Finite Group Databasebeta

← #30641870

Subgroups — #30641870

Order 16256.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861179C211yesmaximal, characteristic
2#22549778C211yesmaximal, characteristic, complemented
3#22549777C211yesmaximal, characteristic, complemented
4#22549776C211yesmaximal, characteristic, complemented
5#22549769C211yesmaximal, characteristic, complemented
6#22549768C211yesmaximal, characteristic, complemented
7#22549767C211yesmaximal, characteristic, complemented
8#22549766C211yesmaximal, characteristic, complemented
9#22549765C211yesmaximal, characteristic, complemented
10#22549754C211yesmaximal, characteristic, complemented
11#22549754C211yesmaximal, characteristic, complemented
12#22549752C211yesmaximal, characteristic, complemented
13#22549731C211yesmaximal, characteristic, complemented
14#22549731C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C24×D127C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C22×D508C2211yescharacteristic
19C22×C22⋊D127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic, complemented
21#16861179C418yescomplemented
22C2×C4⋊Dic127C2211yescharacteristic, complemented
23C2×Dic127⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×(C2×D127)⋊C4C2211yescharacteristic
27C2×(C2×D127)⋊C4C2211yescharacteristic
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, complemented
37C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
39C2×C4×Dic127C2211yescharacteristic, complemented
40C2×C4×Dic127C2211yescharacteristic, complemented
41C127⋊#95C2211yescharacteristic, complemented
42C127⋊#95C2211yescharacteristic, complemented
43C127⋊#95C2211yescharacteristic, complemented
44C127⋊#95C2211yescharacteristic, complemented
45C2×C4⋊C4×C127C2211yescharacteristic, complemented
46C2×C22⋊C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic, complemented
50#95×C127C2211yescharacteristic, complemented
51#95×C127C2211yescharacteristic, complemented
52C23×D127—41no—
53C23×D127—41no—
54C23×D127—41no—
55C23×D127—41no—
56C23×D127—41no—
57C23×D127—41no—
58C23×D127C2×C414yescomplemented
59C23×D127C2311yescharacteristic
60C23×D127C2311yescharacteristic
61C23×D127C2311yescharacteristic
62C2×D508—41no—
63C2×D508—41no—
64C2×D508—41no—
65C2×D508—41no—
66C2×D508C2×C414yes—
67C2×D508C2×C414yes—
68C2×C22⋊D127—41no—
69C2×C22⋊D127—41no—
70C2×C22⋊D127C2×C414yes—
71C22×Dic127C2311yescharacteristic
72C22×Dic127C2311yescharacteristic
73C22×Dic127C2311yescharacteristic
74C22×Dic127C2311yescharacteristic
75C22×Dic127C2311yescharacteristic
76C2×C4×D127—41no—
77C2×C4×D127—41no—
78C2×C4×D127—41no—
79C4⋊Dic127—41no—
80Dic127⋊C4—41no—
81C22⋊Dic127—41no—
82C22⋊Dic127—41no—
83(C2×D127)⋊C4—41no—
84(C2×D127)⋊C4—41no—
85(C2×D127)⋊C4—41no—
86(C2×D127)⋊C4—41no—
87(C2×D127)⋊C4—41no—
88(C2×D127)⋊C4—41no—
89(C2×D127)⋊C4—41no—
90(C2×D127)⋊C4—41no—
91(C2×D127)⋊C4—41no—
92(C2×D127)⋊C4—41no—
93(C2×D127)⋊C4—41no—
94(C2×D127)⋊C4C2×C414yescomplemented
95(C2×D127)⋊C4C2×C414yescomplemented
96C4×Dic127—41no—
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—41no—
106C22⋊C4×C127—41no—
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—21no—
119C22×D127—21no—
120C22×D127—21no—
121C22×D127—21no—
122C22×D127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—21no—
128C22×D127—44no—
129C22×D127—44no—
130C22×D127—44no—
131C22×D127—44no—
132C22×D127—44no—
133C22×D127—44no—
134C22×D127—44no—
135C22×D127C22×C412yes—
136C22×D127C22×C412yes—
137C22×D127C22×C412yes—
138D508—44no—
139D508—44no—
140C22⋊D127—44no—
