Finite Group Databasebeta

← #30648076

Subgroups — #30648076

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#22550674C211yesmaximal, characteristic, complemented
3#22550422C211yesmaximal, characteristic, complemented
4#22550057C212yesmaximal, complemented
5#22550052C212yesmaximal, complemented
6#22550047C212yesmaximal, complemented
7#22550042C212yesmaximal, complemented
8C2×#16861136C211yesmaximal, characteristic, complemented
9C2×#16861135C211yesmaximal, characteristic, complemented
10#22549858C211yesmaximal, characteristic, complemented
11#575×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
22#41×D127—22no—
23#41×D127—22no—
24#16861240C2212yescomplemented
25#16861239—21no—
26#16861235C2212yescomplemented
27#16861234—21no—
28#16861229—22no—
29C2×#14523232C2211yescharacteristic, complemented
30#16861207C2211yescharacteristic, complemented
31#16861205C2211yescharacteristic, complemented
32#16861198C2211yescharacteristic, complemented
33#16861196C2211yescharacteristic, complemented
34#16861181—22no—
35C4×C22⋊D127—22no—
36C2×C4×Dic127C2211yescharacteristic, complemented
37#16861136C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861135C2212yescomplemented
41#16861127C2212yescomplemented
42#16861125C2212yescomplemented
43C2×#41×C127C2211yescharacteristic
44C2×#41×C127C2211yescharacteristic, complemented
45#137×C127—21no—
46#137×C127C2212yescomplemented
47AGL(1,ℤ/8)×C127—21no—
48AGL(1,ℤ/8)×C127C2212yescomplemented
49#135×C127—22no—
50C2×M16×C127C2211yescharacteristic, complemented
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#14523248D412yescomplemented
64#14523248C2312yes—
65#14523247—24no—
66#14523247—22no—
67#14523247—24no—
68#14523247—22no—
69C2×Dic254—22no—
70C2×Dic254D411yescharacteristic, complemented
71C2×Dic254C2311yescharacteristic
72Q8×D127—22no—
73Q8×D127—22no—
74C2×D508—22no—
75C2×D508D411yescharacteristic, complemented
76C2×D508C2311yescharacteristic
77C2×C22⋊D127—24no—
78C2×C22⋊D127—22no—
79C2×C22⋊D127—21no—
80C2×C22⋊D127D412yes—
81D4×D127—24no—
82D4×D127—22no—
83D4×D127—24no—
84D4×D127—22no—
85C22×Dic127—21no—
86C2×C4×D127—24no—
87C2×C4×D127—21no—
88C2×C4×D127D412yes—
89#14523238—22no—
90#14523236—22no—
91#14523235—22no—
92#14523233—22no—
93#14523232—21no—
94#14523232C2312yes—
95C2×C127⋊C8—21no—
96Dic127⋊C4—22no—
97C22⋊Dic127—22no—
98(C2×D127)⋊C4—22no—
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic
101C4×Dic127C2311yescharacteristic
102#41×C127—21no—
103#41×C127—21no—
104#41×C127—21no—
105#41×C127—21no—
106#41×C127—21no—
107#41×C127—21no—
108#41×C127D412yescomplemented
109#41×C127C2312yes—
110C2×Q8×C127D411yescharacteristic, complemented
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127—21no—
114C2×D4×C127D411yescharacteristic, complemented
115C2×D4×C127C2311yescharacteristic
116C22×C4×C127—21no—
117C22×C4×C127—21no—
118C22×C4×C127C2311yescharacteristic
119Q16×C127—22no—
120SD16×C127—22no—
121SD16×C127—22no—
122D8×C127—22no—
123M16×C127—21no—
124M16×C127C2312yes—
125C2×C8×C127—21no—
126C22×D127—24no—
127C22×D127—21no—
128C22×D127C2×D412yes—
129Dic254—24no—
130Dic254—21no—
131Dic254—21no—
132Dic254—21no—
133Dic254C2×D412yes—
134D508—24no—
135D508—21no—
136D508—21no—
