Finite Group Databasebeta

← #388×D127 #30641158

Subgroups — #388×D127 #30641158

Order 16256.

928 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×C4⋊C4×D127C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D127C211yesmaximal, characteristic
3C2×C22⋊C4×D127C211yesmaximal, characteristic
4C2×C22⋊C4×D127C211yesmaximal, characteristic
5C2×C42×D127C211yesmaximal, characteristic, complemented
6#22549770C211yesmaximal, characteristic, complemented
7#22549769C211yesmaximal, characteristic, complemented
8#22549768C211yesmaximal, characteristic, complemented
9#22549767C211yesmaximal, characteristic, complemented
10#22549766C211yesmaximal, characteristic, complemented
11#22549765C211yesmaximal, characteristic, complemented
12#22549764C211yesmaximal, characteristic, complemented
13#95×D127C211yesmaximal, characteristic, complemented
14#95×D127C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C24×D127C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21C22×C4×D127C2211yescharacteristic
22C22×C4×D127C2211yescharacteristic
23C2×C4⋊Dic127C2211yescharacteristic, complemented
24C2×Dic127⋊C4C2211yescharacteristic, complemented
25C2×Dic127⋊C4C2211yescharacteristic, complemented
26C4⋊C4×D127—24no—
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C22⋊Dic127C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×(C2×D127)⋊C4C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C22⋊C4×D127—24no—
37C22⋊C4×D127—24no—
38C22⋊C4×D127C418yescomplemented
39C2×C4×Dic127C2211yescharacteristic, complemented
40C2×C4×Dic127C2211yescharacteristic, complemented
41C2×C4×Dic127C2211yescharacteristic, complemented
42C42×D127—24no—
43C127⋊#95C2211yescharacteristic, complemented
44C127⋊#95C2211yescharacteristic, complemented
45C127⋊#95C2211yescharacteristic, complemented
46C127⋊#95C2211yescharacteristic, complemented
47C127⋊#95C2211yescharacteristic, complemented
48C127⋊#95C2211yescharacteristic, complemented
49C2×C4⋊C4×C127C2211yescharacteristic, complemented
50C2×C22⋊C4×C127C2211yescharacteristic
51C2×C22⋊C4×C127C2211yescharacteristic
52C2×C22⋊C4×C127C2211yescharacteristic
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—22no—
61C23×D127—22no—
62C23×D127C2×C414yes—
63C23×D127C2311yescharacteristic
64C23×D127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C22×Dic127C2311yescharacteristic
70C22×Dic127C2311yescharacteristic
71C2×C4×D127—22no—
72C2×C4×D127—22no—
73C2×C4×D127—22no—
74C2×C4×D127—22no—
75C2×C4×D127—22no—
76C2×C4×D127—22no—
77C2×C4×D127—22no—
78C2×C4×D127—22no—
79C2×C4×D127—22no—
80C2×C4×D127—22no—
81C2×C4×D127—22no—
82C2×C4×D127—22no—
83C2×C4×D127—22no—
84C2×C4×D127—22no—
85C2×C4×D127C2×C414yes—
86C2×C4×D127C2×C414yes—
87C2×C4×D127D414yes—
88C2×C4×D127D414yes—
89C4⋊Dic127—22no—
90Dic127⋊C4—22no—
91Dic127⋊C4—22no—
92C22⋊Dic127—22no—
93C22⋊Dic127—22no—
94C22⋊Dic127C2×C414yescomplemented
95(C2×D127)⋊C4—22no—
96(C2×D127)⋊C4—22no—
97(C2×D127)⋊C4—22no—
