Finite Group Databasebeta

← #30650269

Subgroups — #30650269

Order 16256.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861233C211yesmaximal, characteristic, complemented
2C2×#16861230C211yesmaximal, characteristic, complemented
3#22550546C211yesmaximal, characteristic, complemented
4#22550519C211yesmaximal, characteristic, complemented
5#22550119C211yesmaximal, characteristic, complemented
6#22550109C211yesmaximal, characteristic, complemented
7#22550095C211yesmaximal, characteristic, complemented
8#22550068C211yesmaximal, characteristic, complemented
9#22550035C211yesmaximal, characteristic, complemented
10#22550020C211yesmaximal, characteristic, complemented
11#22550018C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549978C211yesmaximal, characteristic, complemented
14#22549949C211yesmaximal, characteristic, complemented
15#496×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×#14523247C2211yescharacteristic, complemented
19C2×D4×D127C2211yescharacteristic
20#16861233—24no—
21#16861230—24no—
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523233C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic, complemented
29#16861205—21no—
30#16861191—21no—
31#16861185C2211yescharacteristic
32#16861182—21no—
33#16861181—21no—
34#16861181—21no—
35#16861180C2211yescharacteristic
36#16861180C2211yescharacteristic
37#16861180C2211yescharacteristic, complemented
38#16861179—21no—
39C4×Dic254C2211yescharacteristic, complemented
40C4×D508C2211yescharacteristic
41C4×D508C2211yescharacteristic
42D4×Dic127—21no—
43C22⋊C4×D127—21no—
44#16861143C2211yescharacteristic, complemented
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46#16861134C2211yescharacteristic, complemented
47#16861133C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic
49#16861130C2211yescharacteristic
50#16861130C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic
53(C2×D127)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic, complemented
56C2×D8×C127C2211yescharacteristic
57C2×D8×C127C2211yescharacteristic
58#127×C127C2211yescharacteristic, complemented
59#124×C127C2211yescharacteristic
60#97×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64#14523247—22no—
65#14523247—22no—
66#14523247—41no—
67#14523247—41no—
68C2×Dic254C2311yescharacteristic
69C2×D508—21no—
70C2×D508D411yescharacteristic
71C2×D508C2311yescharacteristic
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77C2×C22⋊D127—21no—
78D4×D127—22no—
79D4×D127—41no—
80C22×Dic127—21no—
81C22×Dic127—21no—
82C2×C4×D127—21no—
83C2×C4×D127—21no—
84C2×C4×D127C2311yescharacteristic
85C2×C4×D127C2311yescharacteristic
86#14523238—22no—
87#14523237D414yescomplemented
88#14523236—22no—
89#14523233—22no—
90#14523233—22no—
91#14523231—22no—
92C2×C127⋊C8C2311yescharacteristic
93C2×C127⋊C8C2311yescharacteristic
94C8×D127—22no—
95C4⋊Dic127D411yescharacteristic
96C4⋊Dic127C2311yescharacteristic
97C4⋊Dic127C2311yescharacteristic
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4D411yescharacteristic
106(C2×D127)⋊C4D411yescharacteristic
107C4×Dic127—21no—
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127—21no—
110C2×D4×C127—21no—
111C2×D4×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114SD16×C127—22no—
115SD16×C127—41no—
116D8×C127—22no—
117D8×C127—41no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C22⋊C4×C127—21no—
