Finite Group Databasebeta

← #30648081

Subgroups — #30648081

Order 16256.

724 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550712C211yesmaximal, characteristic, complemented
2#22550676C211yesmaximal, characteristic, complemented
3#22550674C211yesmaximal, characteristic, complemented
4#22550426C211yesmaximal, characteristic, complemented
5#22550060C212yesmaximal, complemented
6#22550055C212yesmaximal, complemented
7#22550050C212yesmaximal, complemented
8#22550045C212yesmaximal, complemented
9C2×#16861137C211yesmaximal, characteristic, complemented
10#22549858C211yesmaximal, characteristic, complemented
11C2×#104×C127C211yesmaximal, characteristic, complemented
12#16861257—21no—
13#16861256—21no—
14#16861256C2212yes—
15#16861255—21no—
16#16861255C2212yes—
17#16861254—22no—
18#16861254—21no—
19C2×#14523248—22no—
20C2×#14523248C2211yescharacteristic
21C2×#14523248C2211yescharacteristic
22C2×#14523248C2211yescharacteristic, complemented
23#41×D127—22no—
24#41×D127—22no—
25#16861242—21no—
26#16861242C2212yescomplemented
27#16861240—21no—
28#16861239C2212yescomplemented
29#16861237—21no—
30#16861237C2212yescomplemented
31#16861235—21no—
32#16861234C2212yescomplemented
33#16861231—22no—
34#16861229—22no—
35C2×#14523232C2211yescharacteristic, complemented
36#16861209C2211yescharacteristic, complemented
37#16861206C2211yescharacteristic, complemented
38#16861197C2211yescharacteristic, complemented
39#16861197C2211yescharacteristic, complemented
40#16861183—22no—
41C4×D508—22no—
42#16861137C2212yescomplemented
43#16861137C2212yescomplemented
44#16861127C2212yescomplemented
45#16861125C2212yescomplemented
46C2×#41×C127C2211yescharacteristic
47C2×M16×C127C2211yescharacteristic, complemented
48C2×C42×C127C2211yescharacteristic, complemented
49#104×C127C2212yescomplemented
50#104×C127C2212yescomplemented
51#14523249—22no—
52#14523249—22no—
53#14523248—24no—
54#14523248—24no—
55#14523248—22no—
56#14523248—22no—
57#14523248—21no—
58#14523248—21no—
59#14523248—21no—
60#14523248—21no—
61#14523248—21no—
62#14523248—21no—
63#14523248—21no—
64#14523248—21no—
65#14523248—21no—
66#14523248—21no—
67#14523248D412yescomplemented
68#14523248C2312yes—
69#14523247—24no—
70#14523247—22no—
71#14523247—24no—
72#14523247—22no—
73C2×Dic254—22no—
74C2×Dic254D411yescharacteristic, complemented
75C2×Dic254C2311yescharacteristic
76C2×Dic254C2311yescharacteristic
77Q8×D127—22no—
78Q8×D127—22no—
79C2×D508—22no—
80C2×D508—22no—
81C2×D508D411yescharacteristic, complemented
82C2×D508C2311yescharacteristic
83C2×D508C2311yescharacteristic
84C2×C22⋊D127—24no—
85C2×C22⋊D127—21no—
86C2×C22⋊D127—21no—
87C2×C22⋊D127D412yes—
88D4×D127—24no—
89D4×D127—22no—
90D4×D127—24no—
91D4×D127—22no—
92C2×C4×D127—24no—
93C2×C4×D127—21no—
94C2×C4×D127—21no—
95C2×C4×D127D412yes—
96Dic508—22no—
97#14523238—22no—
98#14523237—22no—
99#14523237—22no—
100#14523236—22no—
101#14523235—22no—
102D1016—22no—
103#14523233—22no—
104#14523232—21no—
105#14523232C2312yes—
106C2×C127⋊C8—21no—
107C4⋊Dic127—22no—
108(C2×D127)⋊C4—22no—
109(C2×D127)⋊C4—22no—
110#41×C127—21no—
111#41×C127—21no—
112#41×C127D412yescomplemented
113#41×C127C2312yes—
114C2×Q8×C127D411yescharacteristic, complemented
115C2×D4×C127—21no—
116C2×D4×C127D411yescharacteristic, complemented
117C22×C4×C127—21no—
118C22×C4×C127—21no—
119C22×C4×C127C2311yescharacteristic
120M16×C127—21no—
121M16×C127C2312yes—
122C2×C8×C127—21no—
123C42×C127—21no—
124C42×C127C2311yescharacteristic
125C42×C127C2311yescharacteristic
126C22×D127—24no—
127C22×D127—21no—
128C22×D127—21no—
129C22×D127C2×D412yes—
130Dic254—24no—
131Dic254—21no—
132Dic254—21no—
133Dic254—21no—
134Dic254—21no—
