Finite Group Databasebeta

← #30641392

Subgroups — #30641392

Order 16256.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861179C211yesmaximal, characteristic
2C2×C4×D508C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549777C211yesmaximal, characteristic, complemented
5#22549776C211yesmaximal, characteristic, complemented
6#22549769C211yesmaximal, characteristic, complemented
7#22549768C211yesmaximal, characteristic, complemented
8#22549767C211yesmaximal, characteristic, complemented
9#22549754C211yesmaximal, characteristic, complemented
10#22549754C211yesmaximal, characteristic, complemented
11#22549752C211yesmaximal, characteristic, complemented
12#22549747C211yesmaximal, characteristic, complemented
13#22549731C211yesmaximal, characteristic, complemented
14#22549723C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C24×D127C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C22×D508C2211yescharacteristic
19C22×C22⋊D127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21C22×C4×D127C2211yescharacteristic
22C22×C4×D127C2211yescharacteristic, complemented
23#16861179C418yescomplemented
24C4×D508—24no—
25C2×C4⋊Dic127C2211yescharacteristic, complemented
26C2×Dic127⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
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, complemented
38C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
40C22⋊C4×D127—24no—
41C2×C4×Dic127C2211yescharacteristic, complemented
42C2×C4×Dic127C2211yescharacteristic, 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—41no—
57C23×D127—41no—
58C23×D127—41no—
59C23×D127—41no—
60C23×D127C2×C414yescomplemented
61C23×D127C2311yescharacteristic
62C23×D127C2311yescharacteristic
63C23×D127C2311yescharacteristic
64C2×D508—22no—
65C2×D508—22no—
66C2×D508—41no—
67C2×D508—41no—
68C2×D508C2×C414yes—
69C2×D508C2×C414yescomplemented
70C2×C22⋊D127—41no—
71C2×C22⋊D127—41no—
72C2×C22⋊D127C2×C414yescomplemented
73C22×Dic127C2311yescharacteristic
74C22×Dic127C2311yescharacteristic
75C22×Dic127C2311yescharacteristic
76C22×Dic127C2311yescharacteristic
77C22×Dic127C2311yescharacteristic
78C2×C4×D127—22no—
79C2×C4×D127—22no—
80C2×C4×D127—41no—
81C2×C4×D127—41no—
82C2×C4×D127—41no—
83C2×C4×D127—41no—
84C2×C4×D127—41no—
85C2×C4×D127—41no—
86C2×C4×D127D414yescomplemented
87C4⋊Dic127—22no—
88Dic127⋊C4—41no—
89C22⋊Dic127—22no—
90C22⋊Dic127—41no—
91(C2×D127)⋊C4—22no—
92(C2×D127)⋊C4—22no—
93(C2×D127)⋊C4—22no—
94(C2×D127)⋊C4—22no—
95(C2×D127)⋊C4—22no—
96(C2×D127)⋊C4—41no—
97(C2×D127)⋊C4—41no—
98(C2×D127)⋊C4—41no—
99(C2×D127)⋊C4—41no—
100(C2×D127)⋊C4C2×C414yescomplemented
101(C2×D127)⋊C4C2×C414yescomplemented
102C4×Dic127—41no—
103C4×Dic127—41no—
104C24×C127C2311yescharacteristic
105C22×C4×C127C2311yescharacteristic
106C22×C4×C127C2311yescharacteristic
107C22×C4×C127C2311yescharacteristic
108C22×C4×C127C2311yescharacteristic
109C22×C4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111C4⋊C4×C127—41no—
112C22⋊C4×C127—22no—
113C22⋊C4×C127—41no—
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—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—21no—
128C22×D127—21no—
129C22×D127—21no—
130C22×D127—21no—
131C22×D127—21no—
132C22×D127—28no—
133C22×D127—44no—
134C22×D127—44no—
135C22×D127—44no—
136C22×D127—44no—
137C22×D127—44no—
138C22×D127—44no—
139C22×D127C22×C412yes—
140C22×D127C22×C412yes—
141C22×D127C22×C412yes—
142C22×D127C2×D412yes—
143C22×D127#4112yes—
144D508—28no—
145D508—44no—
146C22⋊D127—44no—
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×Dic127C22×C412yes—
161C2×Dic127C2×D412yes—
162C4×D127—44no—
163C4×D127—44no—
164C4×D127—44no—
165C23×C127—21no—
166C23×C127—21no—
167C23×C127—21no—
168C23×C127—21no—
169C23×C127—21no—
170C23×C127—21no—
171C23×C127C22×C412yes—
