Finite Group Databasebeta

← C2×#30631831 #44559565

Subgroups — C2×#30631831 #44559565

Order 32128.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#16860811C211yesmaximal, characteristic, complemented
2C22×#16860795C211yesmaximal, characteristic, complemented
3C2×#22548686C211yesmaximal, characteristic, complemented
4#30631831C218yesmaximal, direct product factor
5C2×#22548639C211yesmaximal, characteristic, complemented
6C2×#22548637C211yesmaximal, characteristic, complemented
7C2×(C2×D251)⋊C8C211yesmaximal, characteristic, complemented
8C22×SD16×C251C211yesmaximal, characteristic, complemented
9C2×#16860811—24no—
10C2×#16860811—22no—
11C2×#16860811—24no—
12C2×#16860811C2214yescomplemented
13C22×D1004—21no—
14C22×D1004C2211yescharacteristic
15C22×C22⋊D251—21no—
16C22×C4×D251—21no—
17C22×C4×D251C2211yescharacteristic
18C2×#16860795—24no—
19C2×#16860795C2214yescomplemented
20C22×C251⋊C8C2211yescharacteristic
21#22548686—22no—
22#22548686C2214yescomplemented
23C2×C4⋊Dic251C2211yescharacteristic
24C2×C22⋊Dic251—21no—
25#22548639C2214yescomplemented
26#22548637C2214yescomplemented
27(C2×D251)⋊C8C2214yescomplemented
28C22×Q8×C251C2211yescharacteristic
29C22×D4×C251C2211yescharacteristic
30C2×SD16×C251—24no—
31C2×SD16×C251C2214yescomplemented
32C22×C8×C251C2211yescharacteristic
33C23×D251—21no—
34C23×D251D411yescharacteristic
35#16860811—216no—
36#16860811—28no—
37#16860811—28no—
38C2×D1004—28no—
39C2×D1004—24no—
40C2×D1004—21no—
41C2×D1004—22no—
42C2×D1004D414yescomplemented
43C2×D1004C2312yes—
44C2×C22⋊D251—24no—
45C2×C22⋊D251—44no—
46C22×Dic251—21no—
47C22×Dic251D411yescharacteristic
48C2×C4×D251—28no—
49C2×C4×D251—24no—
50C2×C4×D251—22no—
51C2×C4×D251—22no—
52C2×C4×D251D412yes—
53C2×C4×D251C2312yes—
54#16860795—28no—
55C2×C251⋊C8—22no—
56C2×C251⋊C8C2312yes—
57C4⋊Dic251—21no—
58C4⋊Dic251C2312yes—
59C22⋊Dic251—24no—
60C24×C251—21no—
61C2×Q8×C251—22no—
62C2×Q8×C251—21no—
63C2×Q8×C251D414yescomplemented
64C2×Q8×C251C2312yes—
65C2×D4×C251—22no—
66C2×D4×C251—22no—
67C2×D4×C251—21no—
68C2×D4×C251C2312yes—
69C22×C4×C251—21no—
70C22×C4×C251C2311yescharacteristic
71SD16×C251—28no—
72C2×C8×C251—22no—
73C2×C8×C251C2312yes—
74C22×D251—22no—
75C22×D251—24no—
76C22×D251—22no—
77C22×D251—22no—
78C22×D251—22no—
79C22×D251—42no—
80C22×D251—42no—
81C22×D251C2×D412yes—
82D1004—216no—
83D1004—24no—
84D1004—22no—
85D1004C2×D414yescomplemented
86C22⋊D251—48no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—42no—
90C2×Dic251C2×D412yes—
91C4×D251—216no—
92C4×D251—24no—
93C4×D251—22no—
94C4×D251—22no—
95C251⋊C8—22no—
96C23×C251—22no—
97C23×C251—42no—
98C23×C251—42no—
99C23×C251—42no—
100C23×C251C2×D411yescharacteristic
101Q8×C251—24no—
102Q8×C251—22no—
103Q8×C251C2×D414yescomplemented
104D4×C251—22no—
105D4×C251—22no—
106D4×C251—42no—
107C2×C4×C251—24no—
108C2×C4×C251—22no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2×D412yes—
111C2×C4×C251C2×D412yes—
112C2×C4×C251C2411yescharacteristic
113C8×C251—22no—
114C22×C251—42no—
115C22×C251—42no—
116C22×C251—42no—
117C22×C251—84no—
118C22×C251#12012yes—
119C22×C251#13512yes—
120C22×C251AGL(1,ℤ/8)12yes—
121C22×C251C22×D411yescharacteristic
122C2×D251—22no—
123C2×D251—24no—
124C2×D251—22no—
125C2×D251—22no—
126C2×D251—42no—
127C2×D251—42no—
128C2×D251—44no—
129C2×D251—48no—
130C4×C251—24no—
131C4×C251#12012yes—
132C4×C251C22×D411yescharacteristic
133C4×C251C22×D411yescharacteristic
134Dic251—22no—
135Dic251—42no—
136C2×C251—82no—
137C2×C251#44914yescomplemented
138C2×C251C2×#12011yescharacteristic
139C2×C251C2×#13511yescharacteristic
140C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
141D251—42no—
142D251—44no—
143C251C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—2511nomaximal, Sylow
145C22×#41—2511no—
