Finite Group Databasebeta

← C2×#22550534 #30652195

Subgroups — C2×#22550534 #30652195

Order 16256.

1620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523247C211yesmaximal, characteristic, complemented
2C22×Q8×D127C211yesmaximal, characteristic, complemented
3#22550534C2116yesmaximal, direct product factor
4C2×#16861195C212yesmaximal, complemented
5C2×#16861186C212yesmaximal, complemented
6C2×#16861185C211yesmaximal, characteristic, complemented
7C2×#16861182C212yesmaximal, complemented
8C2×#16861180C211yesmaximal, characteristic, complemented
9C2×C4×Dic254C211yesmaximal, characteristic, complemented
10C2×C4×D508C211yesmaximal, characteristic, complemented
11C2×(C22⋊D127)⋊C4C212yesmaximal, complemented
12C2×#124×C127C211yesmaximal, characteristic, complemented
13C2×#14523247—28no—
14C2×#14523247C2218yescomplemented
15C22×Dic254C2212yes—
16C22×Dic254C2211yescharacteristic
17C2×Q8×D127—28no—
18C2×Q8×D127C2218yescomplemented
19C22×D508C2211yescharacteristic
20C22×C22⋊D127C2212yes—
21C22×C22⋊D127C2212yes—
22C23×Dic127C2212yes—
23C22×C4×D127C2212yes—
24C22×C4×D127C2211yescharacteristic
25C22×C4×D127C2211yescharacteristic
26#16861195C22116yescomplemented
27#16861186C22116yescomplemented
28#16861185C2218yescomplemented
29#16861182C22116yescomplemented
30#16861180C2218yescomplemented
31C4×Dic254C2218yescomplemented
32C4×D508C2218yescomplemented
33(C22⋊D127)⋊C4C22116yescomplemented
34C2×C4⋊Dic127C2211yescharacteristic
35C2×C4⋊Dic127C2211yescharacteristic
36C2×Dic127⋊C4C2212yes—
37C2×Dic127⋊C4C2212yes—
38C2×C22⋊Dic127C2212yes—
39C2×(C2×D127)⋊C4C2211yescharacteristic
40C2×(C2×D127)⋊C4C2211yescharacteristic
41C2×(C2×D127)⋊C4C2212yes—
42C2×C4×Dic127C2212yes—
43C22×Q8×C127C2211yescharacteristic
44C22×D4×C127C2211yescharacteristic
45#124×C127C2218yescomplemented
46C2×C22⋊C4×C127C2212yes—
47C2×C22⋊C4×C127C2212yes—
48C2×C42×C127C2211yescharacteristic
49C23×D127C2311yescharacteristic
50C23×D127C2311yescharacteristic
51#14523247—232no—
52C2×Dic254—28no—
53C2×Dic254D418yescomplemented
54C2×Dic254C2318yes—
55C2×Dic254C2314yes—
56Q8×D127—232no—
57C2×D508—24no—
58C2×D508C2314yes—
59C2×C22⋊D127—28no—
60C2×C22⋊D127—28no—
61C2×C22⋊D127C2318yes—
62C2×C22⋊D127C2318yes—
63C22×Dic127—28no—
64C22×Dic127C2318yes—
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2312yes—
68C22×Dic127C2312yes—
69C2×C4×D127—28no—
70C2×C4×D127—24no—
71C2×C4×D127—24no—
72C2×C4×D127C2314yes—
73C2×C4×D127C2318yes—
74C2×C4×D127C2314yes—
75C4⋊Dic127C2314yes—
76C4⋊Dic127C2314yes—
77Dic127⋊C4C2318yes—
78Dic127⋊C4C2318yes—
79C22⋊Dic127C2318yes—
80(C2×D127)⋊C4C2318yes—
81(C2×D127)⋊C4C2314yes—
82(C2×D127)⋊C4C2314yes—
83C4×Dic127C2318yes—
84C24×C127C2312yes—
85C2×Q8×C127—24no—
86C2×Q8×C127C2314yes—
87C2×D4×C127—24no—
88C2×D4×C127C2314yes—
89C22×C4×C127C2312yes—
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C22⋊C4×C127C2318yes—
94C22⋊C4×C127C2318yes—
95C42×C127C2314yes—
96C22×D127—22no—
97C22×D127—22no—
98C22×D127—22no—
99C22×D127—22no—
100C22×D127—22no—