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×Dic127—21no—
152C2×Dic127—21no—
153C2×Dic127—21no—
154C2×Dic127—21no—
155C2×Dic127C22×C412yes—
156C4×D127—44no—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—21no—
160C23×C127—21no—
161C23×C127—21no—
162C23×C127—21no—
163C23×C127C22×C412yes—
164C23×C127C2411yescharacteristic
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×C127—21no—
177C2×C4×C127—21no—
178C2×C4×C127—21no—
179C2×C4×C127—21no—
180C2×C4×C127—21no—
181C2×C4×C127C22×C412yes—
182C2×C4×C127C22×C412yes—
183C22×C127—22no—
184C22×C127—22no—
185C22×C127—22no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127C23×C411yescharacteristic
191C22×C127H(5,𝔽2)11yescharacteristic
192C22×C127H(5,𝔽2)11yescharacteristic
193C22×C127H(5,𝔽2)11yescharacteristic
194C22×C127H(5,𝔽2)11yescharacteristic
195C22×C127H(5,𝔽2)11yescharacteristic
196C22×C127H(5,𝔽2)11yescharacteristic
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—22no—
206C2×D127—416no—
207C2×D127—41no—
208C2×D127—41no—
209C2×D127—41no—
210C2×D127—41no—
211C2×D127—41no—
212C2×D127—41no—
213C2×D127—41no—
214C2×D127—41no—
215C2×D127—41no—
216C2×D127—41no—
217C2×D127—41no—
218C2×D127—41no—
219C4×C127—22no—
220C4×C127—22no—
221C4×C127—41no—
222C4×C127—41no—
223C4×C127—41no—
224C4×C127—41no—
225Dic127—22no—
226Dic127—41no—
227Dic127—41no—
228Dic127—41no—
229Dic127—41no—
230C2×C127—42no—
231C2×C127#51811yescharacteristic
232C2×C127#51811yescharacteristic
233C2×C127#51811yescharacteristic
234C2×C127#56011yescharacteristic
235C2×C127#56011yescharacteristic
236C2×C127#56011yescharacteristic
237C2×C127H(3,𝔽4)11yescharacteristic
238D127—42no—
239D127—42no—
240D127—42no—
241#2375—1271nomaximal, Sylow
242C127#237511yescharacteristic, complemented, Sylow
243C2×#120—1271no—
244#390—1271no—
245#390—1271no—
246#390—1271no—
247#388—1271no—
248#388—1271no—
249#388—1271no—
250#388—1271no—
251#388—1271no—
252#388D12711yescharacteristic, complemented
253#386—1271no—
254#386—1271no—
255#386—1271no—
256#379—1271no—
257#379—1271no—
258C25—1271no—
259C22×D4—1271no—
260C22×D4—1271no—
261C22×D4—1271no—
262C23×C4—1271no—
263#120—1278no—
264C2×C4⋊C4—1271no—
265C2×C4⋊C4—1271no—
266C2×C4⋊C4C2×D12711yescharacteristic, complemented
267C2×C22⋊C4—1271no—
268C2×C22⋊C4—1271no—
269C2×C22⋊C4—1271no—
270C2×C22⋊C4—1271no—
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—1271no—
278C2×C22⋊C4—1271no—
279C2×C22⋊C4—1271no—
280C2×C22⋊C4—1271no—
281C2×C22⋊C4—1271no—
282C2×C22⋊C4C2×D12711yescharacteristic
283C2×C22⋊C4C2×D12711yescharacteristic
284C2×C22⋊C4C2×D12711yescharacteristic
285C2×C42—1271no—
286C2×C42—1271no—
287C2×C42C2×D12711yescharacteristic, complemented
288#95—1271no—
289#95—1271no—
290#95—1271no—
291#95—1271no—
292#95C2×D12711yescharacteristic, complemented
293#95C2×D12711yescharacteristic, complemented
294C24—1274no—
295C24—1271no—
296C24—1271no—
297C24—1271no—
298C24—5081no—
299C24—5081no—
300C24—5081no—
301C24—5081no—
302C24—5081no—
303C24—5081no—
304C24C22×D12711yescharacteristic
305C2×D4—1274no—
306C2×D4—1274no—
307C2×D4—1274no—
308C2×D4—5081no—
309C2×D4—5081no—
310C2×D4—5081no—
311C2×D4—5081no—
312C2×D4—5081no—
313C2×D4—5081no—
314C22×C4—1271no—
315C22×C4—1271no—
316C22×C4—1271no—
317C22×C4—1271no—
318C22×C4—1271no—
319C22×C4—5081no—
320C22×C4—5081no—
321C22×C4—5081no—
322C22×C4C22×D12711yescharacteristic
323C22×C4C22×D12711yescharacteristic
324C22×C4C22×D12711yescharacteristic
325C22×C4C22×D12711yescharacteristic
326C22×C4C22×D12711yescharacteristic
327C22×C4C22×D12711yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—5081no—