135Dic254—21no—
136Dic254C2×D412yes—
137D508—24no—
138D508—21no—
139D508—21no—
140D508—21no—
141D508—21no—
142D508—21no—
143D508—44no—
144D508C2×D412yes—
145C22⋊D127—24no—
146C22⋊D127—24no—
147C22⋊D127—22no—
148C22⋊D127—22no—
149C22⋊D127—22no—
150C22⋊D127—22no—
151C22⋊D127—41no—
152C22⋊D127—41no—
153C2×Dic127—24no—
154C2×Dic127—21no—
155C2×Dic127—21no—
156C2×Dic127C2×D412yes—
157C4×D127—24no—
158C4×D127—24no—
159C4×D127—22no—
160C4×D127—22no—
161C4×D127—22no—
162C4×D127—22no—
163C4×D127—41no—
164C4×D127—41no—
165C127⋊C8—22no—
166C23×C127—21no—
167C23×C127C2×D411yescharacteristic
168Q8×C127—21no—
169Q8×C127C2×D412yes—
170D4×C127—22no—
171D4×C127—22no—
172D4×C127—21no—
173D4×C127C2×D412yes—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—22no—
177C2×C4×C127—22no—
178C2×C4×C127—21no—
179C2×C4×C127—42no—
180C2×C4×C127C2×D411yescharacteristic
181C2×C4×C127C2×D411yescharacteristic
182C2×C4×C127C2×D411yescharacteristic
183C2×C4×C127C2×D411yescharacteristic
184C2×C4×C127C2×D411yescharacteristic
185C2×C4×C127C2411yescharacteristic
186C8×C127—22no—
187C22×C127—22no—
188C22×C127—41no—
189C22×C127—81no—
190C22×C127#12011yescharacteristic
191C22×C127#12011yescharacteristic
192C22×C127C22×D411yescharacteristic
193C2×D127—22no—
194C2×D127—22no—
195C2×D127—42no—
196C2×D127—42no—
197C2×D127—41no—
198C2×D127—81no—
199C2×D127—84no—
200C4×C127—22no—
201C4×C127—22no—
202C4×C127—41no—
203C4×C127#12011yescharacteristic
204C4×C127#12011yescharacteristic
205C4×C127C22×D411yescharacteristic
206C4×C127C22×D411yescharacteristic
207Dic127—22no—
208Dic127—22no—
209Dic127—41no—
210C2×C127—21no—
211C2×C127—21no—
212C2×C127—21no—
213C2×C127—42no—
214C2×C127C2×#12011yescharacteristic
215D127—42no—
216D127—42no—
217D127—81no—
218#3018—1271nomaximal, Sylow
219C127#301811yescharacteristic, complemented, Sylow
220#585—1271no—
221#575—1271no—
222#575—1271no—
223#532—1271no—
224#456—1272no—
225#455—1272no—
226#454—1272no—
227#453—1272no—
228C2×#104—1271no—
229C2×#104D12711yescharacteristic, complemented
230#413—1271no—
2312−1+4—1272no—
2322−1+4—2541no—
2332−1+4—2541no—
234H(5,𝔽2)—1272no—
235H(5,𝔽2)—2542no—
236H(5,𝔽2)—2541no—
237H(5,𝔽2)—2541no—
238C2×#41—1271no—
239C2×#41—1271no—
240C2×#41—1271no—
241C2×#41—2542no—
242C2×#41—2542no—
243C2×#41—2542no—
244C2×#41C2×D12711yescharacteristic
245#137—1272no—
246#137—1272no—
247#137—2541no—
248#137—2541no—
249AGL(1,ℤ/8)—1272no—
250AGL(1,ℤ/8)—1272no—
251AGL(1,ℤ/8)—2541no—
252AGL(1,ℤ/8)—2541no—
253#135—2542no—
254#135—2542no—
255C2×M16—1271no—
256C2×M16C2×D12711yescharacteristic, complemented
257C4⋊Q8—1271no—
258#127—1271no—
259#124—1271no—
260#124—1271no—
261#121—2542no—
262C4×D4—2542no—
263C2×C42C2×D12711yescharacteristic, complemented
264#104—1272no—
265#104—1272no—
266#104C2×D12712yescomplemented
267#104C2×D12712yescomplemented
268#101—1272no—
269#100—1272no—
270#41—21no—
271#41—21no—
272#41—1272no—
273#41—1272no—
274#41—2542no—
275#41—2542no—
276#41—2544no—
277#41—2544no—
278#41—2544no—
279#41—2541no—
280#41—2541no—
281#41—2541no—
282#41—2541no—
283#41—2544no—
284#41—2542no—
285#41—2542no—
286#41—2542no—
287#41—2542no—
288#41—2541no—
289#41—2541no—
290#41—2541no—
291#41—2541no—
292#41—2541no—
293#41—2541no—
294#41D50812yescomplemented
295#41C22×D12712yes—
296C2×Q8—1271no—
297C2×Q8—1271no—
298C2×Q8—1271no—
299C2×Q8—2542no—
300C2×Q8—2542no—
301C2×Q8—2542no—
302C2×Q8D50811yescharacteristic, complemented
303C2×D4—21no—
304C2×D4—1272no—
305C2×D4—1271no—
306C2×D4—1271no—
307C2×D4—1271no—
308C2×D4—2542no—
309C2×D4—2542no—
310C2×D4—2544no—
311C2×D4—2544no—
312C2×D4—2542no—
313C2×D4—2541no—
314C2×D4—2544no—
315C2×D4—2541no—
316C2×D4—2542no—
317C2×D4D50811yescharacteristic, complemented