101C22×D127#4114yescomplemented
102C22×D127C2412yes—
103C22×D127C2412yes—
104Dic254—216no—
105Dic254C2×D4116yescomplemented
106D508—28no—
107C22⋊D127—216no—
108C22⋊D127—216no—
109C2×Dic127—22no—
110C2×Dic127—22no—
111C2×Dic127—24no—
112C2×Dic127—216no—
113C2×Dic127C2×D418yes—
114C2×Dic127C2414yes—
115C2×Dic127C2412yes—
116C2×Dic127C2412yes—
117C2×Dic127C2414yes—
118C4×D127—28no—
119C4×D127—28no—
120C4×D127—216no—
121C23×C127—24no—
122C23×C127—24no—
123C23×C127—24no—
124C23×C127C2414yes—
125C23×C127C2411yescharacteristic
126Q8×C127—28no—
127D4×C127—28no—
128C2×C4×C127—24no—
129C2×C4×C127—22no—
130C2×C4×C127—22no—
131C2×C4×C127C2×D414yes—
132C2×C4×C127C2412yes—
133C2×C4×C127C2412yes—
134C2×C4×C127C2414yes—
135C2×C4×C127C2412yes—
136C22×C127—24no—
137C22×C127—24no—
138C22×C127—24no—
139C22×C127—48no—
140C22×C127C22×D412yes—
141C22×C127C2×#4112yes—
142C22×C1272−1+412yes—
143C22×C127C2511yescharacteristic
144C2×D127—28no—
145C2×D127—22no—
146C2×D127—22no—
147C2×D127—22no—
148C2×D127—22no—
149C2×D127—22no—
150C2×D127—44no—
151C2×D127C2×#4114yescomplemented
152C4×C127—24no—
153C4×C127—22no—
154C4×C127—22no—
155C4×C127C22×D414yes—
156Dic127—22no—
157Dic127—22no—
158Dic127—24no—
159Dic127C22×D418yes—
160C2×C127—44no—
161C2×C127#54814yescomplemented
162C2×C127C23×D411yescharacteristic
163C2×C127C22×#4111yescharacteristic
164C2×C127C2×2−1+411yescharacteristic
165D127—24no—
166D127—42no—
167C2×#548—1271nomaximal, Sylow
168C127C2×#54811yescharacteristic, complemented, Sylow
169C22×#41—1271no—
170C23×Q8—1271no—
171#548—12716no—
172C2×#124—1272no—
173C2×#124D12711yescharacteristic, complemented
174C2×C22⋊Q8—1271no—
175C2×C22⋊Q8—1272no—
176C2×#121—1271no—
177C2×#121—1272no—
178C2×C4×Q8—1271no—
179C2×C4×D4—1272no—
180C2×C4×D4—1271no—
181C2×#41—1278no—
182C2×#41—2548no—
183C22×Q8—1278no—
184C22×Q8—1271no—
185C22×Q8—1272no—
186C22×Q8—2548no—
187C22×Q8C2×D12711yescharacteristic
188C22×D4—1272no—
189C22×D4—1272no—
190C22×D4—1271no—
191C22×D4C2×D12711yescharacteristic
192C23×C4—1272no—
193C23×C4—1272no—
194C23×C4—1271no—
195C23×C4—1271no—
196#124—12716no—
197#124C2×D12718yescomplemented
198C22⋊Q8—1278no—
199C22⋊Q8—12716no—
200#121—1278no—
201#121—12716no—
202C4×Q8—1278no—
203C4×D4—12716no—
204C4×D4—1278no—
205C2×C4⋊C4—1272no—
206C2×C4⋊C4—1271no—
207C2×C4⋊C4—1272no—
208C2×C4⋊C4—1271no—
209C2×C22⋊C4—1272no—
210C2×C22⋊C4—1271no—
211C2×C22⋊C4—1271no—
212C2×C22⋊C4—1272no—
213C2×C22⋊C4C2×D12712yes—
214C2×C22⋊C4C2×D12712yes—
215C2×C42—1272no—
216C2×C42C2×D12711yescharacteristic
217C24—1271no—
218C24—1271no—
219C24C22×D12712yes—
220#41—25432no—
221C2×Q8—24no—
222C2×Q8—1278no—
223C2×Q8—1274no—
224C2×Q8—1278no—
225C2×Q8—2548no—
226C2×Q8—25432no—
227C2×Q8C22×D12714yes—
228C2×D4—24no—
229C2×D4—1278no—
230C2×D4—1278no—
231C2×D4—1274no—
232C2×D4—2544no—
233C2×D4—2548no—
234C2×D4—2548no—
235C2×D4C22×D12714yes—
236C22×C4—1272no—
237C22×C4—1278no—
238C22×C4—1272no—
239C22×C4—1271no—
240C22×C4—1271no—
241C22×C4—1278no—
242C22×C4—1274no—
243C22×C4—1274no—
