Finite Group Databasebeta

← #30641712

Subgroups — #30641712

Order 16256.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861179C211yesmaximal, characteristic
2C2×C4×D508C211yesmaximal, characteristic, complemented
3C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic, complemented
4C22×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D127C211yesmaximal, characteristic, complemented
6#22549778C211yesmaximal, characteristic, complemented
7#22549768C211yesmaximal, characteristic, complemented
8#22549767C211yesmaximal, characteristic, complemented
9#22549755C211yesmaximal, characteristic, complemented
10#22549754C211yesmaximal, characteristic, complemented
11#22549754C211yesmaximal, characteristic, complemented
12#22549751C211yesmaximal, characteristic, complemented
13#22549750C211yesmaximal, characteristic, complemented
14C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
16C24×D127C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C22×D508C2211yescharacteristic
19C22×C22⋊D127C2211yescharacteristic
20C23×Dic127C2211yescharacteristic
21C22×C4×D127C2211yescharacteristic
22C22×C4×D127C2211yescharacteristic
23C22×C4×D127C2211yescharacteristic, complemented
24#16861179C418yescomplemented
25C4×D508—24no—
26(C22⋊D127)⋊C4—24no—
27C2×C4⋊Dic127C2211yescharacteristic, complemented
28C2×C4⋊Dic127C2211yescharacteristic, complemented
29C2×Dic127⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic127C2211yescharacteristic
31C2×(C2×D127)⋊C4—24no—
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×(C2×D127)⋊C4C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×(C2×D127)⋊C4C2211yescharacteristic
37C2×(C2×D127)⋊C4C2211yescharacteristic
38C2×(C2×D127)⋊C4C2211yescharacteristic
39C2×(C2×D127)⋊C4C2211yescharacteristic
40C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
43C22⋊C4×D127—24no—
44C2×C4×Dic127C2211yescharacteristic, complemented
45C127⋊#95C2211yescharacteristic, complemented
46C127⋊#95C2211yescharacteristic, complemented
47C127⋊#95C2211yescharacteristic, complemented
48C127⋊#95C2211yescharacteristic, complemented
49C23×C4×C127C2211yescharacteristic
50C2×C22⋊C4×C127C2211yescharacteristic
51C2×C22⋊C4×C127C2211yescharacteristic
52C2×C42×C127C2211yescharacteristic, complemented
53C2×C42×C127C2211yescharacteristic, complemented
54#95×C127C2211yescharacteristic, complemented
55#95×C127C2211yescharacteristic, complemented
56C23×D127—22no—
57C23×D127—22no—
58C23×D127—22no—
59C23×D127—22no—
60C23×D127—41no—
61C23×D127—41no—
62C23×D127C2×C414yescomplemented
63C23×D127C2311yescharacteristic
64C23×D127C2311yescharacteristic
65C23×D127C2311yescharacteristic
66C2×D508—22no—
67C2×D508—22no—
68C2×D508—41no—
69C2×D508—41no—
70C2×D508C2×C414yescomplemented
71C2×D508C2×C414yescomplemented
72C2×C22⋊D127—22no—
73C2×C22⋊D127—22no—
74C2×C22⋊D127C2×C414yescomplemented
75C22×Dic127—22no—
76C22×Dic127—22no—
77C22×Dic127—41no—
78C22×Dic127C2311yescharacteristic
79C22×Dic127C2311yescharacteristic
80C22×Dic127C2311yescharacteristic
81C22×Dic127C2311yescharacteristic
82C22×Dic127C2311yescharacteristic
83C2×C4×D127—22no—
84C2×C4×D127—22no—
85C2×C4×D127—22no—
86C2×C4×D127—22no—
87C2×C4×D127—22no—
88C2×C4×D127—22no—
89C2×C4×D127—41no—
90C2×C4×D127—41no—
91C2×C4×D127—41no—
92C4⋊Dic127—22no—
93C4⋊Dic127—41no—
94Dic127⋊C4—22no—
95C22⋊Dic127—22no—
96(C2×D127)⋊C4—22no—
97(C2×D127)⋊C4—22no—
