Finite Group Databasebeta

← #44560325

Subgroups — #44560325

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#30632256C211yesmaximal, characteristic
2#30632168C211yesmaximal, characteristic, complemented
3#30632014C212yesmaximal, complemented
4#30632013C212yesmaximal, complemented
5#30632007C212yesmaximal, complemented
6#30632002C212yesmaximal, complemented
7#30631723C211yesmaximal, characteristic, complemented
8#30631693C211yesmaximal, characteristic, complemented
9C2×#22548635C211yesmaximal, characteristic, complemented
10#30631655C211yesmaximal, characteristic, complemented
11#428×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14C22×C251⋊C8C2211yescharacteristic
15#22548712—21no—
16#22548703—21no—
17#22548690C2212yes—
18#22548690C2212yes—
19#22548685—22no—
20#22548680—21no—
21#22548677—21no—
22C4×C22⋊D251—22no—
23#22548674C2211yescharacteristic
24#22548673C2212yescomplemented
25C4⋊C4×D251C2212yescomplemented
26#22548648C2211yescharacteristic, complemented
27#22548648C2211yescharacteristic, complemented
28#22548646C2211yescharacteristic, complemented
29#22548646C2211yescharacteristic, complemented
30#22548637C2212yes—
31#22548637C2212yes—
32#22548635C2212yes—
33#22548635C2212yes—
34(C2×D251)⋊C8C2212yescomplemented
35(C2×D251)⋊C8C2212yescomplemented
36C2×M16×C251C2211yescharacteristic
37#117×C251C2211yescharacteristic
38C2×C4⋊C4×C251C2211yescharacteristic
39#107×C251C2212yescomplemented
40#106×C251C2212yescomplemented
41C23×D251—21no—
42#16860810—22no—
43#16860810—22no—
44C2×Dic502D411yescharacteristic
45C2×D1004—22no—
46C2×D1004—22no—
47C2×D1004—21no—
48C2×D1004—22no—
49C2×D1004—22no—
50C2×D1004D411yescharacteristic
51C2×D1004C2312yes—
52C2×C22⋊D251D412yes—
53C2×C4×D251—22no—
54C2×C4×D251C2312yes—
55C2×C251⋊C8—22no—
56C2×C251⋊C8C2311yescharacteristic
57C2×C251⋊C8C2311yescharacteristic
58C4⋊Dic251C2311yescharacteristic
59C4⋊Dic251C2311yescharacteristic
60Dic251⋊C4—22no—
61C22⋊Dic251—21no—
62(C2×D251)⋊C4—22no—
63(C2×D251)⋊C4—22no—
64(C2×D251)⋊C4—22no—
65C4×Dic251—22no—
66C4×Dic251—21no—
67C22×C4×C251—21no—
68C22×C4×C251C2311yescharacteristic
69M16×C251—22no—
70C2×C8×C251C2312yes—
71C4⋊C4×C251—21no—
72C4⋊C4×C251C2311yescharacteristic
73C4⋊C4×C251C2312yes—
74C4⋊C4×C251C2311yescharacteristic
75C22⋊C4×C251—21no—
76C42×C251—21no—
77C22×D251—22no—
78C22×D251—42no—
79C22×D251—81no—
80C22×D251—81no—
81C22×D251C2×D412yes—
82Dic502—22no—
83D1004—22no—
84D1004—22no—
85D1004—22no—
86D1004—42no—
87D1004—82no—
88D1004—82no—
89C22⋊D251—24no—
90C2×Dic251—21no—
91C2×Dic251—21no—
92C2×Dic251C2×D412yes—
93C4×D251—22no—
94C4×D251—22no—
95C4×D251—82no—
96C251⋊C8—21no—
97C251⋊C8—21no—
98C23×C251C2×D411yescharacteristic
99C2×C4×C251—21no—
100C2×C4×C251—21no—
101C2×C4×C251—22no—
102C2×C4×C251—81no—
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#12312yes—
111C22×C251C22×D411yescharacteristic
112C2×D251—22no—
113C2×D251—22no—
114C2×D251—22no—
115C2×D251—42no—
116C2×D251—42no—
117C2×D251—42no—
118C2×D251—84no—
119C4×C251—42no—
120C4×C251—41no—
121C4×C251—41no—
122C4×C251#12312yes—
123C4×C251C2×#4111yescharacteristic
124C4×C251C2×#4111yescharacteristic
125Dic251—22no—
126Dic251—41no—
127Dic251—41no—
128C2×C251—22no—
129C2×C251C2×#12311yescharacteristic
130C2×C251#57611yescharacteristic
