Finite Group Databasebeta

← C2×#22550042 #30648052

Subgroups — C2×#22550042 #30648052

Order 16256.

844 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861255C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C2×#16861205C211yesmaximal, characteristic, complemented
4#22550042C218yesmaximal, direct product factor
5C2×#16861136C211yesmaximal, characteristic, complemented
6C2×#16861135C211yesmaximal, characteristic, complemented
7C2×#16861125C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
9#16861255—22no—
10#16861255—22no—
11#16861255—22no—
12#16861255C2214yescomplemented
13C2×#14523249—21no—
14C2×#14523248—21no—
15C2×#14523248C2211yescharacteristic
16C22×D508—21no—
17C22×D508C2211yescharacteristic
18C22×C22⋊D127—22no—
19C2×D4×D127—22no—
20C2×D4×D127—21no—
21#16861235—22no—
22#16861235C2214yescomplemented
23C2×#14523236—21no—
24C2×#14523233—21no—
25C2×#14523232C2211yescharacteristic, complemented
26#16861205—22no—
27#16861205C2214yescomplemented
28C2×C4×Dic127C2211yescharacteristic, complemented
29#16861136C2214yescomplemented
30#16861135C2214yescomplemented
31#16861125C2214yescomplemented
32C2×#41×C127C2211yescharacteristic
33C22×D4×C127C2211yescharacteristic, complemented
34AGL(1,ℤ/8)×C127—22no—
35AGL(1,ℤ/8)×C127C2214yescomplemented
36C2×SD16×C127—21no—
37C2×D8×C127—21no—
38C2×M16×C127C2211yescharacteristic, complemented
39C23×D127—22no—
40C23×D127D412yes—
41#14523249—24no—
42#14523249—24no—
43#14523248—24no—
44#14523248—24no—
45#14523248—21no—
46#14523248—21no—
47#14523248D412yescomplemented
48#14523248C2312yes—
49C2×Dic254D411yescharacteristic, complemented
50C2×D508—22no—
51C2×D508—24no—
52C2×D508—24no—
53C2×D508—21no—
54C2×D508—21no—
55C2×D508—21no—
56C2×D508—21no—
57C2×D508D412yes—
58C2×D508D411yescharacteristic, complemented
59C2×D508C2312yes—
60C2×C22⋊D127—22no—
61C2×C22⋊D127—28no—
62C2×C22⋊D127—21no—
63C2×C22⋊D127—48no—
64D4×D127—28no—
65D4×D127—24no—
66D4×D127—28no—
67D4×D127—24no—
68C22×Dic127—21no—
69C2×C4×D127—22no—
70C2×C4×D127—21no—
71#14523236—24no—
72#14523233—24no—
73#14523232—21no—
74#14523232C2312yescomplemented
75C2×C127⋊C8—21no—
76C4×Dic127—21no—
77C4×Dic127C2312yescomplemented
78C24×C127—21no—
79#41×C127—21no—
80#41×C127—21no—
81#41×C127D412yescomplemented
82#41×C127C2312yes—
83C2×Q8×C127D411yescharacteristic, complemented
84C2×D4×C127—21no—
85C2×D4×C127—21no—
86C2×D4×C127—21no—
87C2×D4×C127—21no—
88C2×D4×C127—21no—
89C2×D4×C127—21no—
90C2×D4×C127D411yescharacteristic, complemented
91C2×D4×C127C2312yescomplemented
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94SD16×C127—24no—
95D8×C127—24no—
96M16×C127—21no—
97M16×C127C2312yescomplemented
98C2×C8×C127—21no—
99C22×D127—22no—
100C22×D127—21no—
101C22×D127—28no—
102C22×D127—22no—
103C22×D127—44no—
104C22×D127—84no—
105C22×D127C2×D414yes—
106Dic254—21no—
107Dic254C2×D412yescomplemented
108D508—28no—
109D508—22no—
110D508—22no—
111D508—21no—
112D508—22no—
113D508—22no—
114D508C2×D412yescomplemented
115C22⋊D127—28no—
116C22⋊D127—22no—
117C22⋊D127—22no—
118C22⋊D127—416no—
119C2×Dic127—24no—
120C2×Dic127—21no—
121C2×Dic127—22no—
122C4×D127—28no—
123C4×D127—22no—
124C4×D127—22no—
125C127⋊C8—22no—
126C23×C127—21no—
