Finite Group Databasebeta

← #44560321

Subgroups — #44560321

Order 32128.

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

normal subgroups only

This group has 418 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#30632254C211yesmaximal, characteristic
2#30632167C211yesmaximal, characteristic, complemented
3#30632010C212yesmaximal, complemented
4#30632009C212yesmaximal, complemented
5#30632004C212yesmaximal, complemented
6#30631999C212yesmaximal, complemented
7#30631729C211yesmaximal, characteristic, complemented
8#30631723C211yesmaximal, characteristic, complemented
9C2×#22548636C211yesmaximal, characteristic, complemented
10#30631655C211yesmaximal, characteristic, complemented
11#418×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×C251⋊C8C2211yescharacteristic
14#22548711—21no—
15#22548702—21no—
16#22548686C2212yes—
17#22548686C2212yes—
18#22548684—22no—
19C4×Dic502—21no—
20C4×D1004—21no—
21(C22⋊D251)⋊C4—22no—
22#22548674C2211yescharacteristic
23#22548673C2212yescomplemented
24C2×C4⋊Dic251C2211yescharacteristic
25C4⋊C4×D251C2212yescomplemented
26#22548649C2212yescomplemented
27#22548648C2211yescharacteristic, complemented
28#22548648C2211yescharacteristic, complemented
29#22548647C2212yescomplemented
30#22548646C2211yescharacteristic, complemented
31#22548646C2211yescharacteristic, complemented
32#22548636C2212yes—
33#22548636C2212yes—
34(C2×D251)⋊C8C2212yescomplemented
35(C2×D251)⋊C8C2212yescomplemented
36C22×D4×C251C2211yescharacteristic
37C2×M16×C251C2211yescharacteristic
38#117×C251C2211yescharacteristic
39#102×C251C2212yes—
40#102×C251C2212yes—
41#16860810—22no—
42#16860810—22no—
43C2×Dic502D411yescharacteristic
44C2×D1004D411yescharacteristic
45C2×C22⋊D251—22no—
46C2×C22⋊D251—22no—
47C2×C22⋊D251D412yes—
48C22×Dic251—21no—
49C2×C4×D251—22no—
50C2×C4×D251C2312yes—
51C2×C251⋊C8—22no—
52C2×C251⋊C8C2311yescharacteristic
53C2×C251⋊C8C2311yescharacteristic
54C4⋊Dic251—21no—
55C4⋊Dic251C2312yes—
56C4⋊Dic251C2311yescharacteristic
57C4⋊Dic251C2311yescharacteristic
58Dic251⋊C4—22no—
59C22⋊Dic251—22no—
60C22⋊Dic251—22no—
61C22⋊Dic251—21no—
62(C2×D251)⋊C4—22no—
63C4×Dic251—22no—
64C4×Dic251—21no—
65C24×C251—21no—
66C2×D4×C251—21no—
67C2×D4×C251—22no—
68C2×D4×C251—22no—
69C2×D4×C251C2312yes—
70C22×C4×C251C2311yescharacteristic
71M16×C251—22no—
72C2×C8×C251C2312yes—
73C4⋊C4×C251C2311yescharacteristic
74C4⋊C4×C251C2311yescharacteristic
75C22⋊C4×C251—21no—
76C42×C251—21no—
77C22×D251C2×D412yes—
78Dic502—22no—
79D1004—22no—
80C22⋊D251—24no—
81C22⋊D251—82no—
82C22⋊D251—82no—
83C2×Dic251—22no—
84C2×Dic251—21no—
85C2×Dic251—21no—
86C2×Dic251—81no—
87C2×Dic251C2×D412yes—
88C4×D251—22no—
89C4×D251—22no—
90C4×D251—82no—
91C251⋊C8—21no—
92C251⋊C8—21no—
93C23×C251—22no—
94C23×C251—42no—
95C23×C251—81no—
96C23×C251—81no—
97C23×C251C2×D411yescharacteristic
98D4×C251—22no—
99D4×C251—22no—
100D4×C251—42no—
101C2×C4×C251—21no—
102C2×C4×C251—21no—
103C2×C4×C251C2×D411yescharacteristic
104C2×C4×C251#4112yes—
105C2×C4×C251#4112yes—
106C2×C4×C251C2411yescharacteristic
107C8×C251—22no—
108C22×C251—21no—
109C22×C251—21no—
110C22×C251—42no—
111C22×C251—42no—
112C22×C251—42no—
113C22×C251—84no—
114C22×C251#12312yes—
115C22×C251C22×D411yescharacteristic
116C2×D251—22no—
117C2×D251—22no—
118C2×D251—22no—
119C4×C251—41no—
120C4×C251—41no—
121C4×C251#12312yes—
122C4×C251C2×#4111yescharacteristic
123C4×C251C2×#4111yescharacteristic
124Dic251—22no—
125Dic251—42no—
126Dic251—41no—
127Dic251—41no—
128C2×C251—22no—
129C2×C251—82no—
130C2×C251C2×#12311yescharacteristic
131C2×C251#57611yescharacteristic