98(C2×D127)⋊C4—22no—
99(C2×D127)⋊C4C2×C414yescomplemented
100(C2×D127)⋊C4C2×C414yescomplemented
101C4×Dic127—22no—
102C4×Dic127—22no—
103C4×Dic127—22no—
104C24×C127C2311yescharacteristic
105C22×C4×C127C2311yescharacteristic
106C22×C4×C127C2311yescharacteristic
107C22×C4×C127C2311yescharacteristic
108C22×C4×C127C2311yescharacteristic
109C22×C4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111C4⋊C4×C127—22no—
112C22⋊C4×C127—22no—
113C22⋊C4×C127—22no—
114C22⋊C4×C127C2×C414yescomplemented
115C42×C127—22no—
116C22×D127—21no—
117C22×D127—21no—
118C22×D127—21no—
119C22×D127—21no—
120C22×D127—21no—
121C22×D127—21no—
122C22×D127—28no—
123C22×D127—28no—
124C22×D127—28no—
125C22×D127—44no—
126C22×D127—44no—
127C22×D127—44no—
128C22×D127—44no—
129C22×D127C22×C412yes—
130C22×D127C22×C412yes—
131C22×D127C2×D412yes—
132C22×D127C2×D412yes—
133C22×D127#4112yes—
134C22×D127#4112yes—
135C22×D127#4112yes—
136C22×D127#4112yes—
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—21no—
149C2×Dic127—21no—
150C2×Dic127—21no—
151C2×Dic127C22×C412yes—
152C2×Dic127C22×C412yes—
153C2×Dic127C2×D412yes—
154C2×Dic127C2×D412yes—
155C4×D127—28no—
156C4×D127—28no—
157C4×D127—28no—
158C4×D127—28no—
159C4×D127—44no—
160C4×D127—44no—
161C23×C127—21no—
162C23×C127—21no—
163C23×C127—21no—
164C23×C127—21no—
165C23×C127—21no—
166C23×C127—21no—
167C23×C127C22×C412yes—
168C23×C127C2411yescharacteristic
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—21no—
177C2×C4×C127—21no—
178C2×C4×C127—21no—
179C2×C4×C127—21no—
180C2×C4×C127—21no—
181C2×C4×C127—21no—
182C2×C4×C127—21no—
183C2×C4×C127C22×C412yes—
184C2×C4×C127C22×C412yes—
185C2×C4×C127C2×D412yes—
186C2×C4×C127C2×D412yes—
187C22×C127—22no—
188C22×C127—22no—
189C22×C127—22no—
190C22×C127—41no—
191C22×C127—41no—
192C22×C127—41no—
193C22×C127—41no—
194C22×C127C23×C411yescharacteristic
195C22×C127C22×D411yescharacteristic
196C22×C127C22×D411yescharacteristic
197C22×C127C2×#4111yescharacteristic
198C22×C127C2×#4111yescharacteristic
199C22×C127C2×#4111yescharacteristic
200C22×C127C2×#4111yescharacteristic
201C2×D127—22no—
202C2×D127—22no—
203C2×D127—22no—
204C2×D127—41no—
205C2×D127—41no—
206C2×D127—41no—
207C2×D127—41no—
208C2×D127—416no—
209C2×D127#11714yes—
210C2×D127C4×D414yes—
211C2×D127C4×D414yes—
212C2×D127#12114yes—
213C2×D127#12314yes—
214C2×D127#12414yes—
215C2×D127#12614yes—
216C4×C127—22no—
217C4×C127—22no—
218C4×C127—22no—
219C4×C127—22no—
220C4×C127—41no—
221C4×C127—41no—
222Dic127—22no—
223Dic127—22no—
224Dic127—22no—
225Dic127—22no—
226Dic127—41no—
227Dic127—41no—
228C2×C127—42no—
229C2×C127C2×#11711yescharacteristic
230C2×C127C2×C4×D411yescharacteristic
231C2×C127C2×C4×D411yescharacteristic
232C2×C127C2×#12111yescharacteristic
233C2×C127C2×#12311yescharacteristic
234C2×C127C2×#12411yescharacteristic
235C2×C127C2×#12611yescharacteristic
236D127—42no—
237D127#38818yesdirect product factor
238C2×#388—1271nomaximal, Sylow
