Finite Group Databasebeta

← #30641578

Subgroups — #30641578

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×#16861167C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D127C211yesmaximal, characteristic
3C2×C22⋊C4×D127C211yesmaximal, characteristic
4#22549770C211yesmaximal, characteristic, complemented
5#22549769C211yesmaximal, characteristic, complemented
6#22549764C211yesmaximal, characteristic, complemented
7#22549750C211yesmaximal, characteristic, complemented
8#22549747C211yesmaximal, characteristic, complemented
9#22549738C211yesmaximal, characteristic, complemented
10#22549735C211yesmaximal, characteristic, complemented
11#22549731C211yesmaximal, characteristic
12C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
13C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
14#22549723C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C24×D127C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21#16861167—24no—
22C2×C4⋊Dic127C2211yescharacteristic, complemented
23C2×Dic127⋊C4C2211yescharacteristic, complemented
24C2×Dic127⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
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
36C22⋊C4×D127—24no—
37C22⋊C4×D127C418yescomplemented
38C2×C4×Dic127C2211yescharacteristic, complemented
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
47C2×C4⋊C4×C127C2211yescharacteristic, complemented
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C22⋊C4×C127C2211yescharacteristic
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×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C2×C4×D127—22no—
70C2×C4×D127—22no—
71C2×C4×D127—22no—
72C2×C4×D127—22no—
73C2×C4×D127—22no—
74C2×C4×D127—22no—
75C2×C4×D127—41no—
76C2×C4×D127—41no—
77C2×C4×D127—41no—
78C2×C4×D127—41no—
79C2×C4×D127C2×C414yescomplemented
80C2×C4×D127C2×C414yescomplemented
81C4⋊Dic127—22no—
82Dic127⋊C4—41no—
83Dic127⋊C4—41no—
84C22⋊Dic127—22no—
85C22⋊Dic127—41no—
86C22⋊Dic127C2×C414yescomplemented
87(C2×D127)⋊C4—22no—
88(C2×D127)⋊C4—22no—
89(C2×D127)⋊C4—22no—
90(C2×D127)⋊C4—22no—
91(C2×D127)⋊C4—22no—
92(C2×D127)⋊C4C2×C414yes—
93(C2×D127)⋊C4C2×C414yes—
94(C2×D127)⋊C4D414yes—
95C4×Dic127—22no—
96C4×Dic127—22no—
97C4×Dic127—41no—
98C24×C127C2311yescharacteristic
99C22×C4×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102C22×C4×C127C2311yescharacteristic
103C22×C4×C127C2311yescharacteristic
104C22×C4×C127C2311yescharacteristic
105C4⋊C4×C127—22no—
106C22⋊C4×C127—22no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127C2×C414yescomplemented
109C42×C127—41no—
110C22×D127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—21no—
115C22×D127—21no—
116C22×D127—21no—
117C22×D127—21no—
118C22×D127—28no—
119C22×D127—28no—
120C22×D127—44no—
121C22×D127—44no—
122C22×D127—44no—
123C22×D127—44no—
124C22×D127—44no—
125C22×D127C22×C412yes—
126C22×D127C22×C412yes—
127C22×D127C2×D412yes—
128C22×D127#4112yes—
129C22×D127#4112yes—
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×Dic127C22×C412yes—
147C2×Dic127C22×C412yes—
148C2×Dic127C2×D412yes—
149C4×D127—28no—
150C4×D127—44no—
151C4×D127—44no—
152C4×D127—44no—
153C23×C127—21no—
154C23×C127—21no—
155C23×C127—21no—
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127C22×C412yes—
160C23×C127C2411yescharacteristic
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—21no—
176C2×C4×C127C22×C412yes—
177C2×C4×C127C22×C412yes—
178C2×C4×C127C2×D412yes—
179C22×C127—22no—
180C22×C127—22no—
181C22×C127—22no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
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—22no—
203C2×D127—22no—
204C2×D127—22no—
205C2×D127—416no—
206C2×D127#11714yescomplemented
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—42no—
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—24no—
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#388D12711yescharacteristic, complemented
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⋊C4—1271no—
247C25—1271no—
248C23×C4—1271no—