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—1272no—
321C22×C4—2544no—
322C22×C4—2541no—
323C22×C4—2541no—
324C22×C4C22×D12711yescharacteristic
325Q16—2542no—
326Q16—2542no—
327SD16—2542no—
328SD16—2542no—
329SD16—2542no—
330SD16—2542no—
331D8—2542no—
332D8—2542no—
333M16—21no—
334M16—1272no—
335M16—2541no—
336M16C22×D12712yes—
337C2×C8—21no—
338C2×C8—2541no—
339C4⋊C4—2542no—
340C22⋊C4—2542no—
341C22⋊C4—2542no—
342C42—21no—
343C42C22×D12711yescharacteristic
344C42C22×D12711yescharacteristic
345C23—21no—
346C23—1272no—
347C23—2544no—
348C23—2541no—
349C23—2541no—
350C23D4×D12711yescharacteristic
351Q8—21no—
352Q8—1272no—
353Q8—2544no—
354Q8—2541no—
355Q8—2541no—
356Q8—2541no—
357Q8—2541no—
358Q8—2541no—
359Q8C2×D50812yes—
360D4—22no—
361D4—22no—
362D4—21no—
363D4—1272no—
364D4—5084no—
365D4—5081no—
366D4—5081no—
367D4—2544no—
368D4—2544no—
369D4—2544no—
370D4—2542no—
371D4—2542no—
372D4—2542no—
373D4—2542no—
374D4—2541no—
375D4—2541no—
376D4—2541no—
377D4—2541no—
378D4—2541no—
379D4C2×D50812yes—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—42no—
386C2×C4—1272no—
387C2×C4—5081no—
388C2×C4—5081no—
389C2×C4—2544no—
390C2×C4—2544no—
391C2×C4—2541no—
392C2×C4—2544no—
393C2×C4—2541no—
394C2×C4—2542no—
395C2×C4—2542no—
396C2×C4—2542no—
397C2×C4—2542no—
398C2×C4D4×D12711yescharacteristic
399C2×C4D4×D12711yescharacteristic
400C2×C4D4×D12711yescharacteristic
401C2×C4C2×D50811yescharacteristic
402C2×C4C2×D50811yescharacteristic
403C2×C4C23×D12711yescharacteristic
404C8—22no—
405C8—2542no—
406C22—22no—
407C22—41no—
408C22—81no—
409C22—10161no—
410C22—10164no—
411C22—5082no—
412C22—5082no—
413C22—5081no—
414C22—2542no—
415C22—2542no—
416C22#1686117911yescharacteristic
417C22#1686117911yescharacteristic
418C22C2×D4×D12711yescharacteristic
419C4—22no—
420C4—22no—
421C4—41no—
422C4—5081no—
423C4—2542no—
424C4—2542no—
425C4#1686117911yescharacteristic
426C4#1686117911yescharacteristic
427C4C2×D4×D12711yescharacteristic
428C4C22×D50811yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—10161no—
434C2—5082no—
435C2—5082no—
436C2C2×#1686117911yescharacteristic
subgroupG/H
centerC4#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#104×C127C2
socleC2×C127C2×#120
hypercenterC2×#104D127
hypocenterC127#3018
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. #22550676C2semidirect
3. #22550674C2semidirect
4. #22550426C2semidirect
5. #22550060C2semidirect
6. #22550055C2semidirect
7. #22550050C2semidirect
8. #22550045C2semidirect
9. C2×#16861137C2semidirect
10. #22549858C2semidirect
11. C2×#104×C127C2semidirect
22. C2×#14523248C22semidirect
26. #16861242C22semidirect
28. #16861239C22semidirect
30. #16861237C22semidirect
32. #16861234C22semidirect
35. C2×#14523232C22semidirect
36. #16861209C22semidirect
37. #16861206C22semidirect
38. #16861197C22semidirect
39. #16861197C22semidirect
42. #16861137C22semidirect
43. #16861137C22semidirect
44. #16861127C22semidirect
45. #16861125C22semidirect
47. C2×M16×C127C22semidirect
48. C2×C42×C127C22semidirect
49. #104×C127C22semidirect
50. #104×C127C22semidirect
67. #14523248D4semidirect
74. C2×Dic254D4semidirect
81. C2×D508D4semidirect
112. #41×C127D4semidirect
114. C2×Q8×C127D4semidirect
116. C2×D4×C127D4semidirect
219. C127#3018semidirect
229. C2×#104D127semidirect
256. C2×M16C2×D127semidirect
263. C2×C42C2×D127semidirect
266. #104C2×D127semidirect
267. #104C2×D127semidirect
294. #41D508semidirect
302. C2×Q8D508semidirect
317. C2×D4D508semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225506762
#225506742
#225504262
#225500602
#225500552
#225500502
#225500452
C2×#168611372
#225498582
C2×#104×C1272
#3018127