Finite Group Databasebeta

← #30641707

Subgroups — #30641707

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×#16861177C211yesmaximal, characteristic
2C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
3C22×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D127C211yesmaximal, characteristic, complemented
5#22549776C211yesmaximal, characteristic, complemented
6#22549767C211yesmaximal, characteristic, complemented
7#22549766C211yesmaximal, characteristic, complemented
8#22549756C211yesmaximal, characteristic, complemented
9#22549755C211yesmaximal, characteristic, complemented
10#22549752C211yesmaximal, characteristic, complemented
11#22549749C211yesmaximal, characteristic, complemented
12#22549747C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
14C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
15C2×C4×D4×C127C211yesmaximal, characteristic, complemented
16C24×D127C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C22⋊D127C2211yescharacteristic
19C23×Dic127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21C22×C4×D127C2211yescharacteristic
22#16861177C418yescomplemented
23C4×C22⋊D127—24no—
24C2×Dic127⋊C4C2211yescharacteristic, complemented
25C2×Dic127⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic, complemented
29C2×C22⋊Dic127C2211yescharacteristic, complemented
30C2×(C2×D127)⋊C4—24no—
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
39C22⋊C4×D127—24no—
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
48C22×D4×C127C2211yescharacteristic
49C23×C4×C127C2211yescharacteristic
50C23×C4×C127C2211yescharacteristic, complemented
51C4×D4×C127—24no—
52C2×C4⋊C4×C127C2211yescharacteristic, complemented
53C2×C22⋊C4×C127C2211yescharacteristic
54C2×C22⋊C4×C127C2211yescharacteristic, complemented
55C2×C42×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
65C2×C22⋊D127—22no—
66C2×C22⋊D127—22no—
67C2×C22⋊D127—41no—
68C2×C22⋊D127—41no—
69C2×C22⋊D127C2×C414yescomplemented
70C2×C22⋊D127C2×C414yescomplemented
71C22×Dic127—22no—
72C22×Dic127—22no—
73C22×Dic127—41no—
74C22×Dic127C2311yescharacteristic
75C22×Dic127C2311yescharacteristic
76C22×Dic127C2311yescharacteristic
77C22×Dic127C2311yescharacteristic
78C22×Dic127C2311yescharacteristic
79C22×Dic127C2311yescharacteristic
80C2×C4×D127—22no—
81C2×C4×D127—22no—
82C2×C4×D127—22no—
83C2×C4×D127—22no—
84C2×C4×D127—41no—
85C2×C4×D127—41no—
86Dic127⋊C4—22no—
87Dic127⋊C4—41no—
88C22⋊Dic127—22no—
89C22⋊Dic127—22no—
90C22⋊Dic127—41no—
91C22⋊Dic127C2×C414yescomplemented
92(C2×D127)⋊C4—22no—
93(C2×D127)⋊C4—22no—
94(C2×D127)⋊C4—22no—
95(C2×D127)⋊C4—22no—
96(C2×D127)⋊C4—22no—
97(C2×D127)⋊C4C2×C414yescomplemented
98(C2×D127)⋊C4C2×C414yescomplemented
99(C2×D127)⋊C4D414yescomplemented
100C4×Dic127—22no—
101C4×Dic127—41no—
102C24×C127C2311yescharacteristic
103C24×C127C2311yescharacteristic
104C2×D4×C127—22no—
105C2×D4×C127—22no—
106C2×D4×C127C2×C414yescomplemented
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—41no—
112C22×C4×C127—41no—
113C22×C4×C127C2311yescharacteristic
114C22×C4×C127C2311yescharacteristic
115C22×C4×C127C2311yescharacteristic
116C22×C4×C127C2311yescharacteristic
117C22×C4×C127C2311yescharacteristic
118C4⋊C4×C127—22no—
119C22⋊C4×C127—22no—
120C22⋊C4×C127D414yescomplemented
121C42×C127—22no—
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—28no—
134C22×D127—44no—
135C22×D127—44no—
136C22×D127—44no—
137C22×D127—44no—
138C22×D127—44no—
139C22×D127C22×C412yes—
140C22×D127C22×C412yes—
141C22×D127C2×D412yes—
142C22×D127#4112yes—
143C22⋊D127—28no—
144C22⋊D127—44no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—21no—