98(C2×D127)⋊C4—22no—
99(C2×D127)⋊C4—22no—
100(C2×D127)⋊C4—22no—
101(C2×D127)⋊C4—22no—
102(C2×D127)⋊C4—22no—
103(C2×D127)⋊C4—41no—
104(C2×D127)⋊C4C2×C414yescomplemented
105(C2×D127)⋊C4C2×C414yescomplemented
106(C2×D127)⋊C4D414yescomplemented
107C4×Dic127—22no—
108C24×C127C2311yescharacteristic
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—41no—
112C22×C4×C127C2311yescharacteristic
113C22×C4×C127C2311yescharacteristic
114C22×C4×C127C2311yescharacteristic
115C22×C4×C127C2311yescharacteristic
116C22×C4×C127C2311yescharacteristic
117C22×C4×C127C2311yescharacteristic
118C22⋊C4×C127C2×C414yescomplemented
119C22⋊C4×C127D414yescomplemented
120C42×C127—22no—
121C42×C127—41no—
122C22×D127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—28no—
127C22×D127—21no—
128C22×D127—21no—
129C22×D127—21no—
130C22×D127—21no—
131C22×D127—21no—
132C22×D127—21no—
133C22×D127—21no—
134C22×D127—21no—
135C22×D127—21no—
136C22×D127—21no—
137C22×D127—21no—
138C22×D127—21no—
139C22×D127—28no—
140C22×D127—44no—
141C22×D127—44no—
142C22×D127—44no—
143C22×D127—44no—
144C22×D127—44no—
145C22×D127C22×C412yes—
146C22×D127C22×C412yes—
147C22×D127C22×C412yes—
148C22×D127C2×D412yes—
149C22×D127#4112yes—
150D508—28no—
151D508—44no—
152C22⋊D127—28no—
153C2×Dic127—21no—
154C2×Dic127—21no—
155C2×Dic127—21no—
156C2×Dic127—21no—
157C2×Dic127—21no—
158C2×Dic127—21no—
159C2×Dic127—21no—
160C2×Dic127—21no—
161C2×Dic127—21no—
162C2×Dic127—21no—
163C2×Dic127—21no—
164C2×Dic127—21no—
165C2×Dic127—21no—
166C2×Dic127—44no—
167C2×Dic127C22×C412yes—
168C2×Dic127C2×D412yes—
169C4×D127—44no—
170C4×D127—44no—
171C4×D127—44no—
172C23×C127—21no—
173C23×C127—21no—
174C23×C127—21no—
175C23×C127—21no—
176C23×C127C22×C412yes—
177C23×C127C2×D412yes—
178C23×C127#4112yes—
179C23×C127C2411yescharacteristic
180C2×C4×C127—21no—
181C2×C4×C127—21no—
182C2×C4×C127—21no—
183C2×C4×C127—21no—
184C2×C4×C127—21no—
185C2×C4×C127—21no—
186C2×C4×C127—21no—
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—21no—
193C2×C4×C127—44no—
194C2×C4×C127C22×C412yes—
195C2×C4×C127C22×C412yes—
196C2×C4×C127C2×D412yes—
197C2×C4×C127C2×D412yes—
198C2×C4×C127C2×D412yes—
199C22×C127—22no—
200C22×C127—22no—
201C22×C127—22no—
202C22×C127—22no—
203C22×C127—22no—
204C22×C127—22no—
205C22×C127C4×D414yescomplemented
206C22×C127C23×C411yescharacteristic
207C22×C127C22×D411yescharacteristic
208C22×C127C22×D411yescharacteristic
209C22×C127C2×#4111yescharacteristic
210C22×C127C2×#4111yescharacteristic
211C22×C127H(5,𝔽2)11yescharacteristic
212C22×C127H(5,𝔽2)11yescharacteristic
213C2×D127—22no—
214C2×D127—22no—
215C2×D127—22no—
216C2×D127—22no—
217C2×D127—22no—
218C2×D127—22no—
219C2×D127—22no—
220C2×D127—22no—
221C2×D127—22no—
222C2×D127—22no—
223C2×D127—22no—
224C2×D127—22no—
225C2×D127—416no—
226C2×D127—41no—
227C2×D127—41no—
228C2×D127—41no—
229C2×D127—41no—
230C2×D127—41no—
231C2×D127—41no—
232C2×D127—41no—
233C2×D127—41no—
234C2×D127C4×D414yescomplemented
235C4×C127—22no—
236C4×C127—22no—
237C4×C127—22no—
238C4×C127—22no—
239C4×C127—22no—
240C4×C127—41no—
241Dic127—22no—
242Dic127—22no—
243Dic127—41no—
244Dic127—41no—
245Dic127—41no—
246C2×C127—24no—
247C2×C127C2×C4×D411yescharacteristic
248C2×C127C2×C4×D411yescharacteristic
249C2×C127#51811yescharacteristic
250C2×C127D4211yescharacteristic
251C2×C127#54611yescharacteristic
252C2×C127#54611yescharacteristic