121C22⋊C4×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—42no—
126C22×D127—81no—
127C22×D127C2×D411yescharacteristic
128C22×D127C2×D411yescharacteristic
129Dic254—21no—
130Dic254C2×D412yes—
131D508—21no—
132D508—41no—
133D508—42no—
134D508C2×D412yes—
135C22⋊D127—42no—
136C22⋊D127—42no—
137C22⋊D127—42no—
138C22⋊D127—81no—
139C22⋊D127—81no—
140C22⋊D127—81no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—42no—
145C2×Dic127—42no—
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—21no—
151C4×D127—21no—
152C4×D127—42no—
153C4×D127—81no—
154C127⋊C8—21no—
155C127⋊C8—21no—
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—21no—
159Q8×C127—21no—
160Q8×C127—21no—
161D4×C127—21no—
162D4×C127—21no—
163D4×C127—21no—
164D4×C127—21no—
165D4×C127—21no—
166D4×C127—21no—
167D4×C127—42no—
168D4×C127—81no—
169C2×C4×C127—21no—
170C2×C4×C127C2×D411yescharacteristic
171C2×C4×C127C2×D411yescharacteristic
172C2×C4×C127C2411yescharacteristic
173C8×C127—21no—
174C8×C127C2×D412yes—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127C22×D411yescharacteristic
185C2×D127—21no—
186C2×D127—21no—
187C2×D127—22no—
188C2×D127—21no—
189C2×D127—21no—
190C2×D127—21no—
191C2×D127—21no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—84no—
195C4×C127—21no—
196C4×C127—21no—
197C4×C127—41no—
198C4×C127C22×D411yescharacteristic
199C4×C127H(5,𝔽2)11yescharacteristic
200Dic127—21no—
201Dic127—21no—
202Dic127—22no—
203Dic127—41no—
204Dic127—41no—
205C2×C127—81no—
206C2×C127—81no—
207C2×C127—81no—
208C2×C127D4211yescharacteristic
209C2×C127#57511yescharacteristic
210C2×C127#57611yescharacteristic
211D127—41no—
212D127—41no—
213D127—42no—
214#3268—1271nomaximal, Sylow
215C127#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1271no—
217C2×#135—1271no—
218#550—1271no—
219#546—1271no—
220#496D12711yescharacteristic, complemented
221#468—1271no—
222#466—1271no—
223#464—1271no—
224#459—1271no—
225#452—1271no—
226#449—1271no—
227#449—1271no—
228#447—1271no—
229#440—1271no—
230#435—1271no—
231C2×#41—1271no—
232C2×#41—1271no—
233C2×#41—1271no—
234C22×D4—1271no—
235AGL(1,ℤ/8)—2544no—
236#135—2544no—
237C2×Q16—1271no—
238C2×SD16—1271no—
239C2×SD16—1271no—
240C2×SD16C2×D12711yescharacteristic, complemented
241C2×SD16C2×D12711yescharacteristic, complemented
242C2×D8—1271no—
243C2×D8—1271no—
244C2×D8C2×D12711yescharacteristic
245C2×D8C2×D12711yescharacteristic
246C2×M16—1271no—
247C22×C8—1271no—
248#127—2541no—
249#127C2×D12711yescharacteristic, complemented
250#124C2×D12711yescharacteristic
251#123—2541no—
252C22⋊Q8—1271no—
253#121—1271no—
254#121—1271no—
255#121—1271no—
256#121—2541no—
257#121—2541no—
258#121—2541no—
259#120—2541no—
260C4×Q8—1271no—
261C4×D4—1271no—
262C4×D4—1271no—
263C4×D4—2541no—
264C2×C22⋊C4—2541no—
265#107—1271no—
266C4⋊C8—1271no—
267#103—1271no—
268#103—1271no—
269#102—1271no—
270#102—1271no—
271#102—1271no—
272#102—1271no—
273C22⋊C8—1271no—
274C22⋊C8—1271no—
275#97C2×D12711yescharacteristic, complemented
276C24—2541no—
277#41—5081no—
278#41—5081no—
279#41—5081no—
280#41—2542no—
281#41—2542no—
282#41—2542no—
283C2×Q8—1271no—
284C2×Q8C22×D12711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1271no—
288C2×D4—1271no—
289C2×D4—5081no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4—2542no—
293C2×D4—2541no—
294C2×D4—2541no—
295C2×D4—2541no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4C22×D12711yescharacteristic
299C2×D4C22×D12711yescharacteristic