148C2×Dic127—21no—
149C2×Dic127—21no—
150C2×Dic127—21no—
151C2×Dic127—21no—
152C2×Dic127—21no—
153C2×Dic127—21no—
154C2×Dic127—21no—
155C2×Dic127—21no—
156C2×Dic127—21no—
157C2×Dic127—21no—
158C2×Dic127—21no—
159C2×Dic127—21no—
160C2×Dic127—44no—
161C2×Dic127C22×C412yes—
162C2×Dic127C22×C412yes—
163C2×Dic127C2×D412yes—
164C4×D127—44no—
165C4×D127—44no—
166C23×C127—21no—
167C23×C127—21no—
168C23×C127—21no—
169C23×C127—21no—
170C23×C127—21no—
171C23×C127—21no—
172C23×C127—21no—
173C23×C127—21no—
174C23×C127—21no—
175C23×C127—21no—
176C23×C127C22×C412yes—
177C23×C127C22×C412yes—
178C23×C127C2×D412yes—
179C23×C127#4112yes—
180C23×C127C2411yescharacteristic
181D4×C127—28no—
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×C127—44no—
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×C127—22no—
206C22×C127—22no—
207C22×C127—22no—
208C22×C127—41no—
209C22×C127—41no—
210C22×C127—41no—
211C22×C127—41no—
212C22×C127C4×D414yescomplemented
213C22×C127C23×C411yescharacteristic
214C22×C127C22×D411yescharacteristic
215C22×C127C22×D411yescharacteristic
216C22×C127C2×#4111yescharacteristic
217C22×C127C2×#4111yescharacteristic
218C22×C127H(5,𝔽2)11yescharacteristic
219C22×C127H(5,𝔽2)11yescharacteristic
220C2×D127—22no—
221C2×D127—22no—
222C2×D127—22no—
223C2×D127—22no—
224C2×D127—22no—
225C2×D127—22no—
226C2×D127—22no—
227C2×D127—22no—
228C2×D127—22no—
229C2×D127—416no—
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—41no—
240Dic127—22no—
241Dic127—22no—
242Dic127—22no—
243Dic127—41no—
244Dic127—41no—
245Dic127—41no—
246C2×C127—24no—
247C2×C127—42no—
248C2×C127C2×C4×D411yescharacteristic
249C2×C127C2×C4×D411yescharacteristic
250C2×C127#51811yescharacteristic
251C2×C127D4211yescharacteristic
252C2×C127#54611yescharacteristic
253C2×C127#54611yescharacteristic
254C2×C127#55111yescharacteristic
255D127—24no—
256D127—42no—
257#2358—1271nomaximal, Sylow
258C127#235811yescharacteristic, complemented, Sylow
259C2×#120—1271no—
260C2×C4×D4—1271no—
261C2×C4×D4D12711yescharacteristic, complemented
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⋊C4—1271no—
274C25—1271no—
275C22×D4—1271no—
276C22×D4—1271no—
277C22×D4C2×D12711yescharacteristic
278C23×C4—1271no—
279C23×C4—1271no—
280C23×C4—1271no—
281C23×C4C2×D12711yescharacteristic
282C23×C4C2×D12711yescharacteristic, complemented
283#120—1278no—
284C4×D4—24no—
285C4×D4—2544no—
286C2×C4⋊C4—1271no—
287C2×C4⋊C4—1271no—
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, complemented
305C2×C42—1271no—
306C2×C42—1271no—
307C2×C42C2×D12711yescharacteristic, complemented
308#95—1271no—
309#95—1271no—
310#95—1271no—
311#95—1271no—
312#95—1271no—
313#95—1271no—
314C24—1271no—
315C24—1274no—
316C24—1271no—
317C24—5081no—
318C24—5081no—
319C24—2542no—
320C24—2542no—
321C24—2542no—
322C24—2542no—
323C24C22×D12711yescharacteristic
324C24C22×D12711yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—1274no—
328C2×D4—1274no—
329C2×D4—5081no—
330C2×D4—5081no—
331C2×D4—2542no—
332C2×D4—2542no—
333C2×D4C4×D12714yescomplemented
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—5081no—
347C22×C4—5081no—
348C22×C4—5081no—
349C22×C4—2542no—
350C22×C4—2542no—
351C22×C4—2542no—
352C22×C4—2542no—
353C22×C4—2542no—
354C22×C4—2542no—
355C22×C4C22×D12711yescharacteristic
356C22×C4C22×D12711yescharacteristic
357C22×C4C22×D12711yescharacteristic
358C22×C4C22×D12711yescharacteristic
359C22×C4C22×D12711yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—5081no—
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⋊C4C22⋊D12714yescomplemented
377C42—22no—
