Finite Group Databasebeta

← #44559844

Subgroups — #44559844

Order 32128.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#22548743C211yesmaximal, characteristic, complemented
2C2×#22548741C211yesmaximal, characteristic, complemented
3C2×#22548711C211yesmaximal, characteristic, complemented
4#30632155C211yesmaximal, characteristic, complemented
5#30631886C214yesmaximal, complemented
6#30631881C214yesmaximal, complemented
7#30631710C211yesmaximal, characteristic, complemented
8#30631691C211yesmaximal, characteristic, complemented
9#418×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12C22×C22⋊D251—22no—
13#22548743—24no—
14#22548741—24no—
15C2×#16860799C2212yescomplemented
16C2×#16860798C2212yescomplemented
17C2×D2008C2212yescomplemented
18C2×#16860795C2212yescomplemented
19C2×#16860794C2211yescharacteristic
20#22548711—21no—
21#22548711—21no—
22#22548711C2214yes—
23#22548680C2212yescomplemented
24#22548677C2212yescomplemented
25(C22⋊D251)⋊C4—22no—
26#22548672—21no—
27C2×C4×Dic251C2211yescharacteristic
28C42×D251—21no—
29Dic251⋊C8C2214yescomplemented
30#22548635C2214yescomplemented
31C22×D4×C251C2211yescharacteristic
32C2×M16×C251C2211yescharacteristic
33#117×C251C2211yescharacteristic
34#102×C251C2214yescomplemented
35C23×D251—21no—
36#16860810—22no—
37#16860810D414yescomplemented
38C2×Dic502C2311yescharacteristic
39C2×D1004—22no—
40C2×D1004—21no—
41C2×D1004—22no—
42C2×D1004C2312yes—
43C2×D1004C2311yescharacteristic
44C2×C22⋊D251—24no—
45C2×C22⋊D251—24no—
46C2×C22⋊D251—21no—
47C2×C22⋊D251—48no—
48C22×Dic251D412yes—
49C2×C4×D251—22no—
50C2×C4×D251—21no—
51#16860799—24no—
52#16860798—24no—
53D2008—24no—
54#16860795—24no—
55#16860794—22no—
56C2×C251⋊C8C2312yes—
57Dic251⋊C4—22no—
58(C2×D251)⋊C4—22no—
59C4×Dic251—22no—
60C4×Dic251D411yescharacteristic
61C4×Dic251D411yescharacteristic
62C4×Dic251C2312yes—
63C24×C251—21no—
64C2×D4×C251—22no—
65C2×D4×C251—21no—
66C2×D4×C251—22no—
67C2×D4×C251C2312yes—
68C22×C4×C251C2311yescharacteristic
69M16×C251—22no—
70C2×C8×C251C2312yes—
71C4⋊C4×C251C2312yes—
72C22⋊C4×C251—21no—
73C42×C251—21no—
74C22×D251—21no—
75C22×D251—22no—
76C22×D251—44no—
77C22×D251—42no—
78Dic502—21no—
79Dic502C2×D412yescomplemented
80D1004—22no—
81D1004—21no—
82D1004—22no—
83D1004—42no—
84D1004C2×D412yescomplemented
85C22⋊D251—24no—
86C22⋊D251—48no—
87C22⋊D251—48no—
88C2×Dic251—21no—
89C2×Dic251—24no—
90C2×Dic251C2×D412yes—
91C2×Dic251C2×D412yes—
92C4×D251—24no—
93C4×D251—44no—
94C251⋊C8—22no—
95C23×C251—22no—
96C23×C251—44no—
97C23×C251—42no—
98C23×C251C2×D411yescharacteristic
99D4×C251—22no—
100D4×C251—22no—
101D4×C251—42no—
102C2×C4×C251—22no—
103C2×C4×C251C2×D411yescharacteristic
104C2×C4×C251C2×D412yes—
105C2×C4×C251#4112yes—
106C2×C4×C251C2411yescharacteristic
107C8×C251—22no—
108C22×C251—22no—
109C22×C251—42no—
110C22×C251—44no—
111C22×C251—84no—
112C22×C251#12112yes—
113C22×C251C22×D411yescharacteristic
114C2×D251—22no—
115C2×D251—42no—
116C2×D251—42no—
117C2×D251—44no—
118C2×D251—84no—
119C4×C251—42no—
120C4×C251#12112yes—
121C4×C251C22×D411yescharacteristic
122C4×C251C2×#4111yescharacteristic
123Dic251—22no—
124Dic251—22no—
125Dic251AGL(1,ℤ/8)14yescomplemented
126C2×C251—22no—
127C2×C251—82no—
128C2×C251C2×#12111yescharacteristic
129C2×C251C2×AGL(1,ℤ/8)12yes—
130D251—42no—
131D251—82no—
132C251#303811yescharacteristic, complemented, Sylow
133#3038—2511nomaximal, Sylow
