Finite Group Databasebeta

← C2×#30631836 #44559545

Subgroups — C2×#30631836 #44559545

Order 32128.

996 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 173 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×D4×D251C211yesmaximal, characteristic
2C22×#16860797C211yesmaximal, characteristic, complemented
3C2×#22548691C211yesmaximal, characteristic, complemented
4#30631836C218yesmaximal, direct product factor
5C2×#22548637C211yesmaximal, characteristic, complemented
6C2×#22548636C211yesmaximal, characteristic, complemented
7C2×(C2×D251)⋊C8C211yesmaximal, characteristic, complemented
8C22×SD16×C251C211yesmaximal, characteristic, complemented
9C24×D251—21no—
10C22×D1004C2211yescharacteristic
11C22×C22⋊D251—21no—
12C2×D4×D251—24no—
13C2×D4×D251—22no—
14C2×D4×D251—24no—
15C2×D4×D251C2214yes—
16C22×C4×D251C2211yescharacteristic
17C2×#16860797—24no—
18C2×#16860797C2214yescomplemented
19C22×C251⋊C8C2211yescharacteristic, complemented
20#22548691—22no—
21#22548691C2214yescomplemented
22C2×C4⋊Dic251C2211yescharacteristic, complemented
23C2×Dic251⋊C4—21no—
24C2×(C2×D251)⋊C4—21no—
25#22548637C2214yescomplemented
26#22548636C2214yescomplemented
27(C2×D251)⋊C8C2214yescomplemented
28C22×Q8×C251C2211yescharacteristic, complemented
29C22×D4×C251C2211yescharacteristic
30C2×SD16×C251—24no—
31C2×SD16×C251C2214yescomplemented
32C22×C8×C251C2211yescharacteristic, complemented
33C23×D251—28no—
34C23×D251—24no—
35C23×D251—21no—
36C23×D251—44no—
37C23×D251—44no—
38C23×D251D411yescharacteristic
39C2×D1004—21no—
40C2×D1004—22no—
41C2×D1004D414yes—
42C2×D1004C2312yes—
43C2×C22⋊D251—28no—
44C2×C22⋊D251—24no—
45D4×D251—216no—
46D4×D251—28no—
47D4×D251—28no—
48C22×Dic251—21no—
49C22×Dic251D411yescharacteristic
50C2×C4×D251—22no—
51C2×C4×D251—22no—
52C2×C4×D251D412yes—
53C2×C4×D251C2312yes—
54#16860797—28no—
55C2×C251⋊C8—22no—
56C2×C251⋊C8C2312yescomplemented
57C4⋊Dic251—21no—
58C4⋊Dic251C2312yescomplemented
59Dic251⋊C4—24no—
60(C2×D251)⋊C4—24no—
61C24×C251—21no—
62C2×Q8×C251—21no—
63C2×Q8×C251—22no—
64C2×Q8×C251—22no—
65C2×Q8×C251C2312yescomplemented
66C2×D4×C251—21no—
67C2×D4×C251—22no—
68C2×D4×C251D414yes—
69C2×D4×C251C2312yes—
70C22×C4×C251—21no—
71C22×C4×C251C2311yescharacteristic
72SD16×C251—28no—
73C2×C8×C251—22no—
74C2×C8×C251C2312yescomplemented
75C22×D251—22no—
76C22×D251—22no—
77C22×D251—216no—
78C22×D251—24no—
79C22×D251—22no—
80C22×D251—48no—
81C22×D251—432no—
82C22×D251—42no—
83C22×D251—42no—
84C22×D251—48no—
85C22×D251SD1614yescomplemented
86C22×D251C2×D412yes—
87D1004—24no—
88D1004—22no—
89D1004C2×D414yes—
90C22⋊D251—216no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—42no—
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
104Q8×C251—22no—
105Q8×C251—22no—
106Q8×C251—42no—
107D4×C251—24no—
108D4×C251—22no—
109D4×C251C2×D414yes—
110C2×C4×C251—22no—
111C2×C4×C251—42no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2×D412yes—
114C2×C4×C251C2×D412yes—
115C2×C4×C251C2411yescharacteristic
116C8×C251—22no—
117C22×C251—24no—
118C22×C251—42no—
119C22×C251—42no—
120C22×C251—48no—
121C22×C251#12012yes—
122C22×C251C2×SD1612yes—
123C22×C251AGL(1,ℤ/8)12yes—
124C22×C251C22×D411yescharacteristic
125C2×D251—22no—
126C2×D251—22no—
127C2×D251—28no—
128C2×D251—24no—
129C2×D251—432no—
130C2×D251—42no—
131C2×D251—42no—
132C2×D251—48no—
133C2×D251C2×SD1614yescomplemented
134C4×C251—42no—
135C4×C251#12012yes—
136C4×C251C22×D411yescharacteristic
137C4×C251C22×D411yescharacteristic
138Dic251—22no—
139Dic251—42no—
140C2×C251—44no—
141C2×C251#45014yescomplemented
142C2×C251C2×#12011yescharacteristic
143C2×C251C22×SD1611yescharacteristic
144C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
145D251—24no—
146D251—44no—
147C251C2×#45011yescharacteristic, complemented, Sylow
148C2×#450—2511nomaximal, Sylow
149C23×D4—2511no—
150C22×SD16—2511no—
151C22×SD16D25111yescharacteristic, complemented
152C2×C22⋊Q8—2511no—
153#450—2518no—
154C2×#102—2511no—
155C2×#102—2511no—
156C2×C22⋊C8—2511no—
