Finite Group Databasebeta

← #30650335

Subgroups — #30650335

Order 16256.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861233C211yesmaximal, characteristic, complemented
2#22550520C211yesmaximal, characteristic, complemented
3#22550514C211yesmaximal, characteristic, complemented
4#22550257C211yesmaximal, characteristic, complemented
5#22550122C211yesmaximal, characteristic, complemented
6#22550119C211yesmaximal, characteristic, complemented
7#22550093C211yesmaximal, characteristic, complemented
8#22550089C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550035C211yesmaximal, characteristic, complemented
11#22550010C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13D8×Dic127C211yesmaximal, characteristic, complemented
14#22549952C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic, complemented
18#16861233—24no—
19C2×#14523238C2211yescharacteristic, complemented
20C2×#14523237C2211yescharacteristic, complemented
21C2×#14523236C2211yescharacteristic, complemented
22C2×#14523233C2211yescharacteristic
23C2×#14523233C2211yescharacteristic
24C2×#14523231C2211yescharacteristic, complemented
25#16861198C2211yescharacteristic, complemented
26#16861196C2211yescharacteristic, complemented
27#16861192—21no—
28#16861190—21no—
29#16861188C2211yescharacteristic, complemented
30#16861185C2211yescharacteristic, complemented
31#16861183C2211yescharacteristic
32#16861181—21no—
33#16861181—21no—
34#16861180C2211yescharacteristic
35#16861178—21no—
36#16861177—21no—
37D4×Dic127C2211yescharacteristic, complemented
38D4×Dic127C2211yescharacteristic, complemented
39C4×C22⋊D127C2211yescharacteristic
40C2×C22⋊Dic127—21no—
41C2×(C2×D127)⋊C4—21no—
42#16861143C2211yescharacteristic, complemented
43Dic127⋊C8C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861133C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic, complemented
48#16861130C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50C22⋊(C127⋊C8)C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52C8×Dic127C2211yescharacteristic, complemented
53C2×#41×C127C2211yescharacteristic
54C22×D4×C127C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C127—24no—
56C2×SD16×C127C2211yescharacteristic, complemented
57C2×SD16×C127C2211yescharacteristic, complemented
58C2×D8×C127C2211yescharacteristic
59C2×D8×C127C2211yescharacteristic
60C2×M16×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523247—22no—
63#14523247—41no—
64C2×Dic254C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127D411yescharacteristic
71D4×D127—22no—
72D4×D127—41no—
73C22×Dic127—21no—
74C22×Dic127—21no—
75C22×Dic127—21no—
76C2×C4×D127C2311yescharacteristic
77#14523238—41no—
78#14523237—22no—
79#14523236—22no—
80#14523233—22no—
81#14523233—41no—
82#14523231—22no—
83C2×C127⋊C8C2311yescharacteristic
84C2×C127⋊C8C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86C4⋊Dic127C2311yescharacteristic
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95C22⋊Dic127—21no—
96C22⋊Dic127D411yescharacteristic
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4D411yescharacteristic
102C4×Dic127C2311yescharacteristic
103C24×C127—21no—
104#41×C127—22no—
105#41×C127—41no—
106C2×Q8×C127C2311yescharacteristic
107C2×D4×C127—22no—
108C2×D4×C127—21no—
109C2×D4×C127—21no—
110C2×D4×C127—41no—
111C2×D4×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116SD16×C127—22no—
117SD16×C127—22no—
118D8×C127—22no—
119D8×C127D414yescomplemented
120M16×C127—22no—
121C2×C8×C127C2311yescharacteristic
