Finite Group Databasebeta

← #30641706

Subgroups — #30641706

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×#16861171C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
4C22×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D127C211yesmaximal, characteristic, complemented
6#22549777C211yesmaximal, characteristic, complemented
7#22549766C211yesmaximal, characteristic, complemented
8#22549765C211yesmaximal, characteristic, complemented
9#22549754C211yesmaximal, characteristic, complemented
10#22549754C211yesmaximal, characteristic, complemented
11#22549753C211yesmaximal, characteristic, complemented
12#22549747C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
14#22549723C211yesmaximal, characteristic, complemented
15#385×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
25#16861171—24no—
26C4×C22⋊D127—24no—
27C2×Dic127⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic127C2211yescharacteristic
29C2×(C2×D127)⋊C4—24no—
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
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, complemented
39C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
41C22⋊C4×D127—24no—
42C2×C4×Dic127C2211yescharacteristic, complemented
43C2×C4×Dic127C2211yescharacteristic, complemented
44C2×C4×Dic127C2211yescharacteristic, complemented
45C127⋊#95C2211yescharacteristic, complemented
46C127⋊#95C2211yescharacteristic, complemented
47C127⋊#95C2211yescharacteristic, complemented
48C127⋊#95C2211yescharacteristic, complemented
49C23×C4×C127C2211yescharacteristic
50C2×C4⋊C4×C127C2211yescharacteristic, complemented
51C2×C4⋊C4×C127C2211yescharacteristic, complemented
52C2×C22⋊C4×C127C2211yescharacteristic
53C2×C22⋊C4×C127C2211yescharacteristic
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—
92Dic127⋊C4—22no—
93C22⋊Dic127D414yescomplemented
94(C2×D127)⋊C4—22no—
95(C2×D127)⋊C4—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—41no—
102(C2×D127)⋊C4C2×C414yescomplemented
103(C2×D127)⋊C4C2×C414yescomplemented
104(C2×D127)⋊C4D414yescomplemented
105C4×Dic127—22no—
106C4×Dic127—22no—
107C4×Dic127—41no—
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
118C4⋊C4×C127—22no—
119C4⋊C4×C127—41no—
120C22⋊C4×C127—22no—
121C22⋊C4×C127C2×C414yescomplemented
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—44no—
165C2×Dic127C22×C412yes—
166C2×Dic127C2×D412yes—
167C2×Dic127C2×D412yes—
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—21no—
194C2×C4×C127—21no—
195C2×C4×C127—44no—
196C2×C4×C127C22×C412yes—
197C2×C4×C127C22×C412yes—
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—41no—
239C4×C127—41no—
240C4×C127—41no—
241Dic127—22no—
242Dic127—22no—
243Dic127—22no—
244Dic127—22no—
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#385D12711yescharacteristic, complemented
272C4×C22⋊C4—1271no—
273C4×C22⋊C4—1271no—
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⋊C4C2×D12711yescharacteristic, complemented
288C2×C4⋊C4C2×D12711yescharacteristic, complemented
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×C42—1271no—
307C2×C42—1271no—
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—22no—
361C4⋊C4—41no—
362C4⋊C4—2542no—
363C22⋊C4—22no—
364C22⋊C4—1274no—
365C22⋊C4—1274no—
366C22⋊C4—1274no—
367C22⋊C4—1274no—
368C22⋊C4—5081no—
369C22⋊C4—2542no—
370C22⋊C4—2542no—
371C22⋊C4—2542no—
372C22⋊C4—2542no—
373C22⋊C4—2542no—
374C22⋊C4—2542no—
375C22⋊C4—2542no—
376C22⋊C4C4×D12714yescomplemented
377C42—5081no—
378C42—2542no—
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—
413C23D4×D12712yes—
414C23#1452324912yes—
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—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—1272no—
436C2×C4—1272no—
437C2×C4—1272no—
438C2×C4—1272no—
439C2×C4—5084no—
440C2×C4—5084no—
441C2×C4—5084no—
442C2×C4—5084no—
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×C4—2541no—
453C2×C4—2541no—
454C2×C4C2×C4×D12712yes—
455C2×C4C2×C4×D12712yes—
456C2×C4D4×D12712yes—
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—
485C22#1686117114yescomplemented
486C22C22×C4×D12711yescharacteristic
487C22C2×D4×D12711yescharacteristic
488C22C2×D4×D12711yescharacteristic
489C22#41×D12711yescharacteristic
490C22C2×#1452324911yescharacteristic
491C22#1686125411yescharacteristic
492C22#1686125511yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—5081no—
500C4—2542no—
501C4—2542no—
502C4—2542no—
503C4—2542no—
504C2—24no—
505C2—5082no—
506C2—5082no—
507C2—2544no—
508C2C4×D4×D12711yescharacteristic
509C2C2×#1686117111yescharacteristic
510C2#2255034811yescharacteristic
511C2#2255050811yescharacteristic
512C2#2255051711yescharacteristic
513C2#2255051811yescharacteristic
514C2#2255055311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#385×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861179C2
hypercenter#385D127
hypocenterC127#2358
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#16861171C2semidirect
3. C2×C4×C22⋊D127C2semidirect
4. C22×(C2×D127)⋊C4C2semidirect
5. C2×C22⋊C4×D127C2semidirect
6. #22549777C2semidirect
7. #22549766C2semidirect
8. #22549765C2semidirect
9. #22549754C2semidirect
10. #22549754C2semidirect
11. #22549753C2semidirect
12. #22549747C2semidirect
13. C22⋊C4×Dic127C2semidirect
14. #22549723C2semidirect
15. #385×C127C2semidirect
23. C22×C4×D127C22semidirect
24. #16861179C4semidirect
27. C2×Dic127⋊C4C22semidirect
38. C2×(C2×D127)⋊C4C22semidirect
39. C2×(C2×D127)⋊C4C22semidirect
40. C2×(C2×D127)⋊C4C22semidirect
42. C2×C4×Dic127C22semidirect
43. C2×C4×Dic127C22semidirect
44. C2×C4×Dic127C22semidirect
45. C127⋊#95C22semidirect
46. C127⋊#95C22semidirect
47. C127⋊#95C22semidirect
48. C127⋊#95C22semidirect
50. C2×C4⋊C4×C127C22semidirect
51. C2×C4⋊C4×C127C22semidirect
54. #95×C127C22semidirect
55. #95×C127C22semidirect
62. C23×D127C2×C4semidirect
70. C2×D508C2×C4semidirect
71. C2×D508C2×C4semidirect
74. C2×C22⋊D127C2×C4semidirect
93. C22⋊Dic127D4semidirect
102. (C2×D127)⋊C4C2×C4semidirect
103. (C2×D127)⋊C4C2×C4semidirect
104. (C2×D127)⋊C4D4semidirect
121. C22⋊C4×C127C2×C4semidirect
205. C22×C127C4×D4semidirect
234. C2×D127C4×D4semidirect
258. C127#2358semidirect
271. #385D127semidirect
287. C2×C4⋊C4C2×D127semidirect
288. C2×C4⋊C4C2×D127semidirect
312. #95C2×D127semidirect
313. #95C2×D127semidirect
376. C22⋊C4C4×D127semidirect
485. C22#16861171semidirect

Maximal subgroups

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

subgroupindex
C2×#168611792
C2×#168611712
C2×C4×C22⋊D1272
C22×(C2×D127)⋊C42
C2×C22⋊C4×D1272
#225497772
#225497662
#225497652
#225497542
#225497532
#225497472
C22⋊C4×Dic1272
#225497232
#385×C1272
#2358127