132C2×C251#57711yescharacteristic
133D251—42no—
134C251#309611yescharacteristic, complemented, Sylow
135#3096—2511nomaximal, Sylow
136#535—2511no—
137#520—2511no—
138#482—2512no—
139#482—2512no—
140#481—2512no—
141#480—2512no—
142#428—2511no—
143#427—2511no—
144#418D25111yescharacteristic, complemented
145C2×#102—2511no—
146#408—2511no—
147C2×#41—2511no—
148C22×D4C2×D25111yescharacteristic
149C2×M16C2×D25111yescharacteristic
150C22×C8—2511no—
151#127—5021no—
152#124—5021no—
153#121—2512no—
154#121—2512no—
155#120—5022no—
156C4×Q8—5021no—
157C4×D4—5021no—
158C4×D4—5022no—
159#117—2511no—
160#117—2512no—
161#117C2×D25111yescharacteristic
162C2×C4⋊C4—2511no—
163C2×C4⋊C4—2512no—
164#107—2512no—
165#107—2511no—
166#107—2511no—
167#106—2512no—
168#106—2511no—
169#106—2511no—
170#102—2512no—
171#102—2512no—
172#102C2×D25112yes—
173#102C2×D25112yes—
174C22⋊C8—2512no—
175C22⋊C8—2512no—
176C24—21no—
177#41—5022no—
178#41—5022no—
179C2×Q8—2511no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—5022no—
184C2×D4—5022no—
185C2×D4—2512no—
186C2×D4—2511no—
187C2×D4C22×D25112yes—
188C22×C4—5022no—
189C22×C4—5021no—
190C22×C4—2512no—
191C22×C4C22×D25111yescharacteristic
192M16—22no—
193C2×C8—5022no—
194C2×C8—2511no—
195C2×C8—2511no—
196C2×C8C22×D25112yes—
197C4⋊C4—5022no—
198C4⋊C4—5021no—
199C4⋊C4—2512no—
200C4⋊C4—2511no—
201C4⋊C4—2511no—
202C4⋊C4C22×D25111yescharacteristic
203C4⋊C4C22×D25111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—5022no—
206C22⋊C4—5021no—
207C22⋊C4—5022no—
208C22⋊C4—5022no—
209C42—21no—
210C42—5022no—
211C42—5021no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—2512no—
217C23D4×D25111yescharacteristic
218Q8—5022no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—20082no—
223D4—20082no—
224D4—5024no—
225D4—5022no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—20082no—
229C2×C4—20081no—
230C2×C4—5021no—
231C2×C4—5021no—
232C2×C4—5022no—
233C2×C4—5022no—
234C2×C4—5022no—
235C2×C4—2512no—
236C2×C4D4×D25111yescharacteristic
237C2×C4#1686080912yes—
238C2×C4#1686081012yes—
239C2×C4C23×D25111yescharacteristic
240C8—22no—
241C8—5021no—
242C8—5021no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—5022no—
250C22—5022no—
251C22—5022no—
252C22#2254869712yes—
253C22C2×D4×D25111yescharacteristic
254C4—41no—
255C4—41no—
256C4—10042no—
257C4—10041no—
258C4—10041no—
259C4—5022no—
260C4#2254869712yes—
261C4C2×#1686080911yescharacteristic
262C4C2×#1686081011yescharacteristic
263C2—22no—
264C2—82no—
265C2—10042no—
266C2C2×#2254869711yescharacteristic
267C2#3063249111yescharacteristic
268C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
prime radical#30632254C2
hypercenter#418D251
hypocenterC251#3096
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #30632167C2semidirect
3. #30632010C2semidirect
4. #30632009C2semidirect
5. #30632004C2semidirect
6. #30631999C2semidirect
7. #30631729C2semidirect
8. #30631723C2semidirect
9. C2×#22548636C2semidirect
10. #30631655C2semidirect
11. #418×C251C2semidirect
23. #22548673C22semidirect
25. C4⋊C4×D251C22semidirect
26. #22548649C22semidirect
27. #22548648C22semidirect
28. #22548648C22semidirect
29. #22548647C22semidirect
30. #22548646C22semidirect
31. #22548646C22semidirect
34. (C2×D251)⋊C8C22semidirect
35. (C2×D251)⋊C8C22semidirect
134. C251#3096semidirect
144. #418D251semidirect

Maximal subgroups

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

subgroupindex
#306322542
#306321672
#306320102
#306320092
#306320042
#306319992
#306317292
#306317232
C2×#225486362
#306316552
#418×C2512
#3096251