127C23×C127—22no—
128C23×C127—44no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127C2×D411yescharacteristic
132Q8×C127—21no—
133Q8×C127C2×D412yescomplemented
134D4×C127—22no—
135D4×C127—22no—
136D4×C127—21no—
137D4×C127—22no—
138D4×C127—22no—
139D4×C127—41no—
140D4×C127—41no—
141D4×C127C2×D412yescomplemented
142C2×C4×C127—21no—
143C2×C4×C127—22no—
144C2×C4×C127—22no—
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2411yescharacteristic
151C8×C127—22no—
152C22×C127—22no—
153C22×C127—22no—
154C22×C127—44no—
155C22×C127—42no—
156C22×C127—42no—
157C22×C127—82no—
158C22×C127—82no—
159C22×C127#12011yescharacteristic
160C22×C127#12011yescharacteristic
161C22×C127C22×D411yescharacteristic
162C2×D127—24no—
163C2×D127—22no—
164C2×D127—44no—
165C2×D127—42no—
166C2×D127—44no—
167C2×D127—82no—
168C2×D127—88no—
169C4×C127—22no—
170C4×C127#12011yescharacteristic
171C4×C127#12011yescharacteristic
172C4×C127C22×D411yescharacteristic
173C4×C127C22×D411yescharacteristic
174Dic127—24no—
175Dic127—22no—
176C2×C127—21no—
177C2×C127—21no—
178C2×C127—42no—
179C2×C127—82no—
180C2×C127#45312yescomplemented
181C2×C127C2×#12011yescharacteristic
182D127—44no—
183D127—42no—
184C2×#453—1271nomaximal, Sylow
185C127C2×#45311yescharacteristic, complemented, Sylow
186C2×H(5,𝔽2)—1271no—
187C2×AGL(1,ℤ/8)—1271no—
188C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
189C2×#127—1271no—
190#453—1278no—
191C2×#104—1271no—
192C2×#104—1271no—
193C2×#100—1271no—
194H(5,𝔽2)—1274no—
195H(5,𝔽2)—2542no—
196H(5,𝔽2)—2542no—
197H(5,𝔽2)—2542no—
198C2×#41—1271no—
199C2×#41—2541no—
200C2×#41—2541no—
201C2×#41C2×D12711yescharacteristic
202C22×D4—1271no—
203C22×D4—2541no—
204C22×D4—2542no—
205C22×D4—2542no—
206C22×D4—2541no—
207C22×D4C2×D12711yescharacteristic, complemented
208AGL(1,ℤ/8)—22no—
209AGL(1,ℤ/8)—1274no—
210AGL(1,ℤ/8)—2542no—
211AGL(1,ℤ/8)C2×D12714yescomplemented
212C2×SD16—21no—
213C2×SD16—2541no—
214C2×D8—21no—
215C2×D8—2541no—
216C2×M16—1271no—
217C2×M16C2×D12711yescharacteristic, complemented
218#127—1274no—
219#127—2542no—
220C2×C42—1271no—
221#104—1274no—
222#104—1274no—
223#100—1274no—
224C24—21no—
225C24—1272no—
226C24—2542no—
227#41—21no—
228#41—21no—
229#41—1272no—
230#41—1272no—
231#41—2544no—
232#41—2544no—
233#41—2544no—
234#41—2544no—
235#41—2541no—
236#41—2541no—
237#41C22⋊D12712yescomplemented
238#41C22×D12712yes—
239C2×Q8—1271no—
240C2×Q8C22⋊D12711yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—1271no—
248C2×D4—1272no—
249C2×D4—1272no—
250C2×D4—5088no—
251C2×D4—2544no—
252C2×D4—2542no—
253C2×D4—2544no—
254C2×D4—2548no—
255C2×D4—2548no—
256C2×D4—2544no—
257C2×D4—2548no—
258C2×D4—2542no—
259C2×D4—2541no—
260C2×D4—2544no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4C22⋊D12711yescharacteristic, complemented
266C2×D4C22×D12712yescomplemented
267C22×C4—21no—
268C22×C4—2542no—
269C22×C4—2541no—
270C22×C4—2541no—
271C22×C4C22×D12711yescharacteristic
272SD16—24no—
273SD16—2544no—
274D8—24no—
275D8—2544no—
276M16—21no—
277M16—1272no—
278M16—2541no—
279M16C22×D12712yescomplemented
280C2×C8—21no—
281C2×C8—2541no—
282C42—1272no—
283C42—2541no—
284C23—21no—
285C23—22no—
286C23—44no—
287C23—41no—
288C23—41no—
289C23—1274no—
