Finite Group Databasebeta

← #30641324

Subgroups — #30641324

Order 16256.

672 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×D127C211yesmaximal, characteristic
2#22549770C211yesmaximal, characteristic, complemented
3#22549769C211yesmaximal, characteristic, complemented
4#22549768C211yesmaximal, characteristic, complemented
5#22549767C211yesmaximal, characteristic, complemented
6#22549766C211yesmaximal, characteristic, complemented
7#22549765C211yesmaximal, characteristic, complemented
8#22549764C211yesmaximal, characteristic, complemented
9#22549735C211yesmaximal, characteristic, complemented
10#22549733C211yesmaximal, characteristic, complemented
11#22549731C211yesmaximal, characteristic
12#22549731C211yesmaximal, characteristic
13C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
14#22549723C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C24×D127C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic, complemented
20C2×Dic127⋊C4C2211yescharacteristic, complemented
21C2×Dic127⋊C4C2211yescharacteristic, complemented
22C2×C22⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×(C2×D127)⋊C4C2211yescharacteristic
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
35C22⋊C4×D127C418yescomplemented
36C2×C4×Dic127C2211yescharacteristic, complemented
37C2×C4×Dic127C2211yescharacteristic, complemented
38C2×C4×Dic127C2211yescharacteristic, complemented
39C127⋊#95C2211yescharacteristic, complemented
40C127⋊#95C2211yescharacteristic, 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—22no—
53C23×D127—22no—
54C23×D127—41no—
55C23×D127—41no—
56C23×D127—41no—
57C23×D127—41no—
58C23×D127C2×C414yes—
59C23×D127C2311yescharacteristic
60C23×D127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C2×C4×D127—41no—
68C2×C4×D127—41no—
69C2×C4×D127—41no—
70C2×C4×D127—41no—
71C2×C4×D127C2×C414yes—
72C2×C4×D127C2×C414yes—
73C4⋊Dic127—41no—
74Dic127⋊C4—41no—
75Dic127⋊C4—41no—
76C22⋊Dic127—41no—
77C22⋊Dic127—41no—
78C22⋊Dic127C2×C414yes—
79(C2×D127)⋊C4—22no—
80(C2×D127)⋊C4—22no—
81(C2×D127)⋊C4—22no—
82(C2×D127)⋊C4—22no—
83(C2×D127)⋊C4—41no—
84(C2×D127)⋊C4—41no—
85(C2×D127)⋊C4—41no—
86(C2×D127)⋊C4—41no—
87(C2×D127)⋊C4C2×C414yes—
88(C2×D127)⋊C4C2×C414yes—
89C4×Dic127—41no—
90C4×Dic127—41no—
91C4×Dic127—41no—
92C24×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C22×C4×C127C2311yescharacteristic
95C22×C4×C127C2311yescharacteristic
96C22×C4×C127C2311yescharacteristic
97C22×C4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C4⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127C2×C414yes—
103C42×C127—41no—
104C22×D127—21no—
105C22×D127—21no—
106C22×D127—21no—
107C22×D127—21no—
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—28no—
115C22×D127—44no—
116C22×D127—44no—
117C22×D127—44no—
118C22×D127—44no—
119C22×D127—44no—
120C22×D127—44no—
121C22×D127C22×C412yes—
122C22×D127C22×C412yes—
123C22×D127#4112yes—
124C22×D127#4112yes—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—21no—
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×Dic127C22×C412yes—
142C2×Dic127C22×C412yes—
143C4×D127—44no—
144C4×D127—44no—
145C23×C127—21no—
146C23×C127—21no—
147C23×C127—21no—
148C23×C127—21no—
149C23×C127—21no—
150C23×C127—21no—
151C23×C127C22×C412yes—