137D508—21no—
138D508C2×D412yes—
139C22⋊D127—24no—
140C22⋊D127—24no—
141C22⋊D127—22no—
142C22⋊D127—22no—
143C22⋊D127—22no—
144C22⋊D127—22no—
145C22⋊D127—44no—
146C2×Dic127—24no—
147C2×Dic127—21no—
148C2×Dic127—21no—
149C2×Dic127—21no—
150C2×Dic127—42no—
151C2×Dic127C2×D412yes—
152C4×D127—24no—
153C4×D127—24no—
154C4×D127—22no—
155C4×D127—22no—
156C4×D127—22no—
157C4×D127—22no—
158C127⋊C8—22no—
159C23×C127—21no—
160C23×C127—21no—
161C23×C127C2×D411yescharacteristic
162Q8×C127—21no—
163Q8×C127—21no—
164Q8×C127—21no—
165Q8×C127C2×D412yes—
166D4×C127—22no—
167D4×C127—22no—
168D4×C127—21no—
169D4×C127—21no—
170D4×C127—21no—
171D4×C127—41no—
172D4×C127—41no—
173D4×C127C2×D412yes—
174C2×C4×C127—21no—
175C2×C4×C127—22no—
176C2×C4×C127—22no—
177C2×C4×C127—21no—
178C2×C4×C127—41no—
179C2×C4×C127—41no—
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—42no—
190C22×C127—41no—
191C22×C127—81no—
192C22×C127#12011yescharacteristic
193C22×C127#12011yescharacteristic
194C22×C127C22×D411yescharacteristic
195C2×D127—22no—
196C2×D127—22no—
197C2×D127—42no—
198C2×D127—81no—
199C2×D127—84no—
200C4×C127—22no—
201C4×C127—41no—
202C4×C127#12011yescharacteristic
203C4×C127#12011yescharacteristic
204C4×C127C22×D411yescharacteristic
205C4×C127C22×D411yescharacteristic
206Dic127—22no—
207Dic127—22no—
208Dic127—22no—
209Dic127—41no—
210C2×C127—21no—
211C2×C127—21no—
212C2×C127—21no—
213C2×C127—42no—
214C2×C127—81no—
215C2×C127C2×#12011yescharacteristic
216D127—42no—
217D127—42no—
218#3018—1271nomaximal, Sylow
219C127#301811yescharacteristic, complemented, Sylow
220#585—1271no—
221#575—1271no—
222#575D12711yescharacteristic, complemented
223#532—1271no—
224#456—1272no—
225#455—1272no—
226#454—1272no—
227#453—1272no—
228C2×#104—1271no—
229C2×#104—1271no—
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—2542no—
241C2×#41—2542no—
242C2×#41—2542no—
243C2×#41C2×D12711yescharacteristic
244C2×#41C2×D12711yescharacteristic, complemented
245#137—21no—
246#137—1272no—
247#137—2541no—
248#137C2×D12712yescomplemented
249AGL(1,ℤ/8)—21no—
250AGL(1,ℤ/8)—1272no—
251AGL(1,ℤ/8)—2541no—
252AGL(1,ℤ/8)C2×D12712yescomplemented
253#135—22no—
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×C42—1271no—
264#104—1272no—
265#104—1272no—
266#104—1272no—
267#104—1272no—
268#101—1272no—
269#100—1272no—
270#41—21no—
271#41—21no—
272#41—21no—
273#41—21no—
274#41—21no—
275#41—21no—
276#41—1272no—
277#41—1272no—
278#41—2542no—
279#41—2542no—
280#41—2544no—
281#41—2544no—
282#41—2544no—
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#41C22⋊D12712yescomplemented
295#41C22×D12712yes—
296C2×Q8—1271no—
297C2×Q8—1271no—
298C2×Q8—2542no—
299C2×Q8—2542no—
300C2×Q8—2542no—
301C2×Q8C22⋊D12711yescharacteristic, complemented
302C2×Q8C22×D12711yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—1272no—
306C2×D4—1271no—
307C2×D4—1271no—
308C2×D4—2542no—
309C2×D4—2544no—
310C2×D4—2544no—
311C2×D4—2542no—
312C2×D4—2542no—
313C2×D4—2544no—
314C2×D4—2541no—
315C2×D4—2542no—
316C2×D4C22⋊D12711yescharacteristic, complemented
317C2×D4C22×D12711yescharacteristic
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—1272no—
321C22×C4—2544no—
322C22×C4—2541no—
323C22×C4—2541no—