172C23×C127C2411yescharacteristic
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×C127—21no—
184C2×C4×C127—21no—
185C2×C4×C127—21no—
186C2×C4×C127—21no—
187C2×C4×C127—21no—
188C2×C4×C127C22×C412yes—
189C2×C4×C127C22×C412yes—
190C2×C4×C127C2×D412yes—
191C22×C127—22no—
192C22×C127—22no—
193C22×C127—22no—
194C22×C127—41no—
195C22×C127—41no—
196C22×C127—41no—
197C22×C127—41no—
198C22×C127C23×C411yescharacteristic
199C22×C127C22×D411yescharacteristic
200C22×C127C2×#4111yescharacteristic
201C22×C127H(5,𝔽2)11yescharacteristic
202C22×C127H(5,𝔽2)11yescharacteristic
203C22×C127H(5,𝔽2)11yescharacteristic
204C22×C127H(5,𝔽2)11yescharacteristic
205C2×D127—22no—
206C2×D127—22no—
207C2×D127—22no—
208C2×D127—22no—
209C2×D127—22no—
210C2×D127—22no—
211C2×D127—22no—
212C2×D127—22no—
213C2×D127—22no—
214C2×D127—22no—
215C2×D127—22no—
216C2×D127—22no—
217C2×D127—416no—
218C2×D127—41no—
219C2×D127—41no—
220C2×D127—41no—
221C2×D127—41no—
222C2×D127—41no—
223C2×D127—41no—
224C2×D127—41no—
225C2×D127—41no—
226C2×D127C4×D414yes—
227C4×C127—22no—
228C4×C127—22no—
229C4×C127—22no—
230C4×C127—41no—
231C4×C127—41no—
232C4×C127—41no—
233Dic127—22no—
234Dic127—22no—
235Dic127—41no—
236Dic127—41no—
237Dic127—41no—
238C2×C127—42no—
239C2×C127C2×C4×D411yescharacteristic
240C2×C127#51811yescharacteristic
241C2×C127#51811yescharacteristic
242C2×C127#53411yescharacteristic
243C2×C127#53511yescharacteristic
244C2×C127#53811yescharacteristic
245C2×C127#53811yescharacteristic
246D127—24no—
247D127—42no—
248D127—42no—
249#2321—1271nomaximal, Sylow
250C127#232111yescharacteristic, complemented, Sylow
251C2×#120—1271no—
252C2×C4×D4—1271no—
253C22×C22⋊C4—1271no—
254#390—1271no—
255#390—1271no—
256#388—1271no—
257#388—1271no—
258#388—1271no—
259#388D12711yescharacteristic, complemented
260#386—1271no—
261#386—1271no—
262#386—1271no—
263#385—1271no—
264#379—1271no—
265C4×C22⋊C4—1271no—
266C25—1271no—
267C22×D4—1271no—
268C22×D4—1271no—
269C22×D4—1271no—
270C23×C4—1271no—
271C23×C4—1271no—
272C23×C4—1271no—
273#120—1278no—
274C4×D4—2544no—
275C2×C4⋊C4—1271no—
276C2×C4⋊C4—1271no—
277C2×C4⋊C4C2×D12711yescharacteristic, complemented
278C2×C22⋊C4—1271no—
279C2×C22⋊C4—1271no—
280C2×C22⋊C4—1271no—
281C2×C22⋊C4—1271no—
282C2×C22⋊C4—1271no—
283C2×C22⋊C4—1271no—
284C2×C22⋊C4—1271no—
285C2×C22⋊C4—1271no—
286C2×C22⋊C4—1271no—
287C2×C22⋊C4—1271no—
288C2×C22⋊C4—1271no—
289C2×C22⋊C4—1271no—
290C2×C22⋊C4—1271no—
291C2×C22⋊C4—2544no—
292C2×C22⋊C4C2×D12711yescharacteristic
293C2×C22⋊C4C2×D12711yescharacteristic
294C2×C22⋊C4C2×D12711yescharacteristic
295C2×C42—1271no—
296C2×C42—1271no—
297C2×C42C2×D12711yescharacteristic, complemented
298#95—1271no—
299#95—1271no—
300#95—1271no—
301#95—1271no—
302#95C2×D12711yescharacteristic, complemented
303#95C2×D12711yescharacteristic, complemented
304C24—1271no—
305C24—1274no—
306C24—1271no—
307C24—1271no—
308C24—5081no—
309C24—5081no—
310C24—5081no—
311C24—5081no—
312C24—2542no—
313C24—2542no—
314C24C22×D12711yescharacteristic
315C2×D4—1274no—
316C2×D4—1274no—
317C2×D4—1274no—
318C2×D4—5081no—
319C2×D4—5081no—
320C2×D4—5081no—
321C2×D4—5081no—
322C2×D4—2542no—
323C2×D4—2542no—
324C22×C4—1271no—
325C22×C4—1271no—
326C22×C4—1274no—
327C22×C4—1271no—
328C22×C4—1271no—
329C22×C4—1271no—
330C22×C4—5081no—
331C22×C4—5081no—
332C22×C4—5081no—
333C22×C4—5081no—
334C22×C4—5081no—
335C22×C4—5081no—
336C22×C4—2542no—
337C22×C4—2542no—
338C22×C4C22×D12711yescharacteristic
339C22×C4C22×D12711yescharacteristic
340C22×C4C22×D12711yescharacteristic
341C22×C4C22×D12711yescharacteristic
342C22×C4C22×D12711yescharacteristic
343C22×C4C22×D12711yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—5081no—
346C4⋊C4—2542no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—1274no—