157C25—5021no—
158C22×Q8C2×D25111yescharacteristic, complemented
159C22×D4—5024no—
160C22×D4—5024no—
161C22×D4—5022no—
162C22×D4—5021no—
163C22×D4—2514no—
164C22×D4—2511no—
165C22×D4C2×D25111yescharacteristic
166C23×C4—2511no—
167C2×SD16—24no—
168C2×SD16—5024no—
169C2×SD16—2514no—
170C2×SD16C2×D25114yescomplemented
171C22×C8—2511no—
172C22×C8C2×D25111yescharacteristic, complemented
173C22⋊Q8—5022no—
174C22⋊Q8—2514no—
175C2×C4⋊C4—5021no—
176C2×C4⋊C4—2511no—
177C2×C22⋊C4—5021no—
178#102—2514no—
179#102—2514no—
180C22⋊C8—2514no—
181C24—21no—
182C24—10044no—
183C24—10044no—
184C24—5028no—
185C24—5024no—
186C24—5021no—
187C24—2511no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8C22×D25112yescomplemented
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—50216no—
195C2×D4—5028no—
196C2×D4—5028no—
197C2×D4—5028no—
198C2×D4—5024no—
199C2×D4—5021no—
200C2×D4—5022no—
201C2×D4—2514no—
202C2×D4—2512no—
203C2×D4C22⋊D25114yes—
204C2×D4C22×D25112yes—
205C22×C4—21no—
206C22×C4—5021no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4—2511no—
210C22×C4—2512no—
211C22×C4—2512no—
212C22×C4C22×D25111yescharacteristic
213SD16—28no—
214SD16—5028no—
215C2×C8—22no—
216C2×C8—5022no—
217C2×C8—2512no—
218C2×C8C22×D25112yescomplemented
219C4⋊C4—5024no—
220C4⋊C4—5021no—
221C4⋊C4—2512no—
222C22⋊C4—5024no—
223C23—22no—
224C23—24no—
225C23—42no—
226C23—42no—
227C23—10048no—
228C23—100432no—
229C23—10048no—
230C23—10042no—
231C23—10042no—
232C23—50216no—
233C23—5022no—
234C23—5024no—
235C23—5022no—
236C23—5022no—
237C23—2512no—
238C23—2514no—
239C23D4×D25111yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—42no—
243D4—22no—
244D4—24no—
245D4—50216no—
246D4—5022no—
247D4—5024no—
248D4—2514no—
249D4C2×C22⋊D25114yes—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—10042no—
253C2×C4—5022no—
254C2×C4—5022no—
255C2×C4—5022no—
256C2×C4—5022no—
257C2×C4—5024no—
258C2×C4—2512no—
259C2×C4D4×D25112yes—
260C2×C4D4×D25111yescharacteristic
261C2×C4C2×C22⋊D25112yes—
262C2×C4C23×D25111yescharacteristic
263C8—22no—
264C8—5022no—
265C22—24no—
266C22—42no—
267C22—42no—
268C22—48no—
269C22—100432no—
270C22—10042no—
271C22—10042no—
272C22—10048no—
273C22—5024no—
274C22—5022no—
275C22—5028no—
276C22—5022no—
277C22—2514no—
278C22#2254868312yes—
279C22SD16×D25112yes—
280C22#2254874212yes—
281C22C2×D4×D25111yescharacteristic
282C4—42no—
283C4—10042no—
284C4—5022no—
285C4#2254868312yes—
286C4C2×D4×D25111yescharacteristic
287C4C22×C22⋊D25111yescharacteristic
288C2—44no—
289C2—10044no—
290C2—5024no—
291C2#3063183614yesdirect product factor
292C2C2×#2254868311yescharacteristic
293C2C2×SD16×D25111yescharacteristic
294C2C2×#2254874211yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×SD16×C251C2
socleC23×C251C2×D4
prime radicalC22×D4×D251C2
hypercenterC22×SD16D251
hypocenterC251C2×#450
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
2. C22×#16860797C2semidirect
3. C2×#22548691C2semidirect
4. #30631836C2direct (N × H)
5. C2×#22548637C2semidirect
6. C2×#22548636C2semidirect
7. C2×(C2×D251)⋊C8C2semidirect
8. C22×SD16×C251C2semidirect
18. C2×#16860797C22semidirect
19. C22×C251⋊C8C22semidirect
21. #22548691C22semidirect
22. C2×C4⋊Dic251C22semidirect
25. #22548637C22semidirect
26. #22548636C22semidirect
27. (C2×D251)⋊C8C22semidirect
28. C22×Q8×C251C22semidirect
31. C2×SD16×C251C22semidirect
32. C22×C8×C251C22semidirect
56. C2×C251⋊C8C23semidirect
58. C4⋊Dic251C23semidirect
65. C2×Q8×C251C23semidirect
74. C2×C8×C251C23semidirect
85. C22×D251SD16semidirect
133. C2×D251C2×SD16semidirect
141. C2×C251#450semidirect
147. C251C2×#450semidirect
151. C22×SD16D251semidirect
158. C22×Q8C2×D251semidirect
170. C2×SD16C2×D251semidirect
172. C22×C8C2×D251semidirect
191. C2×Q8C22×D251semidirect
218. C2×C8C22×D251semidirect
291. C2#30631836direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D2512
C22×#168607972
C2×#225486912
#306318362
C2×#225486372
C2×#225486362
C2×(C2×D251)⋊C82
C22×SD16×C2512
C2×#450251