Finite Group Databasebeta

← #30648002

Subgroups — #30648002

Order 16256.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861254C211yesmaximal, characteristic
2C2×#16861238C211yesmaximal, characteristic, complemented
3C2×SD16×D127C211yesmaximal, characteristic, complemented
4#22550449C211yesmaximal, characteristic, complemented
5#22550040C211yesmaximal, characteristic, complemented
6#22550039C211yesmaximal, characteristic, complemented
7#22550036C211yesmaximal, characteristic, complemented
8#22550035C211yesmaximal, characteristic, complemented
9#22550028C211yesmaximal, characteristic, complemented
10#22550027C211yesmaximal, characteristic, complemented
11#22550025C211yesmaximal, characteristic, complemented
12#22549879C211yesmaximal, characteristic, complemented
13#22549869C211yesmaximal, characteristic, complemented
14#22549841C211yesmaximal, characteristic, complemented
15#450×C127C211yesmaximal, characteristic, complemented
16#16861254—24no—
17#16861254—24no—
18C2×#14523248C2211yescharacteristic
19C2×#14523248C2211yescharacteristic
20C2×#14523247—21no—
21C2×#14523247C2211yescharacteristic
22C2×#14523247C2211yescharacteristic
23C2×Q8×D127C2211yescharacteristic, complemented
24C22×C22⋊D127—21no—
25C22×C22⋊D127—21no—
26C2×D4×D127—21no—
27C2×D4×D127C2211yescharacteristic
28C2×D4×D127C2211yescharacteristic
29#16861238—24no—
30C2×Dic508C2211yescharacteristic, complemented
31C2×#14523238C2211yescharacteristic, complemented
32C2×#14523237C2211yescharacteristic, complemented
33C2×#14523236C2211yescharacteristic, complemented
34C2×#14523236C2211yescharacteristic, complemented
35C2×#14523235C2211yescharacteristic, complemented
36SD16×D127—24no—
37C2×#14523231C2211yescharacteristic, complemented
38C2×C8×D127C2211yescharacteristic, complemented
39#16861208—21no—
40#16861195—21no—
41#16861194—21no—
42#16861191—21no—
43#16861189C2211yescharacteristic, complemented
44#16861188C2211yescharacteristic, complemented
45#16861185C2211yescharacteristic, complemented
46#16861167C2211yescharacteristic, complemented
47#16861134C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52#16861129C2211yescharacteristic, complemented
53C22⋊(C127⋊C8)C2211yescharacteristic, complemented
54(C2×D127)⋊C8C2211yescharacteristic, complemented
55(C2×D127)⋊C8C2211yescharacteristic, complemented
56C22×D4×C127C2211yescharacteristic
57C2×SD16×C127C2211yescharacteristic, complemented
58C2×SD16×C127C2211yescharacteristic, complemented
59C22⋊Q8×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61#102×C127C2211yescharacteristic, complemented
62C22⋊C8×C127C2211yescharacteristic, complemented
63C23×D127—21no—
64C23×D127—21no—
65#14523248—22no—
66#14523248—22no—
67#14523248—22no—
68#14523248—22no—
69#14523247—28no—
70#14523247—22no—
71#14523247—22no—
72#14523247—22no—
73#14523247D414yes—
74C2×Dic254—21no—
75C2×Dic254D411yescharacteristic
76C2×Dic254C2311yescharacteristic
77C2×Dic254C2311yescharacteristic
78Q8×D127—22no—
79Q8×D127—22no—
80C2×D508D411yescharacteristic
81C2×D508C2311yescharacteristic
82C2×C22⋊D127—28no—
83C2×C22⋊D127—28no—
84C2×C22⋊D127—21no—
