Finite Group Databasebeta

← #30641709

Subgroups — #30641709

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×#16861178C211yesmaximal, characteristic
2C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
3C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic127C211yesmaximal, characteristic, complemented
5#22549776C211yesmaximal, characteristic, complemented
6#22549770C211yesmaximal, characteristic, complemented
7#22549764C211yesmaximal, characteristic, complemented
8#22549755C211yesmaximal, characteristic, complemented
9#22549753C211yesmaximal, characteristic, complemented
10#22549752C211yesmaximal, characteristic, complemented
11#22549749C211yesmaximal, characteristic, complemented
12#22549748C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C22⋊D127C2211yescharacteristic
19C23×Dic127C2211yescharacteristic
20C23×Dic127C2211yescharacteristic
21C22×C4×D127C2211yescharacteristic, complemented
22#16861178C418yescomplemented
23C4×C22⋊D127—24no—
24(C22⋊D127)⋊C4—24no—
25C2×Dic127⋊C4C2211yescharacteristic, complemented
26C2×Dic127⋊C4C2211yescharacteristic, complemented
27C2×Dic127⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic127—24no—
29C2×C22⋊Dic127C2211yescharacteristic
30C2×C22⋊Dic127C2211yescharacteristic
31C2×C22⋊Dic127C2211yescharacteristic
32C2×C22⋊Dic127C2211yescharacteristic
33C2×C22⋊Dic127C2211yescharacteristic
34C2×C22⋊Dic127C2211yescharacteristic
35C2×C22⋊Dic127C2211yescharacteristic
36C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
37C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
39C2×C4×Dic127C2211yescharacteristic, complemented
40C2×C4×Dic127C2211yescharacteristic, complemented
41C2×C4×Dic127C2211yescharacteristic, complemented
42C127⋊#95C2211yescharacteristic, complemented
43C127⋊#95C2211yescharacteristic, complemented
44C127⋊#95C2211yescharacteristic, complemented
45C127⋊#95C2211yescharacteristic, complemented
46C127⋊#95C2211yescharacteristic, complemented
47C127⋊#95C2211yescharacteristic, complemented
48C25×C127C2211yescharacteristic
49C23×C4×C127C2211yescharacteristic
50C23×C4×C127C2211yescharacteristic
51C2×C22⋊C4×C127—24no—
52C2×C22⋊C4×C127C2211yescharacteristic
53C2×C22⋊C4×C127C2211yescharacteristic
54C2×C22⋊C4×C127C2211yescharacteristic
55C2×C22⋊C4×C127C2211yescharacteristic
56C23×D127C2311yescharacteristic
57C2×C22⋊D127—22no—
58C2×C22⋊D127—22no—
59C2×C22⋊D127—22no—
60C2×C22⋊D127—22no—
61C2×C22⋊D127—41no—
62C2×C22⋊D127—41no—
63C2×C22⋊D127C2×C414yescomplemented
64C2×C22⋊D127C2×C414yescomplemented
65C2×C22⋊D127C2×C414yescomplemented
66C22×Dic127—22no—
67C22×Dic127—22no—
68C22×Dic127—22no—
69C22×Dic127—22no—
70C22×Dic127—41no—
71C22×Dic127—41no—
72C22×Dic127C2311yescharacteristic
73C22×Dic127C2311yescharacteristic
74C22×Dic127C2311yescharacteristic
75C22×Dic127C2311yescharacteristic
76C22×Dic127C2311yescharacteristic
77C22×Dic127C2311yescharacteristic
78C22×Dic127C2311yescharacteristic
79C2×C4×D127—22no—
80C2×C4×D127—22no—
81C2×C4×D127—41no—
82Dic127⋊C4—22no—
83Dic127⋊C4—22no—
84Dic127⋊C4—41no—
85C22⋊Dic127—22no—
86C22⋊Dic127—22no—
87C22⋊Dic127—22no—
88C22⋊Dic127C2×C414yescomplemented
89C22⋊Dic127C2×C414yescomplemented
90C22⋊Dic127C2×C414yescomplemented
91C22⋊Dic127D414yescomplemented
92(C2×D127)⋊C4—22no—
93(C2×D127)⋊C4—22no—
94(C2×D127)⋊C4—41no—
95C4×Dic127—22no—
96C4×Dic127—22no—
97C4×Dic127—41no—
98C24×C127—22no—
99C24×C127—22no—
100C24×C127—22no—
101C24×C127—22no—
102C24×C127—41no—
103C24×C127—41no—
104C24×C127C2×C414yescomplemented
105C24×C127C2311yescharacteristic
106C24×C127C2311yescharacteristic
107C24×C127C2311yescharacteristic
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—41no—
113C22×C4×C127—41no—
114C22×C4×C127C2311yescharacteristic
115C22×C4×C127C2311yescharacteristic
116C22×C4×C127C2311yescharacteristic