253C2×C127#55111yescharacteristic
254D127—24no—
255D127—42no—
256D127—42no—
257#2358—1271nomaximal, Sylow
258C127#235811yescharacteristic, complemented, Sylow
259C2×#120—1271no—
260C2×C4×D4—1271no—
261C2×C4×D4—1271no—
262C22×C22⋊C4—1271no—
263C22×C22⋊C4—1271no—
264#390—1271no—
265#388—1271no—
266#388—1271no—
267#386—1271no—
268#386—1271no—
269#386—1271no—
270#385—1271no—
271#385—1271no—
272C4×C22⋊C4—1271no—
273C4×C22⋊C4D12711yescharacteristic, complemented
274C25—1271no—
275C22×D4—1271no—
276C22×D4—1271no—
277C22×D4—1271no—
278C23×C4—1271no—
279C23×C4—1271no—
280C23×C4—1271no—
281C23×C4—1271no—
282C23×C4C2×D12711yescharacteristic
283#120—1278no—
284C4×D4—2544no—
285C4×D4—2544no—
286C2×C4⋊C4—1271no—
287C2×C4⋊C4—1271no—
288C2×C4⋊C4—1271no—
289C2×C22⋊C4—1271no—
290C2×C22⋊C4—1271no—
291C2×C22⋊C4—1271no—
292C2×C22⋊C4—1271no—
293C2×C22⋊C4—1271no—
294C2×C22⋊C4—1271no—
295C2×C22⋊C4—1271no—
296C2×C22⋊C4—1271no—
297C2×C22⋊C4—1271no—
298C2×C22⋊C4—1271no—
299C2×C22⋊C4—1271no—
300C2×C22⋊C4—1271no—
301C2×C22⋊C4—2544no—
302C2×C22⋊C4—2544no—
303C2×C22⋊C4C2×D12711yescharacteristic
304C2×C22⋊C4C2×D12711yescharacteristic
305C2×C42—1271no—
306C2×C42C2×D12711yescharacteristic, complemented
307C2×C42C2×D12711yescharacteristic, complemented
308#95—1271no—
309#95—1271no—
310#95—1271no—
311#95—1271no—
312#95C2×D12711yescharacteristic, complemented
313#95C2×D12711yescharacteristic, complemented
314C24—1271no—
315C24—1274no—
316C24—1271no—
317C24—1271no—
318C24—5081no—
319C24—5081no—
320C24—2542no—
321C24—2542no—
322C24—2542no—
323C24—2542no—
324C24C22×D12711yescharacteristic
325C2×D4—1274no—
326C2×D4—1274no—
327C2×D4—1274no—
328C2×D4—5081no—
329C2×D4—5081no—
330C2×D4—2542no—
331C2×D4—2542no—
332C2×D4—2542no—
333C2×D4—2542no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—1271no—
338C22×C4—1271no—
339C22×C4—1271no—
340C22×C4—1271no—
341C22×C4—1271no—
342C22×C4—5081no—
343C22×C4—5081no—
344C22×C4—5081no—
345C22×C4—5081no—
346C22×C4—2542no—
347C22×C4—2542no—
348C22×C4—2542no—
349C22×C4—2542no—
350C22×C4—2542no—
351C22×C4—2542no—
352C22×C4—2542no—
353C22×C4—2542no—
354C22×C4C22×D12711yescharacteristic
355C22×C4C22×D12711yescharacteristic
356C22×C4C22×D12711yescharacteristic
357C22×C4C22×D12711yescharacteristic
358C22×C4C22×D12711yescharacteristic
359C22×C4C22×D12711yescharacteristic
360C4⋊C4—5081no—
361C4⋊C4—2542no—
362C4⋊C4—2542no—
363C22⋊C4—1274no—
364C22⋊C4—1274no—
365C22⋊C4—1274no—
366C22⋊C4—5081no—
367C22⋊C4—2542no—
368C22⋊C4—2542no—
369C22⋊C4—2542no—
370C22⋊C4—2542no—
371C22⋊C4—2542no—
372C22⋊C4—2542no—
373C22⋊C4—2542no—
374C22⋊C4—2542no—
375C22⋊C4C4×D12714yescomplemented
376C22⋊C4D50814yescomplemented
377C42—22no—
378C42—41no—
379C42—2542no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—1272no—
385C23—1272no—
386C23—1272no—
387C23—1272no—
388C23—1272no—
389C23—5084no—
390C23—5084no—
391C23—5084no—
392C23—5084no—
393C23—5084no—
394C23—2548no—
395C23—2541no—
396C23—2541no—
397C23—2541no—
398C23—2541no—
399C23—2541no—
400C23—2541no—
401C23—2541no—
402C23—2541no—
403C23—2541no—
404C23—2541no—
405C23—2541no—
406C23—2541no—
407C23—2548no—
408C23—2541no—
409C23—2541no—
410C23—2541no—
411C23—2541no—
412C23C2×C4×D12712yes—
413C23C2×D50812yes—
414C23#1452324812yes—
415C23C23×D12711yescharacteristic
416D4—5084no—
417D4—2548no—
418D4—2548no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—44no—
433C2×C4—1272no—