85C2×C22⋊D127—21no—
86C2×C22⋊D127—21no—
87C2×C22⋊D127—21no—
88C2×C22⋊D127—21no—
89C2×C22⋊D127D411yescharacteristic
90C2×C22⋊D127D411yescharacteristic
91D4×D127—28no—
92D4×D127—22no—
93D4×D127—22no—
94D4×D127—22no—
95D4×D127D414yes—
96C22×Dic127—21no—
97C22×Dic127—21no—
98C2×C4×D127—21no—
99C2×C4×D127—21no—
100C2×C4×D127C2311yescharacteristic
101C2×C4×D127C2311yescharacteristic
102Dic508—22no—
103#14523238—22no—
104#14523237—22no—
105#14523236—22no—
106#14523236—22no—
107#14523235—22no—
108#14523231—22no—
109C2×C127⋊C8C2311yescharacteristic
110C2×C127⋊C8C2311yescharacteristic
111C8×D127—22no—
112C4⋊Dic127—21no—
113C4⋊Dic127C2311yescharacteristic
114Dic127⋊C4—21no—
115Dic127⋊C4—21no—
116C22⋊Dic127—21no—
117(C2×D127)⋊C4—21no—
118(C2×D127)⋊C4—21no—
119(C2×D127)⋊C4—21no—
120C4×Dic127—21no—
121C24×C127—21no—
122C2×Q8×C127C2311yescharacteristic
123C2×D4×C127—22no—
124C2×D4×C127—22no—
125C2×D4×C127—21no—
126C2×D4×C127C2311yescharacteristic
127C2×D4×C127C2311yescharacteristic
128C22×C4×C127C2311yescharacteristic
129SD16×C127—22no—
130SD16×C127—22no—
131C2×C8×C127C2311yescharacteristic
132C2×C8×C127C2311yescharacteristic
133C4⋊C4×C127—21no—
134C4⋊C4×C127C2311yescharacteristic
135C22⋊C4×C127—21no—
136C22×D127—21no—
137C22×D127—24no—
138C22×D127—24no—
139C22×D127—81no—
140C22×D127—81no—
141C22×D127—81no—
142C22×D127—81no—
143C22×D127C2×D411yescharacteristic
144C22×D127C2×D411yescharacteristic
145Dic254—21no—
146Dic254—22no—
147Dic254—21no—
148Dic254—21no—
149Dic254—42no—
150Dic254C2×D412yes—
151D508—22no—
152D508—21no—
153D508C2×D412yes—
154C22⋊D127—24no—
155C22⋊D127—24no—
156C22⋊D127—24no—
157C22⋊D127—24no—
158C22⋊D127—22no—
159C22⋊D127—24no—
160C22⋊D127—22no—
161C2×Dic127—24no—
162C2×Dic127—21no—
163C2×Dic127—24no—
164C2×Dic127—21no—
165C2×Dic127—21no—
166C2×Dic127C2×D411yescharacteristic
167C2×Dic127C2×D411yescharacteristic
168C4×D127—24no—
169C4×D127—21no—
170C4×D127—21no—
171C4×D127—42no—
172C4×D127C2×D412yes—
173C4×D127C2×D412yes—
174C127⋊C8—21no—
175C127⋊C8—21no—
176C23×C127—24no—
177C23×C127—21no—
178C23×C127—21no—
179C23×C127—81no—
180C23×C127—81no—
181C23×C127C2×D411yescharacteristic
182Q8×C127—21no—
183Q8×C127—21no—
184D4×C127—24no—
185D4×C127—21no—
186D4×C127—21no—
187D4×C127C2×D412yes—
188D4×C127C2×D412yes—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—21no—
193C2×C4×C127C2×D411yescharacteristic
194C2×C4×C127C2411yescharacteristic
195C8×C127—21no—
196C8×C127—21no—
197C22×C127—21no—
198C22×C127—22no—
199C22×C127—22no—
200C22×C127—21no—
201C22×C127—21no—
202C22×C127—41no—
203C22×C127—41no—
204C22×C127—41no—
205C22×C127—41no—
206C22×C127—84no—
207C22×C127C22×D411yescharacteristic
208C2×D127—21no—
209C2×D127—21no—
210C2×D127—21no—
211C2×D127—22no—
212C2×D127—21no—
213C2×D127—21no—
214C2×D127—41no—
215C2×D127—41no—
216C2×D127—84no—
217C2×D127—84no—
218C2×D127#12012yes—
219C4×C127—21no—
220C4×C127—41no—
221C4×C127—41no—
222C4×C127C22×D411yescharacteristic
223C4×C127C22×D411yescharacteristic
224Dic127—22no—
225Dic127—21no—
226Dic127—41no—
227Dic127—41no—
228Dic127#12012yes—
229C2×C127—42no—
230C2×C127—42no—