117C22×C4×C127C2311yescharacteristic
118C22⋊C4×C127—22no—
119C22⋊C4×C127—22no—
120C22⋊C4×C127—22no—
121C22⋊C4×C127D414yescomplemented
122C22×D127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—21no—
128C22×D127C22×C412yes—
129C22⋊D127—28no—
130C22⋊D127—28no—
131C22⋊D127—44no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—21no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—21no—
148C2×Dic127—44no—
149C2×Dic127—44no—
150C2×Dic127C22×C412yes—
151C2×Dic127C22×C412yes—
152C2×Dic127C22×C412yes—
153C2×Dic127C2×D412yes—
154C2×Dic127C2×D412yes—
155C4×D127—44no—
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—21no—
160C23×C127—21no—
161C23×C127—21no—
162C23×C127—21no—
163C23×C127—21no—
164C23×C127—28no—
165C23×C127—28no—
166C23×C127—21no—
167C23×C127—21no—
168C23×C127—21no—
169C23×C127—21no—
170C23×C127—21no—
171C23×C127—21no—
172C23×C127—44no—
173C23×C127—44no—
174C23×C127—44no—
175C23×C127—44no—
176C23×C127—44no—
177C23×C127C22×C412yes—
178C23×C127C22×C412yes—
179C23×C127C22×C412yes—
180C23×C127C2×D412yes—
181C23×C127C2×D412yes—
182C23×C127#4112yes—
183C23×C127#4112yes—
184C23×C127C2411yescharacteristic
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×C127—44no—
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×C127—22no—
206C22×C127—22no—
207C22×C127—22no—
208C22×C127—22no—
209C22×C127—22no—
210C22×C127—22no—
211C22×C127—22no—
212C22×C127—22no—
213C22×C127—22no—
214C22×C127—416no—
215C22×C127—41no—
216C22×C127—41no—
217C22×C127—41no—
218C22×C127—41no—
219C22×C127C4×D414yescomplemented
220C22×C127C4×D414yescomplemented
221C22×C127C23×C411yescharacteristic
222C22×C127C22×D411yescharacteristic
223C22×C127C22×D411yescharacteristic
224C22×C127C2×#4111yescharacteristic
225C22×C127C2×#4111yescharacteristic
226C22×C127H(5,𝔽2)11yescharacteristic
227C22×C127H(5,𝔽2)11yescharacteristic
228C2×D127—22no—
229C2×D127—22no—
230C2×D127—22no—
231C2×D127—41no—
232C2×D127—41no—
233C2×D127—41no—
234C2×D127—41no—
235C4×C127—22no—
236C4×C127—22no—
237C4×C127—41no—
238C4×C127—41no—
239Dic127—22no—
240Dic127—22no—
241Dic127—22no—
242Dic127—22no—
243Dic127—22no—
244Dic127—41no—
245Dic127—41no—
246C2×C127—24no—
247C2×C127—24no—
248C2×C127—42no—
249C2×C127C2×C4×D411yescharacteristic
250C2×C127C2×C4×D411yescharacteristic
251C2×C127#51811yescharacteristic
252C2×C127D4211yescharacteristic
253C2×C127#54611yescharacteristic
254C2×C127#54611yescharacteristic
255C2×C127#55111yescharacteristic
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⋊C4D12711yescharacteristic, complemented
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⋊C4—1271no—
274C25C2×D12711yescharacteristic
275C22×D4—1271no—
276C22×D4—1271no—
277C22×D4—1271no—
278C23×C4—1271no—
279C23×C4—1271no—
280C23×C4—1271no—
281C23×C4C2×D12711yescharacteristic
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—24no—
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—2544no—
301C2×C22⋊C4C2×D12711yescharacteristic
302C2×C22⋊C4C2×D12711yescharacteristic
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#95—1271no—
313#95—1271no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—1271no—
321C24C4×D12714yescomplemented
322C24C22×D12711yescharacteristic
323C24C22×D12711yescharacteristic
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—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—1271no—
341C22×C4—1271no—
342C22×C4—1271no—
343C22×C4—1271no—
344C22×C4—1271no—
345C22×C4—1271no—
346C22×C4—1271no—
347C22×C4—5081no—
348C22×C4—5081no—
349C22×C4—5081no—
350C22×C4—2542no—
351C22×C4—2542no—
352C22×C4—2542no—
353C22×C4—2542no—
354C22×C4—2542no—
355C22×C4—2542no—
356C22×C4C22×D12711yescharacteristic
357C22×C4C22×D12711yescharacteristic
358C22×C4C22×D12711yescharacteristic
359C22×C4C22×D12711yescharacteristic
360C4⋊C4—5081no—
361C4⋊C4—2542no—