134C2×AGL(1,ℤ/8)—2511no—
135C2×AGL(1,ℤ/8)—2511no—
136C2×#127—2511no—
137C4×#41—2511no—
138#460—2514no—
139#459—2514no—
140#423—2511no—
141#418—2511no—
142#418D25111yescharacteristic, complemented
143C2×#41—2511no—
144C22×D4—5022no—
145C22×D4—2511no—
146C22×D4C2×D25111yescharacteristic
147AGL(1,ℤ/8)—5024no—
148AGL(1,ℤ/8)—5024no—
149C2×SD16—2512no—
150C2×SD16—2512no—
151C2×D8—2512no—
152C2×D8—2512no—
153C2×M16—2511no—
154C2×M16C2×D25111yescharacteristic
155#127—5021no—
156#127—5021no—
157#127—2514no—
158C4×Q8—2512no—
159C4×D4—5022no—
160C4×D4—2512no—
161#117—5021no—
162#117C2×D25111yescharacteristic
163C2×C42—5021no—
164C2×C42—2511no—
165C4⋊C8—2514no—
166#102—2514no—
167#102C2×D25114yescomplemented
168C24—21no—
169C24—5021no—
170#41—5022no—
171#41—2514no—
172C2×Q8—2511no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—10048no—
177C2×D4—5021no—
178C2×D4—5024no—
179C2×D4—5024no—
180C2×D4—5021no—
181C2×D4—5022no—
182C2×D4—5022no—
183C2×D4—2512no—
184C2×D4—2511no—
185C2×D4C22×D25112yes—
186C22×C4—5022no—
187C22×C4—5021no—
188C22×C4—2512no—
189C22×C4C22×D25111yescharacteristic
190SD16—5024no—
191SD16—5024no—
192D8—5024no—
193D8—5024no—
194M16—22no—
195M16—5022no—
196C2×C8—2512no—
197C2×C8C22×D25112yes—
198C4⋊C4—5022no—
199C4⋊C4C22×D25112yes—
200C22⋊C4—21no—
201C22⋊C4—5022no—
202C42—21no—
203C42—5022no—
204C42—2511no—
205C42—2511no—
206C42—2512no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—10044no—
211C23—10042no—
212C23—5022no—
213C23—5021no—
214C23D4×D25111yescharacteristic
215Q8—5021no—
216Q8—2512no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—10048no—
221D4—10048no—
222D4—10042no—
223D4—5024no—
224D4—5022no—
225D4—5022no—
226D4—5021no—
227D4—2512no—
228C2×C4—22no—
229C2×C4—10044no—
230C2×C4—5021no—
231C2×C4—5024no—
232C2×C4—5024no—
233C2×C4—2512no—
234C2×C4—2512no—
235C2×C4D4×D25112yes—
236C2×C4D4×D25111yescharacteristic
237C2×C4#1686081012yes—
238C2×C4C23×D25111yescharacteristic
239C8—22no—
240C8—5022no—
241C22—22no—
242C22—44no—
243C22—42no—
244C22—84no—
245C22—20084no—
246C22—10044no—
247C22—10042no—
248C22—10042no—
249C22—5022no—
250C22#2254868812yes—
251C22C2×D4×D25111yescharacteristic
252C4—42no—
253C4—5022no—
254C4—5022no—
255C4—2514no—
256C4#2254868812yes—
257C4C2×D4×D25111yescharacteristic
258C4C2×#1686081011yescharacteristic
259C2—22no—
260C2—82no—
261C2—20082no—
262C2—10042no—
263C2C2×#2254868811yescharacteristic
264C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
hypercenter#418D251
hypocenterC251#3038
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
1. C2×#22548743C2semidirect
2. C2×#22548741C2semidirect
3. C2×#22548711C2semidirect
4. #30632155C2semidirect
5. #30631886C2semidirect
6. #30631881C2semidirect
7. #30631710C2semidirect
8. #30631691C2semidirect
9. #418×C251C2semidirect
15. C2×#16860799C22semidirect
16. C2×#16860798C22semidirect
17. C2×D2008C22semidirect
18. C2×#16860795C22semidirect
23. #22548680C22semidirect
24. #22548677C22semidirect
29. Dic251⋊C8C22semidirect
30. #22548635C22semidirect
34. #102×C251C22semidirect
37. #16860810D4semidirect
79. Dic502C2×D4semidirect
84. D1004C2×D4semidirect
125. Dic251AGL(1,ℤ/8)semidirect
132. C251#3038semidirect
142. #418D251semidirect
167. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×#225487412
C2×#225487112
#306321552
#306318862
#306318812
#306317102
#306316912
#418×C2512
#3038251