Finite Group Databasebeta

← #488×D251 #44560678

Subgroups — #488×D251 #44560678

Order 32128.

412 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 125 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#125×D251C211yesmaximal, characteristic, complemented
2#124×D251C211yesmaximal, characteristic
3#30632044C211yesmaximal, characteristic, complemented
4#30632043C212yesmaximal, complemented
5#30632042C211yesmaximal, characteristic, complemented
6#30632041C211yesmaximal, characteristic, complemented
7#30632040C212yesmaximal, complemented
8#103×D251C212yesmaximal, complemented
9#102×D251C212yesmaximal
10C4×C8×D251C211yesmaximal, characteristic, complemented
11#488×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251C2211yescharacteristic
13C2×D4×D251C2211yescharacteristic
14C2×C8×D251C2212yes—
15#22548707C2211yescharacteristic, complemented
16#22548706C2212yescomplemented
17#22548705—22no—
18#22548704C2211yescharacteristic
19#22548703C2211yescharacteristic
20#22548702C2211yescharacteristic
21#22548701—22no—
22C4⋊C4×D251—22no—
23C4⋊C4×D251C2212yes—
24C22⋊C4×D251—22no—
25C42×D251C2211yescharacteristic
26#22548640C2212yescomplemented
27#22548639C2212yescomplemented
28#22548638C2212yescomplemented
29#22548637C2212yes—
30#22548636C2212yes—
31#22548635C2212yes—
32C4×C251⋊C8C2211yescharacteristic, complemented
33C8×Dic251C2212yescomplemented
34#125×C251C2211yescharacteristic, complemented
35#124×C251C2211yescharacteristic
36#103×C251C2212yescomplemented
37#102×C251C2212yes—
38C4×C8×C251C2211yescharacteristic, complemented
39C23×D251—21no—
40C2×Dic502—21no—
41C2×Dic502C2311yescharacteristic
42Q8×D251—24no—
43C2×D1004C2311yescharacteristic
44C2×C22⋊D251—21no—
45D4×D251—24no—
46C2×C4×D251—21no—
47C2×C4×D251—22no—
48C2×C4×D251D412yes—
49C2×C4×D251C2311yescharacteristic
50C2×C251⋊C8C2312yes—
51C8×D251—24no—
52C4⋊Dic251—22no—
53C4⋊Dic251C2312yes—
54Dic251⋊C4—22no—
55Dic251⋊C4—22no—
56Dic251⋊C4—22no—
57C22⋊Dic251—22no—
58(C2×D251)⋊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×D251—42no—
71C22×D251—44no—
72C22×D251C2×D411yescharacteristic
73Dic502—22no—
74Dic502—42no—
75D1004—22no—
76C22⋊D251—42no—
77C2×Dic251—21no—
78C2×Dic251—22no—
79C2×Dic251C2×D412yes—
80C2×Dic251C2×D411yescharacteristic
81C4×D251—24no—
82C4×D251—42no—
83C4×D251—44no—
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—41no—
98C2×D251—42no—
99C2×D251—84no—
100C2×D251#12412yes—
101C2×D251#13514yes—
102C4×C251—22no—
103C4×C251—41no—
104C4×C251—42no—
105C4×C251C2×#4112yes—
106Dic251—22no—
107Dic251—41no—
108Dic251—42no—
109Dic251#12412yes—
110C2×C251—81no—
111C2×C251C2×#12411yescharacteristic
112C2×C251C2×#13512yes—
113D251—81no—
114D251#48814yesdirect product factor
115C251C2×#48811yescharacteristic, complemented, Sylow
116C2×#488—2511nomaximal, Sylow
117C2×#125—2511no—
118C2×#124—2511no—
119#488—2511no—
120#488—2511no—
121#488—2512no—
122#488—2512no—
123#488—2511no—
124#488D25111yescharacteristic, direct product factor
125C2×#103—2512no—