324C22×C4C22×D12711yescharacteristic
325Q16—22no—
326Q16—2542no—
327SD16—22no—
328SD16—22no—
329SD16—2542no—
330SD16—2542no—
331D8—22no—
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—1271no—
343C42—1271no—
344C42—2541no—
345C23—21no—
346C23—21no—
347C23—1272no—
348C23—2544no—
349C23—2541no—
350C23D4×D12711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—1272no—
355Q8—2544no—
356Q8—2541no—
357Q8—2541no—
358Q8—2541no—
359Q8C2×C22⋊D12712yes—
360D4—22no—
361D4—22no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—41no—
366D4—41no—
367D4—1272no—
368D4—5084no—
369D4—2544no—
370D4—2544no—
371D4—2544no—
372D4—2542no—
373D4—2542no—
374D4—2542no—
375D4—2542no—
376D4—2541no—
377D4—2541no—
378D4—2541no—
379D4C2×C22⋊D12712yes—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—1272no—
387C2×C4—5082no—
388C2×C4—2544no—
389C2×C4—2544no—
390C2×C4—2541no—
391C2×C4—2544no—
392C2×C4—2541no—
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×C22⋊D12711yescharacteristic
402C2×C4C2×C22⋊D12711yescharacteristic
403C2×C4C23×D12711yescharacteristic
404C8—22no—
405C8—2542no—
406C22—22no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—81no—
411C22—10161no—
412C22—10164no—
413C22—5082no—
414C22—2542no—
415C22—2542no—
416C22#1686117711yescharacteristic
417C22#1686117711yescharacteristic
418C22C2×D4×D12711yescharacteristic
419C4—22no—
420C4—41no—
421C4—5081no—
422C4—2542no—
423C4—2542no—
424C4—2542no—
425C4#1686117711yescharacteristic
426C4#1686117711yescharacteristic
427C4C2×D4×D12711yescharacteristic
428C4C22×C22⋊D12711yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—81no—
434C2—5082no—
435C2—5082no—
436C2C2×#1686117711yescharacteristic
subgroupG/H
centerC4#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#575×C127C2
socleC2×C127C2×#120
hypercenter#575D127
hypocenterC127#3018
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. #22550674C2semidirect
3. #22550422C2semidirect
4. #22550057C2semidirect
5. #22550052C2semidirect
6. #22550047C2semidirect
7. #22550042C2semidirect
8. C2×#16861136C2semidirect
9. C2×#16861135C2semidirect
10. #22549858C2semidirect
11. #575×C127C2semidirect
24. #16861240C22semidirect
26. #16861235C22semidirect
29. C2×#14523232C22semidirect
30. #16861207C22semidirect
31. #16861205C22semidirect
32. #16861198C22semidirect
33. #16861196C22semidirect
36. C2×C4×Dic127C22semidirect
37. #16861136C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861135C22semidirect
41. #16861127C22semidirect
42. #16861125C22semidirect
44. C2×#41×C127C22semidirect
46. #137×C127C22semidirect
48. AGL(1,ℤ/8)×C127C22semidirect
50. C2×M16×C127C22semidirect
63. #14523248D4semidirect
70. C2×Dic254D4semidirect
75. C2×D508D4semidirect
108. #41×C127D4semidirect
110. C2×Q8×C127D4semidirect
114. C2×D4×C127D4semidirect
219. C127#3018semidirect
222. #575D127semidirect
244. C2×#41C2×D127semidirect
248. #137C2×D127semidirect
252. AGL(1,ℤ/8)C2×D127semidirect
256. C2×M16C2×D127semidirect
294. #41C22⋊D127semidirect
301. C2×Q8C22⋊D127semidirect
316. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225506742
#225504222
#225500572
#225500522
#225500472
#225500422
C2×#168611362
C2×#168611352
#225498582
#575×C1272
#3018127