152C23×C127C2411yescharacteristic
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
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×C127C22×C412yes—
170C2×C4×C127C22×C412yes—
171C22×C127—22no—
172C22×C127—22no—
173C22×C127—22no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127C23×C411yescharacteristic
179C22×C127C2×#4111yescharacteristic
180C22×C127C2×#4111yescharacteristic
181C22×C127H(5,𝔽2)11yescharacteristic
182C22×C127H(5,𝔽2)11yescharacteristic
183C22×C127H(5,𝔽2)11yescharacteristic
184C22×C127H(5,𝔽2)11yescharacteristic
185C2×D127—22no—
186C2×D127—22no—
187C2×D127—22no—
188C2×D127—22no—
189C2×D127—22no—
190C2×D127—22no—
191C2×D127—22no—
192C2×D127—22no—
193C2×D127—22no—
194C2×D127—416no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—41no—
199C2×D127#11714yes—
200C4×C127—22no—
201C4×C127—22no—
202C4×C127—41no—
203C4×C127—41no—
204C4×C127—41no—
205C4×C127—41no—
206Dic127—22no—
207Dic127—22no—
208Dic127—41no—
209Dic127—41no—
210Dic127—41no—
211Dic127—41no—
212C2×C127—42no—
213C2×C127C2×#11711yescharacteristic
214C2×C127#51811yescharacteristic
215C2×C127#51811yescharacteristic
216C2×C127#53811yescharacteristic
217C2×C127#53811yescharacteristic
218C2×C127#53811yescharacteristic
219C2×C127#53811yescharacteristic
220D127—24no—
221D127—42no—
222#2312—1271nomaximal, Sylow
223C127#231211yescharacteristic, complemented, Sylow
224C22×C22⋊C4—1271no—
225#388—1271no—
226#388—1271no—
227#388—1271no—
228#388—1271no—
229#388—1271no—
230#388—1271no—
231#388—1271no—
232#388D12711yescharacteristic, complemented
233#380—1271no—
234#380—1271no—
235#379—1271no—
236#379—1271no—
237C4×C22⋊C4—1271no—
238C4×C22⋊C4—1271no—
239C25—1271no—
240C23×C4—1271no—
241C23×C4—1271no—
242C2×C4⋊C4—1271no—
243C2×C4⋊C4—1271no—
244C2×C4⋊C4—1271no—
245C2×C4⋊C4C2×D12711yescharacteristic, complemented
246C2×C22⋊C4—1271no—
247C2×C22⋊C4—1271no—
248C2×C22⋊C4—1271no—
249C2×C22⋊C4—1278no—
250C2×C22⋊C4—1271no—
251C2×C22⋊C4—1271no—
252C2×C22⋊C4—1271no—
253C2×C22⋊C4—1271no—
254C2×C22⋊C4—1271no—
255C2×C22⋊C4—1271no—
256C2×C22⋊C4—1271no—
257C2×C22⋊C4—1271no—
258C2×C22⋊C4—1271no—
259C2×C22⋊C4—1271no—
260C2×C22⋊C4C2×D12711yescharacteristic
261C2×C22⋊C4C2×D12711yescharacteristic
262C2×C22⋊C4C2×D12711yescharacteristic
263C2×C42—1271no—
264C2×C42—1271no—
265C2×C42—1271no—
266C2×C42C2×D12711yescharacteristic, complemented
267#95—1271no—
268#95—1271no—
269#95—1271no—
270#95—1271no—
271#95—1271no—
272#95—1271no—
273#95C2×D12711yescharacteristic, complemented
274#95C2×D12711yescharacteristic, complemented
275C24—1271no—
276C24—1274no—
277C24—1271no—
278C24—5081no—
279C24—5081no—
280C24—5081no—
281C24—5081no—
282C24—2542no—
283C24—2542no—
284C24C22×D12711yescharacteristic
285C22×C4—1271no—
286C22×C4—1271no—
287C22×C4—1274no—
288C22×C4—1274no—
289C22×C4—1271no—
290C22×C4—1271no—
291C22×C4—1271no—
292C22×C4—1271no—
293C22×C4—5081no—
294C22×C4—5081no—
295C22×C4—5081no—
296C22×C4—5081no—
297C22×C4C22×D12711yescharacteristic
298C22×C4C22×D12711yescharacteristic
299C22×C4C22×D12711yescharacteristic
300C22×C4C22×D12711yescharacteristic
301C22×C4C22×D12711yescharacteristic
302C22×C4C22×D12711yescharacteristic
303C4⋊C4—41no—
304C4⋊C4—5081no—
305C4⋊C4—5081no—
306C4⋊C4—5081no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—1274no—
