Finite Group Databasebeta

← #30641573

Subgroups — #30641573

Order 16256.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861166C211yesmaximal, characteristic, complemented
2C22×(C2×D127)⋊C4C211yesmaximal, characteristic
3C2×C22⋊C4×D127C211yesmaximal, characteristic
4#22549767C211yesmaximal, characteristic, complemented
5#22549767C211yesmaximal, characteristic, complemented
6#22549765C211yesmaximal, characteristic, complemented
7#22549765C211yesmaximal, characteristic, complemented
8#22549749C211yesmaximal, characteristic, complemented
9#22549748C211yesmaximal, characteristic, complemented
10#22549738C211yesmaximal, characteristic, complemented
11#22549735C211yesmaximal, characteristic, complemented
12#22549731C211yesmaximal, characteristic
13C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
14C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
16C24×D127C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20#16861166—24no—
21C2×Dic127⋊C4C2211yescharacteristic, complemented
22C2×Dic127⋊C4C2211yescharacteristic, complemented
23C2×Dic127⋊C4C2211yescharacteristic, complemented
24C2×Dic127⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×(C2×D127)⋊C4—24no—
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
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
38C22⋊C4×D127C418yescomplemented
39C2×C4×Dic127C2211yescharacteristic, complemented
40C2×C4×Dic127C2211yescharacteristic, complemented
41C127⋊#95C2211yescharacteristic, complemented
42C127⋊#95C2211yescharacteristic, complemented
43C127⋊#95C2211yescharacteristic, complemented
44C127⋊#95C2211yescharacteristic, complemented
45C127⋊#95C2211yescharacteristic, complemented
46C127⋊#95C2211yescharacteristic, complemented
47C23×C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C42×C127C2211yescharacteristic, complemented
51C2×C42×C127C2211yescharacteristic, complemented
52#95×C127C2211yescharacteristic, complemented
53#95×C127C2211yescharacteristic, complemented
54C23×D127—22no—
55C23×D127—22no—
56C23×D127—22no—
57C23×D127—22no—
58C23×D127—41no—
59C23×D127—41no—
60C23×D127C2×C414yes—
61C23×D127C2311yescharacteristic
62C23×D127C2311yescharacteristic
63C22×Dic127—22no—
64C22×Dic127—22no—
65C22×Dic127—41no—
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C22×Dic127C2311yescharacteristic
70C22×Dic127C2311yescharacteristic
71C22×Dic127C2311yescharacteristic
72C2×C4×D127—22no—
73C2×C4×D127—22no—
74C2×C4×D127—41no—
75C2×C4×D127—41no—
76C2×C4×D127C2×C414yescomplemented
77C2×C4×D127C2×C414yescomplemented
78Dic127⋊C4—22no—
79Dic127⋊C4—22no—
80Dic127⋊C4—41no—
81Dic127⋊C4—41no—
82C22⋊Dic127—22no—
83C22⋊Dic127C2×C414yes—
84(C2×D127)⋊C4—22no—
85(C2×D127)⋊C4—22no—
86(C2×D127)⋊C4—22no—
87(C2×D127)⋊C4—22no—
88(C2×D127)⋊C4—22no—
89(C2×D127)⋊C4—41no—
90(C2×D127)⋊C4—41no—
91(C2×D127)⋊C4C2×C414yescomplemented
92(C2×D127)⋊C4C2×C414yescomplemented
93(C2×D127)⋊C4D414yes—
94C4×Dic127—22no—
95C4×Dic127—41no—
96C24×C127C2311yescharacteristic
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—41no—
100C22×C4×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102C22×C4×C127C2311yescharacteristic
103C22×C4×C127C2311yescharacteristic
104C22×C4×C127C2311yescharacteristic
105C22×C4×C127C2311yescharacteristic
106C22⋊C4×C127—22no—
107C22⋊C4×C127C2×C414yes—
108C42×C127—22no—
109C42×C127—41no—
110C22×D127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—28no—
115C22×D127—21no—
116C22×D127—21no—
117C22×D127—21no—
118C22×D127—21no—
119C22×D127—21no—
120C22×D127—21no—
121C22×D127—28no—
122C22×D127—44no—
123C22×D127—44no—
124C22×D127—44no—
125C22×D127—44no—
126C22×D127—44no—
127C22×D127C22×C412yes—
128C22×D127C22×C412yes—
129C22×D127C2×D412yes—
130C22×D127#4112yes—
131C2×Dic127—21no—
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—44no—
