Finite Group Databasebeta

← C2×#30631833 #44559567

Subgroups — C2×#30631833 #44559567

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×#16860809C211yesmaximal, characteristic, complemented
2C22×#16860799C211yesmaximal, characteristic, complemented
3C22×#16860795C211yesmaximal, characteristic, complemented
4C2×#22548690C211yesmaximal, characteristic, complemented
5#30631833C218yesmaximal, direct product factor
6C2×#22548638C211yesmaximal, characteristic, complemented
7C2×(C2×D251)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C251C211yesmaximal, characteristic, complemented
9C2×#16860809—24no—
10C2×#16860809—22no—
11C2×#16860809—24no—
12C2×#16860809C2214yescomplemented
13C22×Dic502C2211yescharacteristic
14C22×D1004—21no—
15C22×D1004C2211yescharacteristic
16C22×C22⋊D251—21no—
17C23×Dic251—21no—
18C22×C4×D251C2211yescharacteristic
19C2×#16860799—24no—
20C2×#16860799C2214yescomplemented
21C2×#16860795—24no—
22C2×#16860795C2214yescomplemented
23C22×C251⋊C8C2211yescharacteristic
24#22548690—22no—
25#22548690C2214yescomplemented
26C2×(C2×D251)⋊C4—21no—
27#22548638C2214yescomplemented
28(C2×D251)⋊C8C2214yescomplemented
29C22×D4×C251C2211yescharacteristic
30C22×C8×C251C2211yescharacteristic
31C2×C4⋊C4×C251C2211yescharacteristic
32#102×C251C2214yescomplemented
33C23×D251—21no—
34C23×D251D411yescharacteristic
35#16860809—216no—
36#16860809—28no—
37#16860809—28no—
38C2×Dic502—21no—
39C2×Dic502—22no—
40C2×Dic502D414yescomplemented
41C2×Dic502C2312yes—
42C2×D1004—24no—
43C2×D1004—22no—
44C2×D1004—22no—
45C2×D1004—21no—
46C2×D1004—44no—
47C2×D1004C2312yes—
48C2×C22⋊D251—28no—
49C2×C22⋊D251—24no—
50C22×Dic251—28no—
51C22×Dic251—24no—
52C22×Dic251—21no—
53C22×Dic251D411yescharacteristic
54C2×C4×D251—22no—
55C2×C4×D251—22no—
56C2×C4×D251D412yes—
57C2×C4×D251C2312yes—
58#16860799—28no—
59#16860795—28no—
60C2×C251⋊C8—22no—
61C2×C251⋊C8C2312yes—
62(C2×D251)⋊C4—24no—
63C24×C251—21no—
64C2×D4×C251—21no—
65C2×D4×C251—22no—
66C2×D4×C251D414yescomplemented
67C2×D4×C251C2312yes—
68C22×C4×C251—21no—
69C22×C4×C251C2311yescharacteristic
70C2×C8×C251—22no—
71C2×C8×C251C2312yes—
72C4⋊C4×C251—21no—
73C4⋊C4×C251C2312yes—
74C22×D251—22no—
75C22×D251—22no—
76C22×D251—22no—
77C22×D251—22no—
78C22×D251—42no—
79C22×D251—42no—
80C22×D251—42no—
81C22×D251C2×D412yes—
82Dic502—24no—
83Dic502—22no—
84Dic502C2×D414yescomplemented
85D1004—22no—
86D1004—22no—
87D1004—48no—
88D1004—42no—
89C22⋊D251—216no—
90C2×Dic251—216no—
91C2×Dic251—22no—
92C2×Dic251—24no—
93C2×Dic251—22no—
94C2×Dic251C2×D412yes—
95C4×D251—24no—
96C4×D251—22no—
97C4×D251—22no—
98C251⋊C8—22no—
99C23×C251—24no—
100C23×C251—22no—
101C23×C251—42no—
102C23×C251—42no—
103C23×C251C2×D411yescharacteristic
104D4×C251—24no—
105D4×C251—22no—
106D4×C251C2×D414yescomplemented
107C2×C4×C251—22no—
108C2×C4×C251—42no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2×D412yes—
111C2×C4×C251C2×D412yes—
112C2×C4×C251C2411yescharacteristic
113C8×C251—22no—
114C22×C251—24no—
115C22×C251—42no—
116C22×C251—42no—
117C22×C251—48no—
118C22×C251#12012yes—
119C22×C251#13512yes—
120C22×C251AGL(1,ℤ/8)12yes—
121C22×C251C22×D411yescharacteristic
122C2×D251—22no—
123C2×D251—22no—
124C2×D251—22no—
125C2×D251—44no—
126C2×D251—42no—
127C2×D251—42no—
128C2×D251—42no—
129C2×D251—84no—
130C4×C251—42no—
131C4×C251#12012yes—
132C4×C251C22×D411yescharacteristic
133C4×C251C22×D411yescharacteristic
134Dic251—22no—
135Dic251—24no—
136C2×C251—44no—
137C2×C251#44914yescomplemented
138C2×C251C2×#12011yescharacteristic
139C2×C251C2×#13511yescharacteristic
140C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
141D251—42no—
142D251—82no—
143C251C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—2511nomaximal, Sylow
145C22×#41—2511no—
146C22×SD16—2511no—
147C22×D8—2511no—