330C4⋊C4—5081no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—1274no—
334C22⋊C4—1274no—
335C22⋊C4—5081no—
336C22⋊C4—5081no—
337C22⋊C4—5081no—
338C22⋊C4—5081no—
339C22⋊C4—5081no—
340C22⋊C4—5081no—
341C22⋊C4—5081no—
342C22⋊C4—5081no—
343C22⋊C4—5081no—
344C22⋊C4—5081no—
345C22⋊C4—5081no—
346C22⋊C4—5081no—
347C22⋊C4—5081no—
348C22⋊C4C4×D12714yescomplemented
349C42—41no—
350C42—5081no—
351C42—5081no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—1272no—
359C23—1272no—
360C23—1272no—
361C23—5084no—
362C23—5084no—
363C23—5084no—
364C23—5084no—
365C23—5084no—
366C23—5084no—
367C23—5084no—
368C23—2541no—
369C23—2541no—
370C23—2541no—
371C23—2541no—
372C23—2541no—
373C23—2541no—
374C23—2541no—
375C23—2541no—
376C23—2541no—
377C23—2541no—
378C23—2541no—
379C23—2541no—
380C23—2541no—
381C23—2541no—
382C23—2541no—
383C23—2541no—
384C23—2541no—
385C23—2541no—
386C23C2×C4×D12712yes—
387C23C23×D12711yescharacteristic
388D4—5084no—
389D4—5084no—
390D4—5084no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—1272no—
408C2×C4—5084no—
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×C4—2541no—
423C2×C4C2×C4×D12712yes—
424C2×C4C2×C4×D12712yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—50816no—
433C22—5081no—
434C22—5081no—
435C22—5081no—
436C22—5081no—
437C22—5081no—
438C22—5081no—
439C22—5081no—
440C22—5081no—
441C22—5081no—
442C22—5081no—
443C22—5081no—
444C22—5081no—
445C22—2542no—
446C22—2542no—
447C22—2542no—
448C22—2542no—
449C22—2542no—
450C22—2542no—
451C22—2542no—
452C22—2542no—
453C22—2542no—
454C22C22×C4×D12711yescharacteristic
455C22#1686125411yescharacteristic
456C22#1686125411yescharacteristic
457C22#1686125511yescharacteristic
458C22#1686125511yescharacteristic
459C22#1686125511yescharacteristic
460C22#1686125511yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—5081no—
468C4—5081no—
469C4—5081no—
470C4—5081no—
471C4—2542no—
472C2—42no—
473C2—5082no—
474C2—5082no—
475C2—5082no—
476C2#2255034811yescharacteristic
477C2#2255034811yescharacteristic
478C2#2255035011yescharacteristic
479C2#2255061411yescharacteristic
480C2#2255061611yescharacteristic
481C2#2255061711yescharacteristic
482C2#2255061811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861179C2
hypercenter#388D127
hypocenterC127#2375
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #22549778C2semidirect
3. #22549777C2semidirect
4. #22549776C2semidirect
5. #22549769C2semidirect
6. #22549768C2semidirect
7. #22549767C2semidirect
8. #22549766C2semidirect
9. #22549765C2semidirect
10. #22549754C2semidirect
11. #22549754C2semidirect
12. #22549752C2semidirect
13. #22549731C2semidirect
14. #22549731C2semidirect
15. #388×C127C2semidirect
20. C22×C4×D127C22semidirect
21. #16861179C4semidirect
22. C2×C4⋊Dic127C22semidirect
23. 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. C127⋊#95C22semidirect
42. C127⋊#95C22semidirect
43. C127⋊#95C22semidirect
44. C127⋊#95C22semidirect
45. C2×C4⋊C4×C127C22semidirect
49. C2×C42×C127C22semidirect
50. #95×C127C22semidirect
51. #95×C127C22semidirect
58. C23×D127C2×C4semidirect
94. (C2×D127)⋊C4C2×C4semidirect
95. (C2×D127)⋊C4C2×C4semidirect
108. C22⋊C4×C127C2×C4semidirect
242. C127#2375semidirect
252. #388D127semidirect
266. C2×C4⋊C4C2×D127semidirect
287. C2×C42C2×D127semidirect
292. #95C2×D127semidirect
293. #95C2×D127semidirect
348. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611792
#225497782
#225497772
#225497762
#225497692
#225497682
#225497672
#225497662
#225497652
#225497542
#225497522
#225497312
#388×C1272
#2375127