350C22⋊C4—1274no—
351C22⋊C4—5081no—
352C22⋊C4—5081no—
353C22⋊C4—5081no—
354C22⋊C4—5081no—
355C22⋊C4—5081no—
356C22⋊C4—2542no—
357C22⋊C4—2542no—
358C22⋊C4—2542no—
359C22⋊C4—2542no—
360C22⋊C4—2542no—
361C22⋊C4—2542no—
362C22⋊C4C4×D12714yescomplemented
363C42—22no—
364C42—5081no—
365C42—5081no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—1272no—
373C23—1272no—
374C23—1272no—
375C23—1272no—
376C23—1272no—
377C23—5084no—
378C23—5084no—
379C23—5084no—
380C23—5084no—
381C23—5084no—
382C23—5084no—
383C23—2548no—
384C23—2541no—
385C23—2541no—
386C23—2541no—
387C23—2541no—
388C23—2541no—
389C23—2541no—
390C23—2541no—
391C23—2541no—
392C23—2541no—
393C23—2541no—
394C23—2541no—
395C23—2541no—
396C23—2541no—
397C23—2541no—
398C23—2541no—
399C23—2541no—
400C23C2×C4×D12712yes—
401C23C23×D12711yescharacteristic
402D4—5084no—
403D4—5084no—
404D4—2548no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—1272no—
421C2×C4—1272no—
422C2×C4—5084no—
423C2×C4—5084no—
424C2×C4—5084no—
425C2×C4—2541no—
426C2×C4—2541no—
427C2×C4—2541no—
428C2×C4—2541no—
429C2×C4—2541no—
430C2×C4—2541no—
431C2×C4—2541no—
432C2×C4—2541no—
433C2×C4—2541no—
434C2×C4—2541no—
435C2×C4—2541no—
436C2×C4—2541no—
437C2×C4—2541no—
438C2×C4C2×C4×D12712yes—
439C2×C4C2×C4×D12712yes—
440C2×C4D4×D12712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—1274no—
449C22—50816no—
450C22—5081no—
451C22—5081no—
452C22—5081no—
453C22—5081no—
454C22—5081no—
455C22—5081no—
456C22—5081no—
457C22—5081no—
458C22—2542no—
459C22—2542no—
460C22—2542no—
461C22—2542no—
462C22—2542no—
463C22—2542no—
464C22—2542no—
465C22—2542no—
466C22—2542no—
467C22—2542no—
468C22—2542no—
469C22—2542no—
470C22C22×C4×D12711yescharacteristic
471C22C2×D4×D12711yescharacteristic
472C22#41×D12711yescharacteristic
473C22#1686125411yescharacteristic
474C22#1686125511yescharacteristic
475C22#1686125511yescharacteristic
476C22#1686125511yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—5081no—
484C4—5081no—
485C4—5081no—
486C4—2542no—
487C4—2542no—
488C2—42no—
489C2—5082no—
490C2—5082no—
491C2—2544no—
492C2C4×D4×D12711yescharacteristic
493C2#2255034811yescharacteristic
494C2#2255035011yescharacteristic
495C2#2255043711yescharacteristic
496C2#2255044211yescharacteristic
497C2#2255046011yescharacteristic
498C2#2255046511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861179C2
hypercenter#388D127
hypocenterC127#2321
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D508C2semidirect
3. C2×C22⋊C4×D127C2semidirect
4. #22549777C2semidirect
5. #22549776C2semidirect
6. #22549769C2semidirect
7. #22549768C2semidirect
8. #22549767C2semidirect
9. #22549754C2semidirect
10. #22549754C2semidirect
11. #22549752C2semidirect
12. #22549747C2semidirect
13. #22549731C2semidirect
14. #22549723C2semidirect
15. #388×C127C2semidirect
22. C22×C4×D127C22semidirect
23. #16861179C4semidirect
25. C2×C4⋊Dic127C22semidirect
26. C2×Dic127⋊C4C22semidirect
37. C2×(C2×D127)⋊C4C22semidirect
38. C2×(C2×D127)⋊C4C22semidirect
39. C2×(C2×D127)⋊C4C22semidirect
41. C2×C4×Dic127C22semidirect
42. C2×C4×Dic127C22semidirect
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
60. C23×D127C2×C4semidirect
69. C2×D508C2×C4semidirect
72. C2×C22⋊D127C2×C4semidirect
86. C2×C4×D127D4semidirect
100. (C2×D127)⋊C4C2×C4semidirect
101. (C2×D127)⋊C4C2×C4semidirect
114. C22⋊C4×C127C2×C4semidirect
250. C127#2321semidirect
259. #388D127semidirect
277. C2×C4⋊C4C2×D127semidirect
297. C2×C42C2×D127semidirect
302. #95C2×D127semidirect
303. #95C2×D127semidirect
362. 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
C2×C4×D5082
C2×C22⋊C4×D1272
#225497772
#225497762
#225497692
#225497682
#225497672
#225497542
#225497522
#225497472
#225497312
#225497232
#388×C1272
#2321127