122C2×C8×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—42no—
126C22×D127—81no—
127C22×D127C2×D411yescharacteristic
128Dic254—21no—
129Dic254—21no—
130D508—21no—
131D508—21no—
132C22⋊D127—41no—
133C22⋊D127—42no—
134C22⋊D127—42no—
135C22⋊D127—81no—
136C22⋊D127—81no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127—42no—
141C2×Dic127—81no—
142C2×Dic127—81no—
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C2×Dic127C2×D411yescharacteristic
146C4×D127—21no—
147C4×D127—21no—
148C127⋊C8—21no—
149C127⋊C8—21no—
150C23×C127—21no—
151C23×C127—21no—
152C23×C127—21no—
153C23×C127—42no—
154C23×C127—42no—
155C23×C127—81no—
156C23×C127C2×D411yescharacteristic
157Q8×C127—21no—
158Q8×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—21no—
163D4×C127—42no—
164D4×C127—42no—
165D4×C127C2×D412yes—
166D4×C127C2×D412yes—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—42no—
171C2×C4×C127C2×D411yescharacteristic
172C2×C4×C127C2411yescharacteristic
173C8×C127—21no—
174C8×C127C2×D412yes—
175C22×C127—21no—
176C22×C127—22no—
177C22×C127—22no—
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—84no—
188C22×C127C22×D411yescharacteristic
189C2×D127—21no—
190C2×D127—21no—
191C2×D127—21no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—84no—
196C4×C127—21no—
197C4×C127—41no—
198C4×C127C22×D411yescharacteristic
199C4×C127H(5,𝔽2)11yescharacteristic
200Dic127—21no—
201Dic127—21no—
202Dic127—21no—
203Dic127—41no—
204Dic127—41no—
205Dic127—41no—
206C2×C127—42no—
207C2×C127—41no—
208C2×C127—42no—
209C2×C127—81no—
210C2×C127D4211yescharacteristic
211C2×C127#57511yescharacteristic
212C2×C127#57611yescharacteristic
213D127—41no—
214D127—81no—
215#3273—1271nomaximal, Sylow
216C127#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1271no—
218C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
219#546—1271no—
220#546—1271no—
221#495—1271no—
222#468—1271no—
223#468—1271no—
224#463—1271no—
225#463—1271no—
226#452—1271no—
227#452—1271no—
228#447—1271no—
229#447—1271no—
230C4×D8—1271no—
231#436—1271no—
232C2×#41—1271no—
233C2×#41C2×D12711yescharacteristic
234C22×D4—1271no—
235C22×D4C2×D12711yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—2544no—
238C2×Q16—1271no—
239C2×SD16—1271no—
240C2×SD16—1271no—
241C2×SD16C2×D12711yescharacteristic, complemented
242C2×SD16C2×D12711yescharacteristic, complemented
243C2×D8—1271no—
244C2×D8—1271no—
245C2×D8C2×D12711yescharacteristic
246C2×D8C2×D12711yescharacteristic
247C2×M16—1271no—
248C2×M16C2×D12711yescharacteristic, complemented
249#124—1271no—
250#124—1271no—
251#123—2541no—
252#123—2541no—
253C22⋊Q8—1271no—
254C22⋊Q8—1271no—
255#121—1271no—
256#121—1271no—
257#121—2541no—
258#121—2541no—
259#120—2541no—
260#120—2541no—
261C4×D4—1271no—
262C4×D4—1271no—
263C4×D4—1271no—
264C2×C22⋊C4—2541no—
265C2×C22⋊C4—2541no—
266#107—1271no—
267C4⋊C8—1271no—
268#103—1271no—
269#103—1271no—
270#102—1271no—
271#102—1271no—
272#102—1271no—
273#102—1271no—
274C22⋊C8—1271no—
275C22⋊C8—1271no—
276C4×C8—1271no—
277C24—21no—
278C24—2541no—
279#41—22no—
280#41—41no—
281#41—5081no—
282#41—2542no—
283C2×Q8—1271no—
284C2×Q8C22×D12711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—1271no—
290C2×D4—1271no—
291C2×D4—5081no—
292C2×D4—2541no—
293C2×D4—2541no—
294C2×D4—2541no—
295C2×D4—2541no—
296C2×D4—2542no—
297C2×D4C22×D12711yescharacteristic
298C2×D4C22×D12711yescharacteristic
299C2×D4C22×D12711yescharacteristic
300C22×C4—21no—
301C22×C4—1271no—
302C22×C4—2541no—
303C22×C4—2541no—
304C22×C4—2541no—
305C22×C4C22×D12711yescharacteristic
