Finite Group Databasebeta

← #44559843

Subgroups — #44559843

Order 32128.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#22548740C211yesmaximal, characteristic, complemented
2C2×#22548711C211yesmaximal, characteristic, complemented
3#41×Dic251C211yesmaximal, characteristic, complemented
4#30631885C214yesmaximal, complemented
5#30631880C214yesmaximal, complemented
6#30631710C211yesmaximal, characteristic, complemented
7#30631693C211yesmaximal, characteristic, complemented
8#30631692C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×C22⋊D251—22no—
12#22548740—24no—
13C2×#16860797C2212yescomplemented
14C2×#16860795C2212yescomplemented
15C2×#16860794C2211yescharacteristic
16#22548711—21no—
17#22548711—21no—
18#22548711C2214yes—
19Q8×Dic251C2212yescomplemented
20D4×Dic251—22no—
21D4×Dic251C2212yescomplemented
22#22548674—21no—
23#22548674C2211yescharacteristic
24C2×C4×Dic251—21no—
25C2×C4×Dic251C2211yescharacteristic
26Dic251⋊C8C2214yescomplemented
27#22548637C2214yescomplemented
28#22548636C2214yescomplemented
29C2×#41×C251C2211yescharacteristic
30C22×D4×C251C2211yescharacteristic
31AGL(1,ℤ/8)×C251—24no—
32C2×SD16×C251C2212yescomplemented
33C2×D8×C251C2212yescomplemented
34C2×M16×C251C2211yescharacteristic
35C23×D251—21no—
36C2×D1004—22no—
37C2×D1004—21no—
38C2×D1004—22no—
39C2×D1004C2312yes—
40C2×C22⋊D251—24no—
41C2×C22⋊D251—24no—
42C2×C22⋊D251—48no—
43C22×Dic251—22no—
44C22×Dic251D412yes—
45#16860797—24no—
46#16860795—24no—
47#16860794—22no—
48C2×C251⋊C8C2312yes—
49C4⋊Dic251—22no—
50C4⋊Dic251C2312yes—
51C22⋊Dic251—22no—
52C22⋊Dic251—21no—
53C4×Dic251—22no—
54C4×Dic251—21no—
55C4×Dic251D411yescharacteristic
56C4×Dic251D411yescharacteristic
57C4×Dic251C2312yes—
58C24×C251—21no—
59#41×C251—22no—
60#41×C251D414yescomplemented
61C2×Q8×C251C2311yescharacteristic
62C2×D4×C251—21no—
63C2×D4×C251—22no—
64C2×D4×C251—21no—
65C2×D4×C251—22no—
66C2×D4×C251C2312yes—
67C2×D4×C251C2311yescharacteristic
68C22×C4×C251—21no—
69C22×C4×C251C2311yescharacteristic
70SD16×C251—24no—
71D8×C251—24no—
72M16×C251—22no—
73C2×C8×C251C2312yes—
74C22×D251—22no—
75C22×D251—44no—
76C22×D251—42no—
77D1004—22no—
78D1004—22no—
79D1004—42no—
80C22⋊D251—48no—
81C22⋊D251—48no—
82C2×Dic251—24no—
83C2×Dic251—22no—
84C2×Dic251—44no—
85C2×Dic251C2×D412yes—
86C2×Dic251C2×D412yes—
87C251⋊C8—22no—
88C23×C251—21no—
89C23×C251—22no—
90C23×C251—44no—
91C23×C251—42no—
92C23×C251C2×D411yescharacteristic
93Q8×C251—21no—
94Q8×C251C2×D412yescomplemented
95D4×C251—24no—
96D4×C251—22no—
97D4×C251—21no—
98D4×C251—22no—
99D4×C251—42no—
100D4×C251C2×D412yescomplemented
101C2×C4×C251—21no—
102C2×C4×C251—24no—
103C2×C4×C251C2×D411yescharacteristic
104C2×C4×C251C2×D412yes—
105C2×C4×C251#4112yes—
106C2×C4×C251C2411yescharacteristic
107C8×C251—22no—
108C22×C251—22no—
109C22×C251—22no—
110C22×C251—42no—
111C22×C251—42no—
112C22×C251—44no—
113C22×C251—84no—
114C22×C251#12112yes—
115C22×C251C22×D411yescharacteristic
116C2×D251—42no—
117C2×D251—44no—
118C2×D251—84no—
119C4×C251—22no—
120C4×C251#12112yes—
121C4×C251C22×D411yescharacteristic
122C4×C251C2×#4111yescharacteristic
123Dic251—22no—
124Dic251—42no—
125Dic251AGL(1,ℤ/8)14yescomplemented
126C2×C251—22no—
127C2×C251—42no—
128C2×C251—82no—
129C2×C251C2×#12111yescharacteristic
130C2×C251C2×AGL(1,ℤ/8)12yes—
131D251—82no—
132C251#303811yescharacteristic, complemented, Sylow
133#3038—2511nomaximal, Sylow
134C2×AGL(1,ℤ/8)—2511no—