249C23×C4—1271no—
250C23×C4—1271no—
251C23×C4—1271no—
252#117—2544no—
253C2×C4⋊C4—1271no—
254C2×C4⋊C4—1271no—
255C2×C4⋊C4—1271no—
256C2×C4⋊C4C2×D12711yescharacteristic, complemented
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—2544no—
270C2×C22⋊C4C2×D12711yescharacteristic
271C2×C22⋊C4C2×D12711yescharacteristic
272C2×C22⋊C4C2×D12711yescharacteristic
273C2×C42—1271no—
274C2×C42—1271no—
275C2×C42—1271no—
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—1271no—
287C24—1274no—
288C24—5081no—
289C24—5081no—
290C24—2542no—
291C24—2542no—
292C24—2542no—
293C24—2542no—
294C24C22×D12711yescharacteristic
295C22×C4—1271no—
296C22×C4—1271no—
297C22×C4—1271no—
298C22×C4—1274no—
299C22×C4—1274no—
300C22×C4—1271no—
301C22×C4—1271no—
302C22×C4—1271no—
303C22×C4—5081no—
304C22×C4—5081no—
305C22×C4—5081no—
306C22×C4—5081no—
307C22×C4—2542no—
308C22×C4—2542no—
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—22no—
320C4⋊C4—5081no—
321C4⋊C4—5081no—
322C4⋊C4—2542no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—1274no—
326C22⋊C4—1274no—
327C22⋊C4—1274no—
328C22⋊C4—1274no—
329C22⋊C4—5081no—
330C22⋊C4—2542no—
331C22⋊C4—2542no—
332C22⋊C4—2542no—
333C22⋊C4—2542no—
334C22⋊C4—2542no—
335C22⋊C4—2542no—
336C22⋊C4C4×D12714yescomplemented
337C42—41no—
338C42—5081no—
339C42—2542no—
340C42—2542no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—1272no—
348C23—1272no—
349C23—1272no—
350C23—1272no—
351C23—1272no—
352C23—1272no—
353C23—5084no—
354C23—5084no—
355C23—5084no—
356C23—5084no—
357C23—5084no—
358C23—2548no—
359C23—2548no—
360C23—2541no—
361C23—2541no—
362C23—2541no—
363C23—2541no—
364C23—2541no—
365C23—2541no—
366C23—2541no—
367C23—2541no—
368C23C2×C4×D12712yes—
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—1272no—
386C2×C4—1272no—
387C2×C4—1272no—
388C2×C4—5084no—
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—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—1274no—
418C22—1274no—
419C22—50816no—
420C22—2542no—
421C22—2542no—
422C22—2542no—
423C22—2542no—
424C22—2542no—
425C22—2542no—
426C22—2542no—
427C22—2542no—
428C22—2542no—
429C22—2542no—
430C22—2542no—
431C22—2542no—
432C22C22×C4×D12711yescharacteristic
433C22C2×D4×D12711yescharacteristic
434C22#41×D12711yescharacteristic
435C22#41×D12711yescharacteristic
436C22#41×D12711yescharacteristic
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—42no—
452C2—2544no—
453C2—2544no—
454C2#117×D12711yescharacteristic
455C2C4×D4×D12711yescharacteristic
456C2#2255034811yescharacteristic
457C2#2255051511yescharacteristic
458C2#2255051911yescharacteristic
459C2#2255055111yescharacteristic
460C2#2255055411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC24×D127C22
hypercenter#388D127
hypocenterC127#2342
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#16861167C2semidirect
4. #22549770C2semidirect
5. #22549769C2semidirect
6. #22549764C2semidirect
7. #22549750C2semidirect
8. #22549747C2semidirect
9. #22549738C2semidirect
10. #22549735C2semidirect
12. C4×(C2×D127)⋊C4C2semidirect
13. C4×(C2×D127)⋊C4C2semidirect
14. #22549723C2semidirect
15. #388×C127C2semidirect
22. C2×C4⋊Dic127C22semidirect
23. C2×Dic127⋊C4C22semidirect
24. C2×Dic127⋊C4C22semidirect
37. C22⋊C4×D127C4semidirect
38. C2×C4×Dic127C22semidirect
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
47. C2×C4⋊C4×C127C22semidirect
51. C2×C42×C127C22semidirect
52. #95×C127C22semidirect
53. #95×C127C22semidirect
79. C2×C4×D127C2×C4semidirect
80. C2×C4×D127C2×C4semidirect
86. C22⋊Dic127C2×C4semidirect
108. C22⋊C4×C127C2×C4semidirect
206. C2×D127#117semidirect
231. C127#2342semidirect
238. #388D127semidirect
256. C2×C4⋊C4C2×D127semidirect
276. C2×C42C2×D127semidirect
283. #95C2×D127semidirect
284. #95C2×D127semidirect
336. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611672
C2×C22⋊C4×D1272
#225497702
#225497692
#225497642
#225497502
#225497472
#225497382
#225497352
#225497312
C4×(C2×D127)⋊C42
#225497232
#388×C1272
#2342127