148C2×#121—2511no—
149#449—2518no—
150C2×#103—2511no—
151C2×#102D25111yescharacteristic, complemented
152C2×C22⋊C8—2511no—
153C2×#41—5024no—
154C2×#41—5024no—
155C2×#41—5022no—
156C2×#41—2514no—
157C22×Q8—2511no—
158C22×D4—5021no—
159C22×D4—5021no—
160C22×D4—2511no—
161C22×D4C2×D25111yescharacteristic
162C23×C4—5021no—
163C23×C4—2511no—
164C2×SD16—5024no—
165C2×SD16—2514no—
166C2×D8—5024no—
167C2×D8—2514no—
168C22×C8—2511no—
169C22×C8C2×D25111yescharacteristic
170#121—5022no—
171#121—2514no—
172C2×C4⋊C4C2×D25111yescharacteristic
173C2×C22⋊C4—5021no—
174#103—2514no—
175#102C2×D25114yescomplemented
176C22⋊C8—2514no—
177C24—21no—
178C24—5021no—
179C24—2511no—
180#41—50216no—
181#41—5028no—
182#41—5028no—
183C2×Q8—5021no—
184C2×Q8—5022no—
185C2×Q8—2514no—
186C2×Q8—2512no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—10044no—
190C2×D4—5024no—
191C2×D4—5028no—
192C2×D4—5024no—
193C2×D4—5021no—
194C2×D4—5022no—
195C2×D4—5022no—
196C2×D4—2512no—
197C2×D4D100414yescomplemented
198C2×D4C22×D25112yes—
199C22×C4—21no—
200C22×C4—5028no—
201C22×C4—5021no—
202C22×C4—5024no—
203C22×C4—5022no—
204C22×C4—5022no—
205C22×C4—2511no—
206C22×C4—2512no—
207C22×C4—2512no—
208C22×C4C22×D25111yescharacteristic
209SD16—5028no—
210D8—5028no—
211C2×C8—22no—
212C2×C8—5022no—
213C2×C8—2512no—
214C2×C8C22×D25112yes—
215C4⋊C4—21no—
216C4⋊C4C22×D25112yes—
217C22⋊C4—5024no—
218C23—22no—
219C23—24no—
220C23—42no—
221C23—42no—
222C23—10042no—
223C23—10042no—
224C23—10042no—
225C23—5022no—
226C23—5022no—
227C23—5022no—
228C23—5022no—
229C23—2512no—
230C23D4×D25111yescharacteristic
231Q8—5022no—
232Q8—5024no—
233Q8—2514no—
234D4—22no—
235D4—24no—
236D4—10048no—
237D4—10042no—
238D4—50216no—
239D4—5022no—
240D4—5022no—
241D4C2×D100414yescomplemented
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—50216no—
245C2×C4—5022no—
246C2×C4—5022no—
247C2×C4—5024no—
248C2×C4—5022no—
249C2×C4—5022no—
250C2×C4—5024no—
251C2×C4—2512no—
252C2×C4D4×D25112yes—
253C2×C4D4×D25111yescharacteristic
254C2×C4C2×D100412yes—
255C2×C4C23×D25111yescharacteristic
256C8—22no—
257C8—5022no—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—48no—
262C22—20084no—
263C22—10042no—
264C22—10042no—
265C22—10042no—
266C22—10044no—
267C22—5022no—
268C22—5022no—
269C22—5022no—
270C22#2254868512yes—
271C22#2254873612yes—
272C22#2254873912yes—
273C22C2×D4×D25111yescharacteristic
274C4—42no—
275C4—5022no—
276C4—5024no—
277C4#2254868512yes—
278C4C2×D4×D25111yescharacteristic
279C4C22×D100411yescharacteristic
280C2—44no—
281C2—20082no—
282C2—10042no—
283C2#3063183314yesdirect product factor
284C2C2×#2254868511yescharacteristic
285C2C2×#2254873611yescharacteristic
286C2C2×#2254873911yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#102×C251C2
socleC23×C251C2×D4
hypercenterC2×#102D251
hypocenterC251C2×#449
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#16860809C2semidirect
2. C22×#16860799C2semidirect
3. C22×#16860795C2semidirect
4. C2×#22548690C2semidirect
5. #30631833C2direct (N × H)
6. C2×#22548638C2semidirect
7. C2×(C2×D251)⋊C8C2semidirect
8. C2×#102×C251C2semidirect
12. C2×#16860809C22semidirect
20. C2×#16860799C22semidirect
22. C2×#16860795C22semidirect
25. #22548690C22semidirect
27. #22548638C22semidirect
28. (C2×D251)⋊C8C22semidirect
32. #102×C251C22semidirect
40. C2×Dic502D4semidirect
66. C2×D4×C251D4semidirect
84. Dic502C2×D4semidirect
106. D4×C251C2×D4semidirect
137. C2×C251#449semidirect
143. C251C2×#449semidirect
151. C2×#102D251semidirect
175. #102C2×D251semidirect
197. C2×D4D1004semidirect
241. D4C2×D1004semidirect
283. C2#30631833direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168608092
C22×#168607992
C22×#168607952
C2×#225486902
#306318332
C2×#225486382
C2×(C2×D251)⋊C82
C2×#102×C2512
C2×#449251