239C127C2×#38811yescharacteristic, complemented, Sylow
240C22×C4⋊C4—1271no—
241C22×C22⋊C4—1271no—
242C22×C22⋊C4—1271no—
243C22×C22⋊C4—1271no—
244C22×C42—1271no—
245#388—1271no—
246#388—1271no—
247#388—1271no—
248#388—1271no—
249#388—1271no—
250#388—1271no—
251#388—1271no—
252#388D12711yescharacteristic, direct product factor
253C2×#95—1271no—
254C2×#95—1271no—
255C25—1271no—
256C23×C4—1271no—
257C23×C4—1271no—
258C23×C4—1271no—
259C23×C4—1271no—
260C23×C4—1271no—
261C23×C4—1271no—
262C2×C4⋊C4—1271no—
263C2×C4⋊C4—1271no—
264C2×C4⋊C4—1271no—
265C2×C4⋊C4—2544no—
266C2×C4⋊C4C2×D12711yescharacteristic, complemented
267C2×C22⋊C4—1271no—
268C2×C22⋊C4—1271no—
269C2×C22⋊C4—1271no—
270C2×C22⋊C4—1278no—
271C2×C22⋊C4—1271no—
272C2×C22⋊C4—1271no—
273C2×C22⋊C4—1271no—
274C2×C22⋊C4—1271no—
275C2×C22⋊C4—1271no—
276C2×C22⋊C4—1271no—
277C2×C22⋊C4—2544no—
278C2×C22⋊C4—2544no—
279C2×C22⋊C4C2×D12711yescharacteristic
280C2×C22⋊C4C2×D12711yescharacteristic
281C2×C22⋊C4C2×D12711yescharacteristic
282C2×C42—1271no—
283C2×C42—1271no—
284C2×C42—1271no—
285C2×C42—2544no—
286C2×C42C2×D12711yescharacteristic, complemented
287#95—1271no—
288#95—1271no—
289#95—1271no—
290#95—1271no—
291#95—1271no—
292#95—1271no—
293#95C2×D12711yescharacteristic, complemented
294#95C2×D12711yescharacteristic, complemented
295C24—1271no—
296C24—1274no—
297C24—1271no—
298C24—2542no—
299C24—2542no—
300C24—2542no—
301C24—2542no—
302C24—2542no—
303C24—2542no—
304C24C22×D12711yescharacteristic
305C22×C4—1271no—
306C22×C4—1271no—
307C22×C4—1271no—
308C22×C4—1271no—
309C22×C4—1274no—
310C22×C4—1274no—
311C22×C4—1274no—
312C22×C4—1274no—
313C22×C4—1271no—
314C22×C4—1271no—
315C22×C4—2542no—
316C22×C4—2542no—
317C22×C4—2542no—
318C22×C4—2542no—
319C22×C4—2542no—
320C22×C4—2542no—
321C22×C4—2542no—
322C22×C4—2542no—
323C22×C4—2542no—
324C22×C4—2542no—
325C22×C4—2542no—
326C22×C4—2542no—
327C22×C4—2542no—
328C22×C4—2542no—
329C22×C4C22×D12711yescharacteristic
330C22×C4C22×D12711yescharacteristic
331C22×C4C22×D12711yescharacteristic
332C22×C4C22×D12711yescharacteristic
333C22×C4C22×D12711yescharacteristic
334C22×C4C22×D12711yescharacteristic
335C4⋊C4—22no—
336C4⋊C4—2542no—
337C4⋊C4—2542no—
338C4⋊C4—2542no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—1274no—
342C22⋊C4—1274no—
343C22⋊C4—1274no—
344C22⋊C4—2542no—
345C22⋊C4—2542no—
346C22⋊C4—2542no—
347C22⋊C4—2542no—
348C22⋊C4—2542no—
349C22⋊C4—2542no—
350C22⋊C4C4×D12714yescomplemented
351C42—22no—
352C42—2542no—
353C42—2542no—
354C42—2542no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—1272no—
362C23—1272no—
363C23—1272no—
364C23—1272no—
365C23—1272no—
366C23—1272no—
367C23—1272no—
368C23—1272no—
369C23—5084no—
370C23—5084no—
371C23—5084no—
372C23—5084no—
373C23—2548no—
374C23—2548no—
375C23—2548no—
376C23—2541no—
377C23—2541no—
378C23—2541no—
379C23—2541no—
380C23—2541no—
381C23—2541no—
382C23C2×C4×D12712yes—
383C23C23×D12711yescharacteristic