306Q16—5081no—
307SD16—22no—
308SD16—22no—
309SD16—2542no—
310SD16—2542no—
311D8—22no—
312D8—5081no—
313D8—2542no—
314D8C22⋊D12714yescomplemented
315M16—22no—
316M16—2542no—
317C2×C8—1271no—
318C2×C8—1271no—
319C2×C8C22×D12711yescharacteristic
320C2×C8C22×D12711yescharacteristic
321C4⋊C4—1271no—
322C4⋊C4—1271no—
323C4⋊C4—1271no—
324C4⋊C4—2541no—
325C4⋊C4—2541no—
326C22⋊C4—1271no—
327C22⋊C4—1271no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C22⋊C4—2541no—
331C22⋊C4—2541no—
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C22⋊C4—2541no—
337C22⋊C4—2541no—
338C42—1271no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—1271no—
346C23—10161no—
347C23—5082no—
348C23—5082no—
349C23—2541no—
350C23D4×D12711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2541no—
354Q8—2541no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—10161no—
362D4—10161no—
363D4—5082no—
364D4—5082no—
365D4—5081no—
366D4—2541no—
367D4—2541no—
368D4C2×C22⋊D12712yes—
369D4C2×C22⋊D12712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—1271no—
375C2×C4—1271no—
376C2×C4—1271no—
377C2×C4—10161no—
378C2×C4—10161no—
379C2×C4—5082no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4—2541no—
385C2×C4D4×D12711yescharacteristic
386C2×C4C23×D12711yescharacteristic
387C8—21no—
388C8—2541no—
389C8—2541no—
390C8C2×C22⋊D12712yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—10164no—
405C22—5081no—
406C22—5081no—
407C22—5081no—
408C22—2541no—
409C22—2541no—
410C22—2541no—
411C22C2×D4×D12711yescharacteristic
412C4—21no—
413C4—41no—
414C4—5081no—
415C4—5081no—
416C4—5081no—
417C4—2541no—
418C4—2541no—
419C4—2541no—
420C4C22×C22⋊D12711yescharacteristic
421C4#1686125411yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—10161no—
427C2—5081no—
428C2D4×C22⋊D12711yescharacteristic
429C2#2255067211yescharacteristic
430C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3273
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. #22550520C2semidirect
3. #22550514C2semidirect
4. #22550257C2semidirect
5. #22550122C2semidirect
6. #22550119C2semidirect
7. #22550093C2semidirect
8. #22550089C2semidirect
9. #22550038C2semidirect
10. #22550035C2semidirect
11. #22550010C2semidirect
12. #22550008C2semidirect
13. D8×Dic127C2semidirect
14. #22549952C2semidirect
15. C2×AGL(1,ℤ/8)×C127C2semidirect
17. C2×D4×D127C22semidirect
19. C2×#14523238C22semidirect
20. C2×#14523237C22semidirect
21. C2×#14523236C22semidirect
24. C2×#14523231C22semidirect
25. #16861198C22semidirect
26. #16861196C22semidirect
29. #16861188C22semidirect
30. #16861185C22semidirect
37. D4×Dic127C22semidirect
38. D4×Dic127C22semidirect
42. #16861143C22semidirect
43. Dic127⋊C8C22semidirect
44. #16861134C22semidirect
45. #16861133C22semidirect
46. #16861131C22semidirect
47. #16861130C22semidirect
48. #16861130C22semidirect
49. #16861130C22semidirect
50. C22⋊(C127⋊C8)C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. C8×Dic127C22semidirect
54. C22×D4×C127C22semidirect
56. C2×SD16×C127C22semidirect
57. C2×SD16×C127C22semidirect
60. C2×M16×C127C22semidirect
119. D8×C127D4semidirect
216. C127#3273semidirect
218. C2×AGL(1,ℤ/8)D127semidirect
235. C22×D4C2×D127semidirect
241. C2×SD16C2×D127semidirect
242. C2×SD16C2×D127semidirect
248. C2×M16C2×D127semidirect
314. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
#225505202
#225505142
#225502572
#225501222
#225501192
#225500932
#225500892
#225500382
#225500352
#225500102
#225500082
D8×Dic1272
#225499522
C2×AGL(1,ℤ/8)×C1272
#3273127