135C2×AGL(1,ℤ/8)D25111yescharacteristic, complemented
136C2×#127—2511no—
137C4×#41—2511no—
138#460—2514no—
139#459—2514no—
140#423—2511no—
141#418—2511no—
142#418—2511no—
143C2×#41C2×D25111yescharacteristic
144C22×D4—5022no—
145C22×D4—2511no—
146C22×D4C2×D25111yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—5024no—
149C2×SD16—2512no—
150C2×SD16C2×D25112yescomplemented
151C2×D8—2512no—
152C2×D8C2×D25112yescomplemented
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#117—2511no—
163C2×C42—5021no—
164C2×C42—2511no—
165C4⋊C8—2514no—
166#102—2514no—
167#102—2514no—
168C24—21no—
169C24—5021no—
170#41—22no—
171#41C22⋊D25114yescomplemented
172C2×Q8C22×D25111yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—10048no—
178C2×D4—5024no—
179C2×D4—5024no—
180C2×D4—5021no—
181C2×D4—5022no—
182C2×D4—5022no—
183C2×D4—2512no—
184C2×D4C22×D25112yes—
185C2×D4C22×D25111yescharacteristic
186C22×C4—21no—
187C22×C4—5022no—
188C22×C4—2512no—
189C22×C4C22×D25111yescharacteristic
190SD16—24no—
191SD16—5024no—
192D8—24no—
193D8—5024no—
194M16—22no—
195M16—5022no—
196C2×C8—2512no—
197C2×C8C22×D25112yes—
198C4⋊C4—5022no—
199C4⋊C4—2512no—
200C22⋊C4—5021no—
201C22⋊C4—5022no—
202C42—5022no—
203C42—5021no—
204C42—2511no—
205C42—2511no—
206C42—2512no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—10044no—
212C23—10042no—
213C23—5022no—
214C23D4×D25111yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D25112yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—10048no—
223D4—10048no—
224D4—10042no—
225D4—5022no—
226D4—5022no—
227D4C2×C22⋊D25112yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—10044no—
231C2×C4—5022no—
232C2×C4—5024no—
233C2×C4—2512no—
234C2×C4—2512no—
235C2×C4D4×D25111yescharacteristic
236C2×C4C2×C22⋊D25112yes—
237C2×C4#1686080912yes—
238C2×C4C23×D25111yescharacteristic
239C8—22no—
240C8—5022no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—84no—
247C22—20084no—
248C22—10044no—
249C22—10042no—
250C22#2254868712yes—
251C22C2×D4×D25111yescharacteristic
252C4—22no—
253C4—10042no—
254C4—5022no—
255C4—2514no—
256C4#2254868712yes—
257C4C22×C22⋊D25111yescharacteristic
258C4C2×#1686080911yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—20082no—
263C2C2×#2254868711yescharacteristic
264C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×AGL(1,ℤ/8)×C251C2
socleC22×C251C22×D4
hypercenterC2×AGL(1,ℤ/8)D251
hypocenterC251#3038
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22548740C2semidirect
2. C2×#22548711C2semidirect
3. #41×Dic251C2semidirect
4. #30631885C2semidirect
5. #30631880C2semidirect
6. #30631710C2semidirect
7. #30631693C2semidirect
8. #30631692C2semidirect
9. C2×AGL(1,ℤ/8)×C251C2semidirect
13. C2×#16860797C22semidirect
14. C2×#16860795C22semidirect
19. Q8×Dic251C22semidirect
21. D4×Dic251C22semidirect
26. Dic251⋊C8C22semidirect
27. #22548637C22semidirect
28. #22548636C22semidirect
32. C2×SD16×C251C22semidirect
33. C2×D8×C251C22semidirect
60. #41×C251D4semidirect
94. Q8×C251C2×D4semidirect
100. D4×C251C2×D4semidirect
125. Dic251AGL(1,ℤ/8)semidirect
132. C251#3038semidirect
135. C2×AGL(1,ℤ/8)D251semidirect
150. C2×SD16C2×D251semidirect
152. C2×D8C2×D251semidirect
171. #41C22⋊D251semidirect
216. Q8C2×C22⋊D251semidirect
227. D4C2×C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487402
C2×#225487112
#41×Dic2512
#306318852
#306318802
#306317102
#306316932
#306316922
C2×AGL(1,ℤ/8)×C2512
#3038251