146C22×SD16D25111yescharacteristic, complemented
147C22×D8—2511no—
148C2×#121—2511no—
149#449—2518no—
150C2×#103—2511no—
151C2×#102—2511no—
152C2×C22⋊C8—2511no—
153C2×#41—5024no—
154C2×#41—5024no—
155C2×#41—5022no—
156C2×#41—2514no—
157C22×Q8C2×D25111yescharacteristic
158C22×D4—5021no—
159C22×D4—5021no—
160C22×D4—2511no—
161C22×D4C2×D25111yescharacteristic
162C23×C4—5021no—
163C23×C4—2511no—
164C2×SD16—24no—
165C2×SD16C2×D25114yescomplemented
166C2×D8—5024no—
167C2×D8—2514no—
168C22×C8—2511no—
169C22×C8C2×D25111yescharacteristic
170#121—5022no—
171#121—2514no—
172C2×C4⋊C4—2511no—
173C2×C22⋊C4—5021no—
174#103—2514no—
175#102—2514no—
176C22⋊C8—2514no—
177C24—21no—
178C24—5021no—
179C24—2511no—
180#41—50216no—
181#41—5028no—
182#41—5028no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8C22⋊D25114yescomplemented
186C2×Q8C22×D25112yes—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—10044no—
191C2×D4—5028no—
192C2×D4—5024no—
193C2×D4—5024no—
194C2×D4—5021no—
195C2×D4—5022no—
196C2×D4—2514no—
197C2×D4—2512no—
198C2×D4C22×D25112yes—
199C22×C4—21no—
200C22×C4—5028no—
201C22×C4—5024no—
202C22×C4—5021no—
203C22×C4—5022no—
204C22×C4—5022no—
205C22×C4—2511no—
206C22×C4—2512no—
207C22×C4—2512no—
208C22×C4C22×D25111yescharacteristic
209SD16—28no—
210D8—5028no—
211C2×C8—22no—
212C2×C8—5022no—
213C2×C8—2512no—
214C2×C8C22×D25112yes—
215C4⋊C4—5021no—
216C4⋊C4—2512no—
217C22⋊C4—5024no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—10042no—
223C23—10042no—
224C23—5022no—
225C23—5022no—
226C23—5022no—
227C23—5022no—
228C23—5024no—
229C23—2512no—
230C23D4×D25111yescharacteristic
231Q8—22no—
232Q8—24no—
233Q8C2×C22⋊D25114yescomplemented
234D4—22no—
235D4—22no—
236D4—42no—
237D4—10048no—
238D4—50216no—
239D4—5022no—
240D4—5024no—
241D4—2514no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—10042no—
245C2×C4—50216no—
246C2×C4—5022no—
247C2×C4—5022no—
248C2×C4—5022no—
249C2×C4—5022no—
250C2×C4—5024no—
251C2×C4—2512no—
252C2×C4D4×D25112yes—
253C2×C4D4×D25111yescharacteristic
254C2×C4C2×C22⋊D25112yes—
255C2×C4C23×D25111yescharacteristic
256C8—22no—
257C8—5022no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—84no—
262C22—10042no—
263C22—10042no—
264C22—10048no—
265C22—10044no—
266C22—5022no—
267C22—5022no—
268C22—5022no—
269C22—5024no—
270C22#2254868312yes—
271C22#2254873612yes—
272C22#2254874212yes—
273C22C2×D4×D25111yescharacteristic
274C4—24no—
275C4—10042no—
276C4—5022no—
277C4#2254868312yes—
278C4C2×D4×D25111yescharacteristic
279C4C22×C22⋊D25111yescharacteristic
280C2—82no—
281C2—10044no—
282C2—10042no—
283C2#3063183114yesdirect product factor
284C2C2×#2254868311yescharacteristic
285C2C2×#2254873611yescharacteristic
286C2C2×#2254874211yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×SD16×C251C2
socleC23×C251C2×D4
hypercenterC22×SD16D251
hypocenterC251C2×#449
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#16860811C2semidirect
2. C22×#16860795C2semidirect
3. C2×#22548686C2semidirect
4. #30631831C2direct (N × H)
5. C2×#22548639C2semidirect
6. C2×#22548637C2semidirect
7. C2×(C2×D251)⋊C8C2semidirect
8. C22×SD16×C251C2semidirect
12. C2×#16860811C22semidirect
19. C2×#16860795C22semidirect
22. #22548686C22semidirect
25. #22548639C22semidirect
26. #22548637C22semidirect
27. (C2×D251)⋊C8C22semidirect
31. C2×SD16×C251C22semidirect
42. C2×D1004D4semidirect
63. C2×Q8×C251D4semidirect
85. D1004C2×D4semidirect
103. Q8×C251C2×D4semidirect
137. C2×C251#449semidirect
143. C251C2×#449semidirect
146. C22×SD16D251semidirect
165. C2×SD16C2×D251semidirect
185. C2×Q8C22⋊D251semidirect
233. Q8C2×C22⋊D251semidirect
283. C2#30631831direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168608112
C22×#168607952
C2×#225486862
#306318312
C2×#225486392
C2×#225486372
C2×(C2×D251)⋊C82
C22×SD16×C2512
C2×#449251