231C2×C127—41no—
232C2×C127C2×#12011yescharacteristic
233C2×C127#57711yescharacteristic
234C2×C127#57711yescharacteristic
235D127—22no—
236D127—41no—
237D127—42no—
238#3009—1271nomaximal, Sylow
239C127#300911yescharacteristic, complemented, Sylow
240C2×H(5,𝔽2)—1271no—
241C2×#137—1271no—
242C22×SD16—1271no—
243#536—1271no—
244#452—1271no—
245#452—1271no—
246#452—1271no—
247#452—1271no—
248#450—1271no—
249#450—1271no—
250#450—1271no—
251#450D12711yescharacteristic, complemented
252#418—1271no—
253#416—1271no—
254#408—1271no—
255H(5,𝔽2)—2544no—
256H(5,𝔽2)—2544no—
257C2×#41—1271no—
258C2×#41—1271no—
259C2×#41—1271no—
260C2×#41—1271no—
261C2×#41—2541no—
262C22×Q8—1271no—
263C22×D4—1271no—
264C22×D4—1271no—
265C22×D4—2541no—
266C22×D4—2541no—
267C22×D4—2541no—
268C22×D4C2×D12711yescharacteristic
269#137—2544no—
270C2×Q16—1271no—
271C2×Q16—1271no—
272C2×SD16—1271no—
273C2×SD16—1271no—
274C2×SD16—1271no—
275C2×SD16—1271no—
276C2×SD16—2544no—
277C2×SD16C2×D12711yescharacteristic, complemented
278C2×SD16C2×D12711yescharacteristic, complemented
279C2×M16—1271no—
280C22×C8—1271no—
281C4⋊Q8—2541no—
282#124—2541no—
283#123—2541no—
284#123—2541no—
285C22⋊Q8—1271no—
286C22⋊Q8—1271no—
287C22⋊Q8—1271no—
288C22⋊Q8C2×D12711yescharacteristic, complemented
289#117—1271no—
290#103—1271no—
291#103—1271no—
292#102—1271no—
293#102—1271no—
294#102—1271no—
295#102—1271no—
296#102C2×D12711yescharacteristic, complemented
297#102C2×D12711yescharacteristic, complemented
298C22⋊C8—1271no—
299C22⋊C8—1271no—
300C22⋊C8—1271no—
301C22⋊C8C2×D12711yescharacteristic, complemented
302C24—21no—
303C24—2541no—
304C24—2541no—
305#41—1274no—
306#41—2542no—
307#41—2542no—
308#41—2542no—
309#41—2542no—
310#41—2542no—
311#41—2542no—
312#41—2548no—
313#41—2542no—
314C2×Q8—1271no—
315C2×Q8—1271no—
316C2×Q8—1271no—
317C2×Q8—2541no—
318C2×Q8—2542no—
319C2×Q8—2542no—
320C2×Q8C22×D12711yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—1274no—
325C2×D4—1271no—
326C2×D4—1271no—
327C2×D4—1271no—
328C2×D4—1271no—
329C2×D4—2542no—
330C2×D4—2541no—
331C2×D4—2541no—
332C2×D4—2542no—
333C2×D4—2548no—
334C2×D4—2541no—
335C2×D4—2541no—
336C2×D4—2548no—
337C2×D4—2548no—
338C2×D4—2541no—
339C2×D4—2542no—
340C2×D4C22×D12711yescharacteristic
341C2×D4C22×D12711yescharacteristic
342C22×C4—1271no—
343C22×C4—1271no—
344C22×C4—2541no—
345C22×C4—2541no—
346C22×C4—2541no—
347C22×C4—2541no—
348C22×C4C22×D12711yescharacteristic
349Q16—2542no—
350Q16—2542no—
351SD16—22no—
352SD16—22no—
353SD16—2542no—
354SD16—2542no—
355SD16—2542no—
356SD16—2542no—
357M16—2542no—
358C2×C8—1271no—
359C2×C8—1271no—
360C2×C8—2542no—
361C2×C8C22×D12711yescharacteristic
362C2×C8C22×D12711yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—1271no—
365C4⋊C4—2541no—
366C4⋊C4—2541no—
367C4⋊C4—2541no—
368C4⋊C4C22×D12711yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—2541no—
371C22⋊C4—2541no—
372C22⋊C4—2541no—
373C22⋊C4—2541no—
374C42—2541no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—1271no—
381C23—1271no—
382C23—10161no—
383C23—10161no—
384C23—10161no—
385C23—10161no—
386C23—2544no—
387C23—2544no—