300C2×D4C22×D12711yescharacteristic
301C22×C4—1271no—
302C22×C4—1271no—
303C22×C4—2541no—
304C22×C4—2541no—
305C22×C4—2541no—
306C22×C4—2541no—
307Q16—2542no—
308SD16—22no—
309SD16—41no—
310SD16—1274no—
311SD16—2542no—
312D8—22no—
313D8—41no—
314D8—2542no—
315D8—2542no—
316M16—2542no—
317C2×C8—1271no—
318C2×C8—1271no—
319C2×C8—2542no—
320C2×C8C22×D12711yescharacteristic
321C2×C8C22×D12711yescharacteristic
322C4⋊C4—1271no—
323C4⋊C4—1271no—
324C4⋊C4—1271no—
325C4⋊C4—2541no—
326C4⋊C4—2541no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—1271no—
330C22⋊C4—1271no—
331C22⋊C4—2541no—
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C42—2541no—
337C42C22×D12711yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—1271no—
342C23—1271no—
343C23—10161no—
344C23—5082no—
345C23—5082no—
346C23—2541no—
347Q8—21no—
348Q8—21no—
349Q8—1272no—
350Q8—2541no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—1272no—
360D4—10161no—
361D4—10161no—
362D4—10161no—
363D4—5081no—
364D4—5082no—
365D4—5082no—
366D4—5082no—
367D4—5082no—
368D4—2541no—
369C2×C4—21no—
370C2×C4—1271no—
371C2×C4—1271no—
372C2×C4—10161no—
373C2×C4—5082no—
374C2×C4—5082no—
375C2×C4—5082no—
376C2×C4—2541no—
377C2×C4—2541no—
378C2×C4—2541no—
379C2×C4—2541no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4D4×D12711yescharacteristic
384C2×C4D4×D12711yescharacteristic
385C2×C4C23×D12711yescharacteristic
386C8—21no—
387C8—2541no—
388C8—2541no—
389C8D4×D12712yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—10164no—
400C22—5081no—
401C22—5081no—
402C22—2541no—
403C22—2541no—
404C22—2541no—
405C22—2541no—
406C22—2542no—
407C22—2541no—
408C22—2541no—
409C22C2×D4×D12711yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5081no—
414C4—5081no—
415C4—2541no—
416C4—2541no—
417C4—2542no—
418C4C2×D4×D12711yescharacteristic
419C4#1686125411yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—5081no—
424C2—5082no—
425C2—5081no—
426C2#2255051111yescharacteristic
427C2#2255067211yescharacteristic
428C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#496×C127C2
socleC22×C127C22×D4
hypercenter#496D127
hypocenterC127#3268
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×#16861230C2semidirect
3. #22550546C2semidirect
4. #22550519C2semidirect
5. #22550119C2semidirect
6. #22550109C2semidirect
7. #22550095C2semidirect
8. #22550068C2semidirect
9. #22550035C2semidirect
10. #22550020C2semidirect
11. #22550018C2semidirect
12. #22550008C2semidirect
13. #22549978C2semidirect
14. #22549949C2semidirect
15. #496×C127C2semidirect
18. C2×#14523247C22semidirect
22. C2×#14523238C22semidirect
28. C2×C8×D127C22semidirect
37. #16861180C22semidirect
39. C4×Dic254C22semidirect
44. #16861143C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. #16861134C22semidirect
47. #16861133C22semidirect
50. #16861130C22semidirect
51. #16861130C22semidirect
53. (C2×D127)⋊C8C22semidirect
54. C2×SD16×C127C22semidirect
55. C2×SD16×C127C22semidirect
58. #127×C127C22semidirect
60. #97×C127C22semidirect
87. #14523237D4semidirect
215. C127#3268semidirect
220. #496D127semidirect
240. C2×SD16C2×D127semidirect
241. C2×SD16C2×D127semidirect
249. #127C2×D127semidirect
275. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
C2×#168612302
#225505462
#225505192
#225501192
#225501092
#225500952
#225500682
#225500352
#225500202
#225500182
#225500082
#225499782
#225499492
#496×C1272
#3268127