434C2×C4—1272no—
435C2×C4—5084no—
436C2×C4—5084no—
437C2×C4—5084no—
438C2×C4—5084no—
439C2×C4—2541no—
440C2×C4—2541no—
441C2×C4—2541no—
442C2×C4—2541no—
443C2×C4—2541no—
444C2×C4—2541no—
445C2×C4—2541no—
446C2×C4—2541no—
447C2×C4—2541no—
448C2×C4—2541no—
449C2×C4—2541no—
450C2×C4—2541no—
451C2×C4—2541no—
452C2×C4C2×C4×D12712yes—
453C2×C4C2×C4×D12712yes—
454C2×C4D4×D12712yes—
455C2×C4C2×D50812yes—
456C2×C4C2×D50812yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—1274no—
464C22—50816no—
465C22—5081no—
466C22—5081no—
467C22—5081no—
468C22—5081no—
469C22—5081no—
470C22—5081no—
471C22—5081no—
472C22—5081no—
473C22—2542no—
474C22—2542no—
475C22—2542no—
476C22—2542no—
477C22—2542no—
478C22—2542no—
479C22—2542no—
480C22—2542no—
481C22—2542no—
482C22—2542no—
483C22—2542no—
484C22—2542no—
485C22C4×D50814yescomplemented
486C22C22×C4×D12711yescharacteristic
487C22C2×D4×D12711yescharacteristic
488C22C22×D50811yescharacteristic
489C22#41×D12711yescharacteristic
490C22C2×#1452324811yescharacteristic
491C22#1686125411yescharacteristic
492C22#1686125511yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—5081no—
500C4—5081no—
501C4—5081no—
502C4—2542no—
503C4—2542no—
504C2—24no—
505C2—5082no—
506C2—5082no—
507C2—2544no—
508C2C4×D4×D12711yescharacteristic
509C2C2×C4×D50811yescharacteristic
510C2#2255034811yescharacteristic
511C2D4×D50811yescharacteristic
512C2#2255051611yescharacteristic
513C2#2255052111yescharacteristic
514C2#2255055011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC4×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861179C2
hypercenterC4×C22⋊C4D127
hypocenterC127#2358
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D508C2semidirect
3. C2×(C22⋊D127)⋊C4C2semidirect
4. C22×(C2×D127)⋊C4C2semidirect
5. C2×C22⋊C4×D127C2semidirect
6. #22549778C2semidirect
7. #22549768C2semidirect
8. #22549767C2semidirect
9. #22549755C2semidirect
10. #22549754C2semidirect
11. #22549754C2semidirect
12. #22549751C2semidirect
13. #22549750C2semidirect
14. C4×(C2×D127)⋊C4C2semidirect
15. C4×C22⋊C4×C127C2semidirect
23. C22×C4×D127C22semidirect
24. #16861179C4semidirect
27. C2×C4⋊Dic127C22semidirect
28. C2×C4⋊Dic127C22semidirect
29. C2×Dic127⋊C4C22semidirect
40. C2×(C2×D127)⋊C4C22semidirect
41. C2×(C2×D127)⋊C4C22semidirect
42. C2×(C2×D127)⋊C4C22semidirect
44. C2×C4×Dic127C22semidirect
45. C127⋊#95C22semidirect
46. C127⋊#95C22semidirect
47. C127⋊#95C22semidirect
48. C127⋊#95C22semidirect
52. C2×C42×C127C22semidirect
53. C2×C42×C127C22semidirect
54. #95×C127C22semidirect
55. #95×C127C22semidirect
62. C23×D127C2×C4semidirect
70. C2×D508C2×C4semidirect
71. C2×D508C2×C4semidirect
74. C2×C22⋊D127C2×C4semidirect
104. (C2×D127)⋊C4C2×C4semidirect
105. (C2×D127)⋊C4C2×C4semidirect
106. (C2×D127)⋊C4D4semidirect
118. C22⋊C4×C127C2×C4semidirect
119. C22⋊C4×C127D4semidirect
205. C22×C127C4×D4semidirect
234. C2×D127C4×D4semidirect
258. C127#2358semidirect
273. C4×C22⋊C4D127semidirect
306. C2×C42C2×D127semidirect
307. C2×C42C2×D127semidirect
312. #95C2×D127semidirect
313. #95C2×D127semidirect
375. C22⋊C4C4×D127semidirect
376. C22⋊C4D508semidirect
485. C22C4×D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168611792
C2×C4×D5082
C2×(C22⋊D127)⋊C42
C22×(C2×D127)⋊C42
C2×C22⋊C4×D1272
#225497782
#225497682
#225497672
#225497552
#225497542
#225497512
#225497502
C4×(C2×D127)⋊C42
C4×C22⋊C4×C1272
#2358127