290C23—10164no—
291C23—5084no—
292C23—2548no—
293C23—2542no—
294C23—2541no—
295C23—2542no—
296C23D4×D12711yescharacteristic
297Q8—21no—
298Q8—1272no—
299Q8—2541no—
300Q8C2×C22⋊D12712yescomplemented
301D4—22no—
302D4—22no—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—41no—
307D4—41no—
308D4—1272no—
309D4—50816no—
310D4—2548no—
311D4—2548no—
312D4—2542no—
313D4—2542no—
314D4—2542no—
315D4—2542no—
316D4—2541no—
317D4—2542no—
318D4—2542no—
319D4C2×C22⋊D12712yescomplemented
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—2548no—
324C2×C4—2544no—
325C2×C4—2542no—
326C2×C4—2541no—
327C2×C4—2542no—
328C2×C4—2542no—
329C2×C4D4×D12711yescharacteristic
330C2×C4D4×D12711yescharacteristic
331C2×C4D4×D12711yescharacteristic
332C2×C4C2×C22⋊D12711yescharacteristic
333C2×C4C2×C22⋊D12711yescharacteristic
334C2×C4C23×D12711yescharacteristic
335C8—22no—
336C8—2542no—
337C22—22no—
338C22—22no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—10162no—
345C22—10168no—
346C22—5084no—
347C22—5082no—
348C22—5084no—
349C22—2542no—
350C22—2544no—
351C22#1686117711yescharacteristic
352C22#1686117711yescharacteristic
353C22C2×D4×D12711yescharacteristic
354C4—22no—
355C4—2544no—
356C4—2542no—
357C4#1686117711yescharacteristic
358C4#1686117711yescharacteristic
359C4C2×D4×D12711yescharacteristic
360C4C22×C22⋊D12711yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—82no—
365C2—5082no—
366C2—5084no—
367C2#2255004212yesdirect product factor
368C2C2×#1686117711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#120
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127C2×#453
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
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×#16861255C2semidirect
2. C2×#16861235C2semidirect
3. C2×#16861205C2semidirect
4. #22550042C2direct (N × H)
5. C2×#16861136C2semidirect
6. C2×#16861135C2semidirect
7. C2×#16861125C2semidirect
8. C2×AGL(1,ℤ/8)×C127C2semidirect
12. #16861255C22semidirect
22. #16861235C22semidirect
25. C2×#14523232C22semidirect
27. #16861205C22semidirect
28. C2×C4×Dic127C22semidirect
29. #16861136C22semidirect
30. #16861135C22semidirect
31. #16861125C22semidirect
33. C22×D4×C127C22semidirect
35. AGL(1,ℤ/8)×C127C22semidirect
38. C2×M16×C127C22semidirect
47. #14523248D4semidirect
49. C2×Dic254D4semidirect
58. C2×D508D4semidirect
74. #14523232C23semidirect
77. C4×Dic127C23semidirect
81. #41×C127D4semidirect
83. C2×Q8×C127D4semidirect
90. C2×D4×C127D4semidirect
91. C2×D4×C127C23semidirect
97. M16×C127C23semidirect
107. Dic254C2×D4semidirect
114. D508C2×D4semidirect
133. Q8×C127C2×D4semidirect
141. D4×C127C2×D4semidirect
180. C2×C127#453semidirect
185. C127C2×#453semidirect
188. C2×AGL(1,ℤ/8)D127semidirect
207. C22×D4C2×D127semidirect
211. AGL(1,ℤ/8)C2×D127semidirect
217. C2×M16C2×D127semidirect
237. #41C22⋊D127semidirect
240. C2×Q8C22⋊D127semidirect
265. C2×D4C22⋊D127semidirect
266. C2×D4C22×D127semidirect
279. M16C22×D127semidirect
300. Q8C2×C22⋊D127semidirect
319. D4C2×C22⋊D127semidirect
367. C2#22550042direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612352
C2×#168612052
#225500422
C2×#168611362
C2×#168611352
C2×#168611252
C2×AGL(1,ℤ/8)×C1272
C2×#453127