126C2×#102—2512no—
127C2×C4×C8—2511no—
128C22×Q8—2511no—
129C22×D4—2511no—
130C22×C8—2512no—
131#125—5022no—
132#125—2512no—
133#125—2511no—
134#125C2×D25111yescharacteristic, complemented
135#124—5022no—
136#124—2511no—
137#124—2511no—
138#124—2511no—
139#124C2×D25111yescharacteristic
140C2×C4⋊C4—5022no—
141C2×C4⋊C4—2512no—
142C2×C22⋊C4—5022no—
143C2×C42—2511no—
144#103—2512no—
145#103—2512no—
146#103—2512no—
147#103C2×D25112yescomplemented
148#102—2512no—
149#102—2512no—
150#102—2512no—
151#102C2×D25112yes—
152C4×C8—2512no—
153C4×C8—2511no—
154C4×C8C2×D25111yescharacteristic, complemented
155C24—5021no—
156C2×Q8—5021no—
157C2×Q8—5024no—
158C2×Q8—2511no—
159C2×Q8C22×D25111yescharacteristic
160C2×D4—5024no—
161C2×D4—5021no—
162C2×D4—2511no—
163C2×D4C22×D25111yescharacteristic
164C22×C4—5021no—
165C22×C4—5022no—
166C22×C4—2512no—
167C22×C4—2511no—
168C2×C8—5024no—
169C2×C8—2512no—
170C2×C8C22×D25112yes—
171C4⋊C4—22no—
172C4⋊C4—5022no—
173C4⋊C4—5022no—
174C4⋊C4—5022no—
175C4⋊C4—5022no—
176C4⋊C4—2512no—
177C4⋊C4C22×D25112yes—
178C22⋊C4—22no—
179C22⋊C4—5022no—
180C22⋊C4—5022no—
181C22⋊C4—5022no—
182C42—2512no—
183C42—2511no—
184C42C22×D25111yescharacteristic
185C23—21no—
186C23—10042no—
187C23—10044no—
188C23—5021no—
189C23—2511no—
190Q8—22no—
191Q8—10042no—
192Q8—5022no—
193D4—22no—
194D4—10042no—
195D4—5022no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—10042no—
199C2×C4—10044no—
200C2×C4—5024no—
201C2×C4—5021no—
202C2×C4—5022no—
203C2×C4—2512no—
204C2×C4—2511no—
205C2×C4—2514no—
206C2×C4D4×D25112yes—
207C2×C4C23×D25111yescharacteristic
208C8—22no—
209C8—5022no—
210C22—41no—
211C22—42no—
212C22—20084no—
213C22—10041no—
214C22—10042no—
215C22—2512no—
216C22—2514no—
217C22C2×D4×D25111yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—10041no—
222C4—10042no—
223C4—5022no—
224C4—2512no—
225C4#41×D25112yes—
226C2—81no—
227C2—20081no—
228C2—2514no—
229C2#124×D25111yescharacteristic
230C2#135×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#488×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#488D251
hypocenterC251C2×#488
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
1. #125×D251C2semidirect
3. #30632044C2semidirect
4. #30632043C2semidirect
5. #30632042C2semidirect
6. #30632041C2semidirect
7. #30632040C2semidirect
8. #103×D251C2semidirect
10. C4×C8×D251C2semidirect
11. #488×C251C2semidirect
15. #22548707C22semidirect
16. #22548706C22semidirect
26. #22548640C22semidirect
27. #22548639C22semidirect
28. #22548638C22semidirect
32. C4×C251⋊C8C22semidirect
33. C8×Dic251C22semidirect
34. #125×C251C22semidirect
36. #103×C251C22semidirect
38. C4×C8×C251C22semidirect
114. D251#488direct (N × H)
115. C251C2×#488semidirect
124. #488D251direct (N × H)
134. #125C2×D251semidirect
147. #103C2×D251semidirect
154. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#125×D2512
#124×D2512
#306320442
#306320432
#306320422
#306320412
#306320402
#103×D2512
#102×D2512
C4×C8×D2512
#488×C2512
C2×#488251