244C22×C4—2548no—
245C22×C4—2548no—
246C22×C4—2544no—
247C22×C4—2544no—
248C22×C4C22×D12711yescharacteristic
249C22×C4C22×D12711yescharacteristic
250C22×C4C22×D12712yes—
251C22×C4C22×D12711yescharacteristic
252C4⋊C4—1278no—
253C4⋊C4—1278no—
254C4⋊C4—1274no—
255C4⋊C4—1274no—
256C22⋊C4—1278no—
257C22⋊C4—1274no—
258C22⋊C4—1274no—
259C22⋊C4—1278no—
260C22⋊C4C22×D12718yes—
261C22⋊C4C22×D12718yes—
262C42—1278no—
263C42C22×D12714yes—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—1274no—
268C23—1272no—
269C23—1272no—
270C23—2542no—
271C23—2542no—
272C23—2542no—
273C23—2542no—
274C23—2542no—
275C23C23×D12714yes—
276C23C23×D12711yescharacteristic
277Q8—28no—
278Q8—12716no—
279Q8—25416no—
280D4—28no—
281D4—2548no—
282D4—25416no—
283D4—25416no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—1278no—
288C2×C4—1274no—
289C2×C4—1274no—
290C2×C4—1272no—
291C2×C4—1272no—
292C2×C4—25416no—
293C2×C4—25416no—
294C2×C4—2542no—
295C2×C4—2542no—
296C2×C4—2544no—
297C2×C4—2548no—
298C2×C4—2548no—
299C2×C4D4×D12714yes—
300C2×C4C23×D12712yes—
301C2×C4C23×D12712yes—
302C2×C4C23×D12714yes—
303C2×C4C23×D12712yes—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—48no—
308C22—1274no—
309C22—5084no—
310C22—2542no—
311C22—2542no—
312C22—2542no—
313C22—2548no—
314C22—2542no—
315C22—2542no—
316C22C2×D4×D12712yes—
317C22#41×D12712yes—
318C22#1686125612yes—
319C22C24×D12711yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—1278no—
324C4—2542no—
325C4—2542no—
326C4—2544no—
327C4C2×D4×D12714yes—
328C2—44no—
329C2—5082no—
330C2—2544no—
331C2#2255053414yesdirect product factor
332C2C22×D4×D12711yescharacteristic
333C2C2×#41×D12711yescharacteristic
334C2C2×#1686125611yescharacteristic
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC2×#124×C127C2
socleC23×C127C24
hypercenterC2×#124D127
hypocenterC127C2×#548
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×#14523247C2semidirect
2. C22×Q8×D127C2semidirect
3. #22550534C2direct (N × H)
4. C2×#16861195C2semidirect
5. C2×#16861186C2semidirect
6. C2×#16861185C2semidirect
7. C2×#16861182C2semidirect
8. C2×#16861180C2semidirect
9. C2×C4×Dic254C2semidirect
10. C2×C4×D508C2semidirect
11. C2×(C22⋊D127)⋊C4C2semidirect
12. C2×#124×C127C2semidirect
14. C2×#14523247C22semidirect
18. C2×Q8×D127C22semidirect
26. #16861195C22semidirect
27. #16861186C22semidirect
28. #16861185C22semidirect
29. #16861182C22semidirect
30. #16861180C22semidirect
31. C4×Dic254C22semidirect
32. C4×D508C22semidirect
33. (C22⋊D127)⋊C4C22semidirect
45. #124×C127C22semidirect
53. C2×Dic254D4semidirect
101. C22×D127#41semidirect
105. Dic254C2×D4semidirect
151. C2×D127C2×#41semidirect
161. C2×C127#548semidirect
168. C127C2×#548semidirect
173. C2×#124D127semidirect
197. #124C2×D127semidirect
331. C2#22550534direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232472
C22×Q8×D1272
#225505342
C2×#168611952
C2×#168611862
C2×#168611852
C2×#168611822
C2×#168611802
C2×C4×Dic2542
C2×C4×D5082
C2×(C22⋊D127)⋊C42
C2×#124×C1272
C2×#548127