147C2×Dic127C22×C412yes—
148C2×Dic127C22×C412yes—
149C2×Dic127C2×D412yes—
150C4×D127—28no—
151C4×D127—44no—
152C23×C127—21no—
153C23×C127—21no—
154C23×C127—21no—
155C23×C127—21no—
156C23×C127C22×C412yes—
157C23×C127#4112yes—
158C23×C127#4112yes—
159C23×C127C2411yescharacteristic
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
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—44no—
176C2×C4×C127C22×C412yes—
177C2×C4×C127C22×C412yes—
178C2×C4×C127C2×D412yes—
179C22×C127—22no—
180C22×C127—22no—
181C22×C127—22no—
182C22×C127—22no—
183C22×C127—22no—
184C22×C127—22no—
185C22×C127#11714yescomplemented
186C22×C127C23×C411yescharacteristic
187C22×C127C22×D411yescharacteristic
188C22×C127C2×#4111yescharacteristic
189C22×C127C2×#4111yescharacteristic
190C22×C127C2×#4111yescharacteristic
191C22×C127H(5,𝔽2)11yescharacteristic
192C22×C127H(5,𝔽2)11yescharacteristic
193C2×D127—22no—
194C2×D127—22no—
195C2×D127—22no—
196C2×D127—22no—
197C2×D127—22no—
198C2×D127—22no—
199C2×D127—22no—
200C2×D127—22no—
201C2×D127—22no—
202C2×D127—416no—
203C2×D127—41no—
204C2×D127—41no—
205C2×D127—41no—
206C2×D127—41no—
207C2×D127C4×D414yes—
208C4×C127—22no—
209C4×C127—22no—
210C4×C127—22no—
211C4×C127—41no—
212C4×C127—41no—
213C4×C127—41no—
214Dic127—22no—
215Dic127—22no—
216Dic127—22no—
217Dic127—41no—
218Dic127—41no—
219Dic127—41no—
220C2×C127—24no—
221C2×C127C2×#11711yescharacteristic
222C2×C127C2×C4×D411yescharacteristic
223C2×C127#51811yescharacteristic
224C2×C127#54611yescharacteristic
225C2×C127#54611yescharacteristic
226C2×C127#55111yescharacteristic
227C2×C127#55111yescharacteristic
228D127—24no—
229D127—42no—
230#2342—1271nomaximal, Sylow
231C127#234211yescharacteristic, complemented, Sylow
232C2×#117—1271no—
233C22×C22⋊C4—1271no—
234C22×C22⋊C4—1271no—
235#388—1271no—
236#388—1271no—
237#388—1271no—
238#388—1271no—
239#385—1271no—
240#385—1271no—
241#381—1271no—
242#380—1271no—
243#379—1271no—
244C4×C22⋊C4—1271no—
245C4×C22⋊C4—1271no—
246C4×C22⋊C4D12711yescharacteristic, complemented
247C25—1271no—
248C23×C4—1271no—
249C23×C4—1271no—
250C23×C4—1271no—
251C23×C4C2×D12711yescharacteristic
252#117—2544no—
253C2×C4⋊C4—1271no—
254C2×C4⋊C4—1271no—
255C2×C4⋊C4—1271no—
256C2×C4⋊C4—1271no—
257C2×C22⋊C4—1271no—
258C2×C22⋊C4—1271no—
259C2×C22⋊C4—1271no—
260C2×C22⋊C4—1271no—
261C2×C22⋊C4—1278no—
262C2×C22⋊C4—1271no—
263C2×C22⋊C4—1271no—
264C2×C22⋊C4—1271no—
265C2×C22⋊C4—1271no—
266C2×C22⋊C4—1271no—
267C2×C22⋊C4—1271no—
268C2×C22⋊C4—1271no—
269C2×C22⋊C4—1271no—
270C2×C22⋊C4—2544no—
271C2×C22⋊C4C2×D12711yescharacteristic
272C2×C22⋊C4C2×D12711yescharacteristic
273C2×C42—1271no—
274C2×C42—1271no—
275C2×C42C2×D12711yescharacteristic, complemented
276C2×C42C2×D12711yescharacteristic, complemented
277#95—1271no—
278#95—1271no—
279#95—1271no—
280#95—1271no—
281#95—1271no—
282#95—1271no—
283#95C2×D12711yescharacteristic, complemented
284#95C2×D12711yescharacteristic, complemented
285C24—1271no—
286C24—1274no—
287C24—1271no—
288C24—5081no—
289C24—5081no—
290C24—2542no—
291C24—2542no—
292C24—2542no—
293C24—2542no—
294C24C22×D12711yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—1271no—
299C22×C4—1271no—
300C22×C4—1271no—
301C22×C4—1274no—
302C22×C4—1274no—
303C22×C4—1271no—
304C22×C4—1271no—
305C22×C4—1271no—
306C22×C4—5081no—
307C22×C4—5081no—
308C22×C4—5081no—
309C22×C4—2542no—
310C22×C4—2542no—
311C22×C4—2542no—
312C22×C4—2542no—
313C22×C4C22×D12711yescharacteristic
314C22×C4C22×D12711yescharacteristic
315C22×C4C22×D12711yescharacteristic
316C22×C4C22×D12711yescharacteristic
317C22×C4C22×D12711yescharacteristic
318C22×C4C22×D12711yescharacteristic
319C4⋊C4—5081no—
320C4⋊C4—5081no—
321C4⋊C4—2542no—
322C4⋊C4—2542no—
323C22⋊C4—22no—
324C22⋊C4—1274no—
325C22⋊C4—1274no—
326C22⋊C4—1274no—