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—1272no—
399C2×C4—1272no—
400C2×C4—1272no—
401C2×C4—1272no—
402C2×C4—5084no—
403C2×C4—5084no—
404C2×C4—2548no—
405C2×C4—2548no—
406C2×C4—2548no—
407C2×C4—2548no—
408C2×C4—2541no—
409C2×C4—2541no—
410C2×C4—2541no—
411C2×C4—2541no—
412C2×C4—2541no—
413C2×C4—2541no—
414C2×C4—2541no—
415C2×C4—2541no—
416C2×C4—2541no—
417C2×C4—2541no—
418C2×C4—2541no—
419C2×C4—2541no—
420C2×C4—2541no—
421C2×C4—2541no—
422C2×C4C2×C4×D12712yes—
423C2×C4C2×C4×D12712yes—
424C2×C4D4×D12712yes—
425C2×C4D4×D12712yes—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—1274no—
434C22—1274no—
435C22—1274no—
436C22—1274no—
437C22—1274no—
438C22—1274no—
439C22—1274no—
440C22—50816no—
441C22—5081no—
442C22—5081no—
443C22—5081no—
444C22—5081no—
445C22—2542no—
446C22—2542no—
447C22—2542no—
448C22C22×C4×D12711yescharacteristic
449C22C2×D4×D12711yescharacteristic
450C22C2×D4×D12711yescharacteristic
451C22#41×D12711yescharacteristic
452C22#41×D12711yescharacteristic
453C22#41×D12711yescharacteristic
454C22#41×D12711yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—41no—
460C4—41no—
461C4—5081no—
462C4—5081no—
463C4—2542no—
464C4—2542no—
465C4—2542no—
466C4—2542no—
467C2—42no—
468C2—1278no—
469C2—5082no—
470C2#117×D12711yescharacteristic
471C2C4×D4×D12711yescharacteristic
472C2C4×D4×D12711yescharacteristic
473C2#121×D12711yescharacteristic
474C2#123×D12711yescharacteristic
475C2#124×D12711yescharacteristic
476C2#126×D12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC24×D127C22
hypercenter#388D127
hypocenterC127C2×#388
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D127C2semidirect
5. C2×C42×D127C2semidirect
6. #22549770C2semidirect
7. #22549769C2semidirect
8. #22549768C2semidirect
9. #22549767C2semidirect
10. #22549766C2semidirect
11. #22549765C2semidirect
12. #22549764C2semidirect
13. #95×D127C2semidirect
14. #95×D127C2semidirect
15. #388×C127C2semidirect
23. C2×C4⋊Dic127C22semidirect
24. C2×Dic127⋊C4C22semidirect
25. C2×Dic127⋊C4C22semidirect
38. C22⋊C4×D127C4semidirect
39. C2×C4×Dic127C22semidirect
40. C2×C4×Dic127C22semidirect
41. C2×C4×Dic127C22semidirect
43. C127⋊#95C22semidirect
44. C127⋊#95C22semidirect
45. C127⋊#95C22semidirect
46. C127⋊#95C22semidirect
47. C127⋊#95C22semidirect
48. C127⋊#95C22semidirect
49. C2×C4⋊C4×C127C22semidirect
53. C2×C42×C127C22semidirect
54. #95×C127C22semidirect
55. #95×C127C22semidirect
94. C22⋊Dic127C2×C4semidirect
99. (C2×D127)⋊C4C2×C4semidirect
100. (C2×D127)⋊C4C2×C4semidirect
114. C22⋊C4×C127C2×C4semidirect
237. D127#388direct (N × H)
239. C127C2×#388semidirect
252. #388D127direct (N × H)
266. C2×C4⋊C4C2×D127semidirect
286. C2×C42C2×D127semidirect
293. #95C2×D127semidirect
294. #95C2×D127semidirect
350. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1272
C2×C22⋊C4×D1272
C2×C42×D1272
#225497702
#225497692
#225497682
#225497672
#225497662
#225497652
#225497642
#95×D1272
#388×C1272
C2×#388127