378C42—5081no—
379C42—2542no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—1272no—
391C23—1272no—
392C23—1272no—
393C23—1272no—
394C23—5084no—
395C23—5084no—
396C23—5084no—
397C23—5084no—
398C23—5084no—
399C23—2548no—
400C23—2541no—
401C23—2541no—
402C23—2541no—
403C23—2541no—
404C23—2541no—
405C23—2541no—
406C23—2548no—
407C23—2541no—
408C23—2541no—
409C23—2541no—
410C23—2541no—
411C23C2×C4×D12712yes—
412C23C2×C4×D12712yes—
413C23C2×C22⋊D12712yes—
414C23#1452324812yes—
415C23C23×D12711yescharacteristic
416D4—28no—
417D4—5084no—
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—44no—
431C2×C4—44no—
432C2×C4—1272no—
433C2×C4—1272no—
434C2×C4—1272no—
435C2×C4—5084no—
436C2×C4—5084no—
437C2×C4—5084no—
438C2×C4—2541no—
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×C4C2×C4×D12712yes—
454C2×C4D4×D12712yes—
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—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—1274no—
471C22—50816no—
472C22—5081no—
473C22—5081no—
474C22—5081no—
475C22—5081no—
476C22—2542no—
477C22—2542no—
478C22—2542no—
479C22—2542no—
480C22—2542no—
481C22—2542no—
482C22—2542no—
483C22—2542no—
484C22—2542no—
485C22C4×C22⋊D12714yescomplemented
486C22C22×C4×D12711yescharacteristic
487C22C2×D4×D12711yescharacteristic
488C22C22×C22⋊D12711yescharacteristic
489C22#41×D12711yescharacteristic
490C22C2×#1452324811yescharacteristic
491C22#1686125411yescharacteristic
492C22#1686125511yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—5081no—
499C4—5081no—
500C4—5081no—
501C4—2542no—
502C4—2542no—
503C4—2542no—
504C2—24no—
505C2—42no—
506C2—5082no—
507C2—2544no—
508C2C4×D4×D12711yescharacteristic
509C2C2×C4×C22⋊D12711yescharacteristic
510C2#2255034811yescharacteristic
511C2D4×C22⋊D12711yescharacteristic
512C2#2255051411yescharacteristic
513C2#2255052211yescharacteristic
514C2#2255055511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×C4×D4×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861177C2
hypercenterC2×C4×D4D127
hypocenterC127#2358
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D127C2semidirect
3. C22×(C2×D127)⋊C4C2semidirect
4. C2×C22⋊C4×D127C2semidirect
5. #22549776C2semidirect
6. #22549767C2semidirect
7. #22549766C2semidirect
8. #22549756C2semidirect
9. #22549755C2semidirect
10. #22549752C2semidirect
11. #22549749C2semidirect
12. #22549747C2semidirect
13. C22⋊C4×Dic127C2semidirect
14. C4×(C2×D127)⋊C4C2semidirect
15. C2×C4×D4×C127C2semidirect
22. #16861177C4semidirect
24. C2×Dic127⋊C4C22semidirect
25. C2×Dic127⋊C4C22semidirect
28. C2×C22⋊Dic127C22semidirect
29. C2×C22⋊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
50. C23×C4×C127C22semidirect
52. C2×C4⋊C4×C127C22semidirect
54. C2×C22⋊C4×C127C22semidirect
55. C2×C42×C127C22semidirect
62. C23×D127C2×C4semidirect
69. C2×C22⋊D127C2×C4semidirect
70. C2×C22⋊D127C2×C4semidirect
91. C22⋊Dic127C2×C4semidirect
97. (C2×D127)⋊C4C2×C4semidirect
98. (C2×D127)⋊C4C2×C4semidirect
99. (C2×D127)⋊C4D4semidirect
106. C2×D4×C127C2×C4semidirect
120. C22⋊C4×C127D4semidirect
212. C22×C127C4×D4semidirect
234. C2×D127C4×D4semidirect
258. C127#2358semidirect
261. C2×C4×D4D127semidirect
282. C23×C4C2×D127semidirect
288. C2×C4⋊C4C2×D127semidirect
304. C2×C22⋊C4C2×D127semidirect
307. C2×C42C2×D127semidirect
333. C2×D4C4×D127semidirect
376. C22⋊C4C22⋊D127semidirect
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×#168611772
C2×C4×C22⋊D1272
C22×(C2×D127)⋊C42
C2×C22⋊C4×D1272
#225497762
#225497672
#225497662
#225497562
#225497552
#225497522
#225497492
#225497472
C22⋊C4×Dic1272
C4×(C2×D127)⋊C42
C2×C4×D4×C1272
#2358127