362C4⋊C4—2542no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—1274no—
367C22⋊C4—1274no—
368C22⋊C4—1274no—
369C22⋊C4—1274no—
370C22⋊C4—5081no—
371C22⋊C4—2542no—
372C22⋊C4—2542no—
373C22⋊C4—2542no—
374C22⋊C4—2542no—
375C22⋊C4—2542no—
376C22⋊C4C22⋊D12714yescomplemented
377C42—5081no—
378C42—2542no—
379C42—2542no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—28no—
395C23—28no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—1272no—
402C23—2541no—
403C23—2541no—
404C23—2541no—
405C23—2541no—
406C23—2541no—
407C23—2541no—
408C23C2×C4×D12712yes—
409C23C2×C4×D12712yes—
410C23C2×C4×D12712yes—
411C23D4×D12712yes—
412C23C2×C22⋊D12712yes—
413C23#1452324712yes—
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—44no—
430C2×C4—44no—
431C2×C4—1272no—
432C2×C4—1272no—
433C2×C4—1272no—
434C2×C4—1272no—
435C2×C4—1272no—
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×C4—2541no—
453C2×C4—2541no—
454C2×C4—2541no—
455C2×C4C2×C22⋊D12712yes—
456C2×C4C2×C22⋊D12712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—416no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—5081no—
478C22—5081no—
479C22—5081no—
480C22—5081no—
481C22—2542no—
482C22—2542no—
483C22—2542no—
484C22(C22⋊D127)⋊C414yescomplemented
485C22C4×C22⋊D12714yescomplemented
486C22C22×C4×D12711yescharacteristic
487C22C2×D4×D12711yescharacteristic
488C22C22×C22⋊D12711yescharacteristic
489C22C2×#1452324711yescharacteristic
490C22C2×#1452324811yescharacteristic
491C22#1686125411yescharacteristic
492C22#1686125411yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—5081no—
498C4—5081no—
499C4—2542no—
500C4—2542no—
501C4—2542no—
502C4—2542no—
503C4—2542no—
504C2—24no—
505C2—24no—
506C2—42no—
507C2—5082no—
508C2C2×(C22⋊D127)⋊C411yescharacteristic
509C2C2×C4×C22⋊D12711yescharacteristic
510C2#2255034911yescharacteristic
511C2D4×C22⋊D12711yescharacteristic
512C2#2255051611yescharacteristic
513C2#2255052011yescharacteristic
514C2#2255055211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC22×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861178C2
hypercenterC22×C22⋊C4D127
hypocenterC127#2358
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D127C2semidirect
3. C2×(C22⋊D127)⋊C4C2semidirect
4. C22×C22⋊Dic127C2semidirect
5. #22549776C2semidirect
6. #22549770C2semidirect
7. #22549764C2semidirect
8. #22549755C2semidirect
9. #22549753C2semidirect
10. #22549752C2semidirect
11. #22549749C2semidirect
12. #22549748C2semidirect
13. C22⋊C4×Dic127C2semidirect
14. C4×C22⋊Dic127C2semidirect
15. C22×C22⋊C4×C127C2semidirect
21. C22×C4×D127C22semidirect
22. #16861178C4semidirect
25. C2×Dic127⋊C4C22semidirect
26. C2×Dic127⋊C4C22semidirect
27. C2×Dic127⋊C4C22semidirect
36. C2×(C2×D127)⋊C4C22semidirect
37. C2×(C2×D127)⋊C4C22semidirect
38. C2×(C2×D127)⋊C4C22semidirect
39. C2×C4×Dic127C22semidirect
40. C2×C4×Dic127C22semidirect
41. C2×C4×Dic127C22semidirect
42. C127⋊#95C22semidirect
43. C127⋊#95C22semidirect
44. C127⋊#95C22semidirect
45. C127⋊#95C22semidirect
46. C127⋊#95C22semidirect
47. C127⋊#95C22semidirect
63. C2×C22⋊D127C2×C4semidirect
64. C2×C22⋊D127C2×C4semidirect
65. C2×C22⋊D127C2×C4semidirect
88. C22⋊Dic127C2×C4semidirect
89. C22⋊Dic127C2×C4semidirect
90. C22⋊Dic127C2×C4semidirect
91. C22⋊Dic127D4semidirect
104. C24×C127C2×C4semidirect
121. C22⋊C4×C127D4semidirect
219. C22×C127C4×D4semidirect
220. C22×C127C4×D4semidirect
258. C127#2358semidirect
263. C22×C22⋊C4D127semidirect
321. C24C4×D127semidirect
376. C22⋊C4C22⋊D127semidirect
484. C22(C22⋊D127)⋊C4semidirect
485. C22C4×C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611782
C2×C4×C22⋊D1272
C2×(C22⋊D127)⋊C42
C22×C22⋊Dic1272
#225497762
#225497702
#225497642
#225497552
#225497532
#225497522
#225497492
#225497482
C22⋊C4×Dic1272
C4×C22⋊Dic1272
C22×C22⋊C4×C1272
#2358127