388C23—2541no—
389C23D4×D12711yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—1272no—
393Q8—5082no—
394Q8—2542no—
395Q8—2541no—
396Q8—2541no—
397Q8—2541no—
398D4—24no—
399D4—21no—
400D4—21no—
401D4—1272no—
402D4—2542no—
403D4—2542no—
404D4—2544no—
405D4—2542no—
406D4—2544no—
407D4—2544no—
408D4—2544no—
409D4—2544no—
410D4—2541no—
411D4D4×D12712yes—
412D4D4×D12712yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—1271no—
418C2×C4—1272no—
419C2×C4—1272no—
420C2×C4—1271no—
421C2×C4—5082no—
422C2×C4—2544no—
423C2×C4—2544no—
424C2×C4—2541no—
425C2×C4—2541no—
426C2×C4—2544no—
427C2×C4—2541no—
428C2×C4—2541no—
429C2×C4—2541no—
430C2×C4D4×D12711yescharacteristic
431C2×C4C23×D12711yescharacteristic
432C8—21no—
433C8—21no—
434C8—2541no—
435C8—2541no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—84no—
446C22—1272no—
447C22—10164no—
448C22—10164no—
449C22—5081no—
450C22—5081no—
451C22—2542no—
452C22—2541no—
453C22—2541no—
454C22—2541no—
455C22—2541no—
456C22—2541no—
457C22C2×D4×D12711yescharacteristic
458C4—21no—
459C4—41no—
460C4—41no—
461C4—1272no—
462C4—5081no—
463C4—5081no—
464C4—2541no—
465C4—2542no—
466C4C2×D4×D12711yescharacteristic
467C4C2×D4×D12711yescharacteristic
468C2—41no—
469C2—42no—
470C2—42no—
471C2—5082no—
472C2—5081no—
473C2—2542no—
474C2#120×D12711yescharacteristic
475C2#2255068211yescharacteristic
476C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861254C2
hypercenter#450D127
hypocenterC127#3009
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#16861238C2semidirect
3. C2×SD16×D127C2semidirect
4. #22550449C2semidirect
5. #22550040C2semidirect
6. #22550039C2semidirect
7. #22550036C2semidirect
8. #22550035C2semidirect
9. #22550028C2semidirect
10. #22550027C2semidirect
11. #22550025C2semidirect
12. #22549879C2semidirect
13. #22549869C2semidirect
14. #22549841C2semidirect
15. #450×C127C2semidirect
23. C2×Q8×D127C22semidirect
30. C2×Dic508C22semidirect
31. C2×#14523238C22semidirect
32. C2×#14523237C22semidirect
33. C2×#14523236C22semidirect
34. C2×#14523236C22semidirect
35. C2×#14523235C22semidirect
37. C2×#14523231C22semidirect
38. C2×C8×D127C22semidirect
43. #16861189C22semidirect
44. #16861188C22semidirect
45. #16861185C22semidirect
46. #16861167C22semidirect
47. #16861134C22semidirect
48. #16861132C22semidirect
49. #16861131C22semidirect
50. #16861130C22semidirect
51. #16861130C22semidirect
52. #16861129C22semidirect
53. C22⋊(C127⋊C8)C22semidirect
54. (C2×D127)⋊C8C22semidirect
55. (C2×D127)⋊C8C22semidirect
57. C2×SD16×C127C22semidirect
58. C2×SD16×C127C22semidirect
59. C22⋊Q8×C127C22semidirect
60. #102×C127C22semidirect
61. #102×C127C22semidirect
62. C22⋊C8×C127C22semidirect
239. C127#3009semidirect
251. #450D127semidirect
277. C2×SD16C2×D127semidirect
278. C2×SD16C2×D127semidirect
288. C22⋊Q8C2×D127semidirect
296. #102C2×D127semidirect
297. #102C2×D127semidirect
301. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168612382
C2×SD16×D1272
#225504492
#225500402
#225500392
#225500362
#225500352
#225500282
#225500272
#225500252
#225498792
#225498692
#225498412
#450×C1272
#3009127