327C22⋊C4—1274no—
328C22⋊C4—5081no—
329C22⋊C4—5081no—
330C22⋊C4—2542no—
331C22⋊C4—2542no—
332C22⋊C4—2542no—
333C22⋊C4—2542no—
334C22⋊C4—2542no—
335C22⋊C4—2542no—
336C22⋊C4C4×D12714yes—
337C42—22no—
338C42—41no—
339C42—5081no—
340C42—2542no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—1272no—
346C23—1272no—
347C23—1272no—
348C23—1272no—
349C23—5084no—
350C23—5084no—
351C23—5084no—
352C23—5084no—
353C23—5084no—
354C23—2548no—
355C23—2541no—
356C23—2541no—
357C23—2541no—
358C23—2541no—
359C23—2541no—
360C23—2541no—
361C23—2548no—
362C23—2541no—
363C23—2541no—
364C23—2541no—
365C23—2541no—
366C23C2×C4×D12712yes—
367C23#1452324812yes—
368C23#1452324812yes—
369C23C23×D12711yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—44no—
386C2×C4—1272no—
387C2×C4—1272no—
388C2×C4—1272no—
389C2×C4—5084no—
390C2×C4—5084no—
391C2×C4—2548no—
392C2×C4—2541no—
393C2×C4—2541no—
394C2×C4—2541no—
395C2×C4—2541no—
396C2×C4—2541no—
397C2×C4—2541no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4—2541no—
401C2×C4—2541no—
402C2×C4—2541no—
403C2×C4—2541no—
404C2×C4—2541no—
405C2×C4—2541no—
406C2×C4—2541no—
407C2×C4C2×C4×D12712yes—
408C2×C4C2×C4×D12712yes—
409C2×C4D4×D12712yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—1274no—
417C22—50816no—
418C22—5081no—
419C22—5081no—
420C22—5081no—
421C22—5081no—
422C22—2542no—
423C22—2542no—
424C22—2542no—
425C22—2542no—
426C22—2542no—
427C22—2542no—
428C22—2542no—
429C22—2542no—
430C22—2542no—
431C22#1686116614yescomplemented
432C22C22×C4×D12711yescharacteristic
433C22C2×D4×D12711yescharacteristic
434C22#41×D12711yescharacteristic
435C22C2×#1452324811yescharacteristic
436C22C2×#1452324811yescharacteristic
437C22#1686125411yescharacteristic
438C22#1686125511yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—5081no—
446C4—5081no—
447C4—5081no—
448C4—2542no—
449C4—2542no—
450C4—2542no—
451C2—24no—
452C2—5082no—
453C2—2544no—
454C2C2×#1686116611yescharacteristic
455C2C4×D4×D12711yescharacteristic
456C2#2255034811yescharacteristic
457C2#2255051611yescharacteristic
458C2#2255052211yescharacteristic
459C2#2255055011yescharacteristic
460C2#2255055511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC4×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC24×D127C22
hypercenterC4×C22⋊C4D127
hypocenterC127#2342
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16861166C2semidirect
4. #22549767C2semidirect
5. #22549767C2semidirect
6. #22549765C2semidirect
7. #22549765C2semidirect
8. #22549749C2semidirect
9. #22549748C2semidirect
10. #22549738C2semidirect
11. #22549735C2semidirect
13. C22⋊C4×Dic127C2semidirect
14. C4×(C2×D127)⋊C4C2semidirect
15. C4×C22⋊C4×C127C2semidirect
21. C2×Dic127⋊C4C22semidirect
22. C2×Dic127⋊C4C22semidirect
23. C2×Dic127⋊C4C22semidirect
24. C2×Dic127⋊C4C22semidirect
38. C22⋊C4×D127C4semidirect
39. C2×C4×Dic127C22semidirect
40. C2×C4×Dic127C22semidirect
41. C127⋊#95C22semidirect
42. C127⋊#95C22semidirect
43. C127⋊#95C22semidirect
44. C127⋊#95C22semidirect
45. C127⋊#95C22semidirect
46. C127⋊#95C22semidirect
50. C2×C42×C127C22semidirect
51. C2×C42×C127C22semidirect
52. #95×C127C22semidirect
53. #95×C127C22semidirect
76. C2×C4×D127C2×C4semidirect
77. C2×C4×D127C2×C4semidirect
91. (C2×D127)⋊C4C2×C4semidirect
92. (C2×D127)⋊C4C2×C4semidirect
185. C22×C127#117semidirect
231. C127#2342semidirect
246. C4×C22⋊C4D127semidirect
275. C2×C42C2×D127semidirect
276. C2×C42C2×D127semidirect
283. #95C2×D127semidirect
284. #95C2×D127semidirect
431. C22#16861166semidirect

Maximal subgroups

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

subgroupindex
C2×#168611662
C22×(C2×D127)⋊C42
C2×C22⋊C4×D1272
#225497672
#225497652
#225497492
#225497482
#225497382
#225497352
#225497312
C22⋊C4×Dic1272
C4×(C2×D127)⋊C42
C4×C22⋊C4×C1272
#2342127