no—
129C2×M16—2512no—
130C4⋊Q8—2511no—
131#127—2511no—
132#125—5022no—
133#125—2512no—
134#125—2511no—
135#125C2×D25111yescharacteristic
136#124—2511no—
137#124C2×D25111yescharacteristic
138#121—5022no—
139C4×D4—5022no—
140C2×C4⋊C4—5022no—
141C2×C4⋊C4—2512no—
142C2×C42—2511no—
143#103—2512no—
144#103—2512no—
145#103—2512no—
146#103C2×D25112yes—
147#102—2512no—
148#102—2512no—
149#102—2512no—
150#102C2×D25112yes—
151#97—2512no—
152C4×C8—2511no—
153C4×C8C2×D25111yescharacteristic
154#41—5024no—
155#41—5024no—
156C2×Q8—2511no—
157C2×Q8C22×D25111yescharacteristic
158C2×D4—5021no—
159C2×D4—5021no—
160C2×D4—5022no—
161C2×D4—2511no—
162C2×D4C22×D25111yescharacteristic
163C22×C4—5022no—
164C22×C4—5021no—
165C22×C4—5021no—
166C22×C4—2512no—
167C22×C4—2511no—
168M16—5024no—
169C2×C8—2512no—
170C2×C8C22×D25112yes—
171C4⋊C4—22no—
172C4⋊C4—5022no—
173C4⋊C4—5022no—
174C4⋊C4—5022no—
175C4⋊C4—5022no—
176C4⋊C4—5022no—
177C4⋊C4—2512no—
178C4⋊C4C22×D25112yes—
179C22⋊C4—22no—
180C22⋊C4—5022no—
181C42—2512no—
182C42—2511no—
183C42C22×D25111yescharacteristic
184C23—21no—
185C23—5021no—
186C23—2511no—
187Q8—22no—
188Q8—5022no—
189D4—22no—
190D4—10042no—
191D4—10042no—
192D4—10044no—
193D4—5022no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—10042no—
197C2×C4—10044no—
198C2×C4—10042no—
199C2×C4—10042no—
200C2×C4—5021no—
201C2×C4—5022no—
202C2×C4—2512no—
203C2×C4—2511no—
204C2×C4—2514no—
205C2×C4D4×D25112yes—
206C2×C4C23×D25111yescharacteristic
207C8—22no—
208C8—5022no—
209C22—41no—
210C22—42no—
211C22—10041no—
212C22—10042no—
213C22—5022no—
214C22—2512no—
215C22C2×D4×D25111yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—10041no—
220C4—10042no—
221C4—5022no—
222C4—2512no—
223C4#41×D25112yes—
224C2—81no—
225C2—20081no—
226C2—5022no—
227C2#124×D25111yescharacteristic
228C2#3063248412yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#488×C251C2
socleC22×C251C22×D4
prime radical#30632236C2
hypercenter#488D251
hypocenterC251#3142
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D251C2semidirect
3. #30632054C2semidirect
4. #30632050C2semidirect
5. #30632044C2semidirect
6. #30632042C2semidirect
7. #30632041C2semidirect
8. #30631687C2semidirect
9. #30631685C2semidirect
10. C4×#16860793C2semidirect
11. #488×C251C2semidirect
114. C251#3142semidirect
123. #488D251semidirect

Maximal subgroups

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

subgroupindex
#306322362
#125×D2512
#306320542
#306320502
#306320442
#306320422
#306320412
#306316872
#306316852
C4×#168607932
#488×C2512
#3142251