310C22⋊C4—1274no—
311C22⋊C4—1274no—
312C22⋊C4—5081no—
313C22⋊C4—5081no—
314C22⋊C4—5081no—
315C22⋊C4—5081no—
316C22⋊C4—5081no—
317C22⋊C4—5081no—
318C22⋊C4—2542no—
319C22⋊C4—2542no—
320C22⋊C4—2542no—
321C22⋊C4—2542no—
322C22⋊C4C4×D12714yes—
323C42—41no—
324C42—5081no—
325C42—5081no—
326C42—5081no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—1272no—
334C23—1272no—
335C23—1272no—
336C23—1272no—
337C23—5084no—
338C23—5084no—
339C23—5084no—
340C23—5084no—
341C23—5084no—
342C23—5084no—
343C23—2548no—
344C23—2541no—
345C23—2541no—
346C23—2541no—
347C23—2541no—
348C23—2541no—
349C23—2541no—
350C23—2541no—
351C23—2541no—
352C23—2541no—
353C23—2541no—
354C23C2×C4×D12712yes—
355C23C23×D12711yescharacteristic
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—1272no—
373C2×C4—1272no—
374C2×C4—5084no—
375C2×C4—5084no—
376C2×C4—2541no—
377C2×C4—2541no—
378C2×C4—2541no—
379C2×C4—2541no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4—2541no—
385C2×C4—2541no—
386C2×C4—2541no—
387C2×C4—2541no—
388C2×C4—2541no—
389C2×C4—2541no—
390C2×C4—2541no—
391C2×C4—2541no—
392C2×C4C2×C4×D12712yes—
393C2×C4C2×C4×D12712yes—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—1274no—
402C22—50816no—
403C22—5081no—
404C22—5081no—
405C22—5081no—
406C22—5081no—
407C22—2542no—
408C22—2542no—
409C22—2542no—
410C22—2542no—
411C22—2542no—
412C22—2542no—
413C22—2542no—
414C22—2542no—
415C22—2542no—
416C22C22×C4×D12711yescharacteristic
417C22#41×D12711yescharacteristic
418C22#41×D12711yescharacteristic
419C22#1686125411yescharacteristic
420C22#1686125411yescharacteristic
421C22#1686125511yescharacteristic
422C22#1686125511yescharacteristic
423C4—22no—
424C4—22no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—5081no—
430C4—5081no—
431C4—5081no—
432C4—5081no—
433C4—2542no—
434C4—2542no—
435C2—42no—
436C2—5082no—
437C2—2544no—
438C2#117×D12711yescharacteristic
439C2#2255034811yescharacteristic
440C2#2255034811yescharacteristic
441C2#2255045811yescharacteristic
442C2#2255046011yescharacteristic
443C2#2255046211yescharacteristic
444C2#2255046511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC24×D127C22
hypercenter#388D127
hypocenterC127#2312
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #22549770C2semidirect
3. #22549769C2semidirect
4. #22549768C2semidirect
5. #22549767C2semidirect
6. #22549766C2semidirect
7. #22549765C2semidirect
8. #22549764C2semidirect
9. #22549735C2semidirect
10. #22549733C2semidirect
13. C4×(C2×D127)⋊C4C2semidirect
14. #22549723C2semidirect
15. #388×C127C2semidirect
19. C2×C4⋊Dic127C22semidirect
20. C2×Dic127⋊C4C22semidirect
21. C2×Dic127⋊C4C22semidirect
35. C22⋊C4×D127C4semidirect
36. C2×C4×Dic127C22semidirect
37. C2×C4×Dic127C22semidirect
38. C2×C4×Dic127C22semidirect
39. C127⋊#95C22semidirect
40. C127⋊#95C22semidirect
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
223. C127#2312semidirect
232. #388D127semidirect
245. C2×C4⋊C4C2×D127semidirect
266. C2×C42C2×D127semidirect
273. #95C2×D127semidirect
274. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1272
#225497702
#225497692
#225497682
#225497672
#225497662
#225497652
#225497642
#225497352
#225497332
#225497312
C4×(C2×D127)⋊C42
#225497232
#388×C1272
#2312127