131C2×C251#57711yescharacteristic
132D251—42no—
133D251—82no—
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#428D25111yescharacteristic, complemented
143#427—2511no—
144#418—2511no—
145C2×#102—2511no—
146#408—2511no—
147C2×#41—2511no—
148C22×D4—2511no—
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—2512no—
163C2×C4⋊C4C2×D25111yescharacteristic
164#107—2511no—
165#107—2511no—
166#107C2×D25112yescomplemented
167#106—2511no—
168#106—2511no—
169#106C2×D25112yescomplemented
170#102—2512no—
171#102—2512no—
172#102—2512no—
173#102—2512no—
174C22⋊C8—2512no—
175C22⋊C8—2512no—
176C24—5021no—
177#41—5022no—
178#41—5022no—
179C2×Q8—2511no—
180C2×D4—5022no—
181C2×D4—5022no—
182C2×D4—5021no—
183C2×D4—5022no—
184C2×D4—5022no—
185C2×D4—2512no—
186C2×D4—2511no—
187C2×D4—2512no—
188C22×C4—21no—
189C22×C4—5022no—
190C22×C4—2512no—
191C22×C4C22×D25111yescharacteristic
192M16—22no—
193C2×C8—5022no—
194C2×C8—2511no—
195C2×C8—2511no—
196C2×C8C22×D25112yes—
197C4⋊C4—21no—
198C4⋊C4—5022no—
199C4⋊C4—2511no—
200C4⋊C4—2511no—
201C4⋊C4C22×D25112yes—
202C4⋊C4C22×D25111yescharacteristic
203C4⋊C4C22×D25111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—5022no—
206C22⋊C4—5022no—
207C22⋊C4—5022no—
208C22⋊C4—5021no—
209C42—21no—
210C42—5022no—
211C42—5021no—
212C23—20081no—
213C23—20081no—
214C23—10042no—
215C23—5022no—
216C23—2512no—
217C23D4×D25111yescharacteristic
218Q8—5022no—
219D4—20082no—
220D4—20082no—
221D4—10042no—
222D4—5024no—
223D4—5022no—
224D4—5022no—
225D4—5022no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—81no—
230C2×C4—20082no—
231C2×C4—5021no—
232C2×C4—5021no—
233C2×C4—5022no—
234C2×C4—5022no—
235C2×C4—2512no—
236C2×C4D4×D25111yescharacteristic
237C2×C4#1686081012yes—
238C2×C4#1686081112yes—
239C2×C4C23×D25111yescharacteristic
240C8—22no—
241C8—5021no—
242C8—5021no—
243C22—21no—
244C22—21no—
245C22—20084no—
246C22—10042no—
247C22—10042no—
248C22—10042no—
249C22—5022no—
250C22—5022no—
251C22—5022no—
252C22#2254870012yes—
253C22C2×D4×D25111yescharacteristic
254C4—42no—
255C4—41no—
256C4—41no—
257C4—10041no—
258C4—10041no—
259C4—5022no—
260C4#2254870012yes—
261C4C2×#1686081011yescharacteristic
262C4C2×#1686081111yescharacteristic
263C2—22no—
264C2—20082no—
265C2—10042no—
266C2C2×#2254870011yescharacteristic
267C2#3063249111yescharacteristic
268C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
prime radical#30632256C2
hypercenter#428D251
hypocenterC251#3096
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #30632168C2semidirect
3. #30632014C2semidirect
4. #30632013C2semidirect
5. #30632007C2semidirect
6. #30632002C2semidirect
7. #30631723C2semidirect
8. #30631693C2semidirect
9. C2×#22548635C2semidirect
10. #30631655C2semidirect
11. #428×C251C2semidirect
24. #22548673C22semidirect
25. C4⋊C4×D251C22semidirect
26. #22548648C22semidirect
27. #22548648C22semidirect
28. #22548646C22semidirect
29. #22548646C22semidirect
34. (C2×D251)⋊C8C22semidirect
35. (C2×D251)⋊C8C22semidirect
39. #107×C251C22semidirect
40. #106×C251C22semidirect
134. C251#3096semidirect
142. #428D251semidirect
166. #107C2×D251semidirect
169. #106C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322562
#306321682
#306320142
#306320132
#306320072
#306320022
#306317232
#306316932
C2×#225486352
#306316552
#428×C2512
#3096251