Finite Group Databasebeta

← #30648103

Subgroups — #30648103

Order 16256.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550712C211yesmaximal, characteristic, complemented
2C2×#16861237C211yesmaximal, characteristic, complemented
3C2×#16861234C211yesmaximal, characteristic, complemented
4#22550445C211yesmaximal, characteristic, complemented
5#22550050C214yesmaximal, complemented
6#22550045C214yesmaximal, complemented
7#22549894C211yesmaximal, characteristic, complemented
8#22549858C211yesmaximal, characteristic, complemented
9#421×C127C211yesmaximal, characteristic, complemented
10#16861257—21no—
11#16861256—21no—
12#16861256C2212yescomplemented
13#16861255—21no—
14#16861255C2212yescomplemented
15#16861254—22no—
16#16861254—21no—
17C2×#14523248—22no—
18C2×#14523248C2211yescharacteristic
19C2×#14523248C2211yescharacteristic
20#41×D127—22no—
21#41×D127—22no—
22C22×D508C2211yescharacteristic, complemented
23#16861237—22no—
24#16861237C2214yescomplemented
25#16861234—22no—
26#16861234C2214yescomplemented
27C2×#14523237—21no—
28C2×#14523235—21no—
29C2×D1016—21no—
30C2×#14523233—21no—
31C2×#14523232C2211yescharacteristic, complemented
32#16861206C2212yescomplemented
33#16861197C2212yescomplemented
34#16861184—22no—
35#16861179—22no—
36#16861137C2214yescomplemented
37#16861127C2212yescomplemented
38#16861125C2212yescomplemented
39C2×#41×C127C2211yescharacteristic
40C2×M16×C127C2211yescharacteristic, complemented
41#117×C127C2211yescharacteristic, complemented
42#104×C127C2214yescomplemented
43C23×D127—21no—
44#14523249—22no—
45#14523249—22no—
46#14523248—24no—
47#14523248—24no—
48#14523248—22no—
49#14523248—22no—
50#14523248—21no—
51#14523248—21no—
52#14523248—21no—
53#14523248—21no—
54#14523248—21no—
55#14523248—21no—
56#14523248D412yescomplemented
57#14523248C2312yes—
58#14523247—24no—
59#14523247—22no—
60#14523247—24no—
61#14523247—22no—
62C2×Dic254—22no—
63C2×Dic254D411yescharacteristic, complemented
64C2×Dic254C2311yescharacteristic
65Q8×D127—22no—
66Q8×D127—22no—
67C2×D508—22no—
68C2×D508—24no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508—21no—
73C2×D508—21no—
74C2×D508D411yescharacteristic, complemented
75C2×D508C2312yes—
76C2×D508C2311yescharacteristic
77C2×C22⋊D127—24no—
78C2×C22⋊D127—21no—
79C2×C22⋊D127D412yes—
80D4×D127—24no—
81D4×D127—22no—
82D4×D127—24no—
83D4×D127—22no—
84C2×C4×D127—24no—
85C2×C4×D127—21no—
86C2×C4×D127D412yes—
87#14523237—24no—
88#14523235—24no—
89D1016—24no—
90#14523233—24no—
91#14523232—21no—
92#14523232C2312yes—
93C2×C127⋊C8—21no—
94(C2×D127)⋊C4—24no—
95#41×C127—21no—
96#41×C127—21no—
97#41×C127D412yescomplemented
98#41×C127C2312yes—
99C2×Q8×C127D411yescharacteristic, complemented
100C2×D4×C127—21no—
101C2×D4×C127D411yescharacteristic, complemented
102C22×C4×C127—21no—
103C22×C4×C127C2311yescharacteristic
104M16×C127—21no—
105M16×C127C2312yes—
106C2×C8×C127—21no—
107C4⋊C4×C127—21no—
108C22⋊C4×C127—21no—
109C42×C127C2312yes—
110C22×D127—24no—
111C22×D127—21no—
112C22×D127—22no—
113C22×D127—41no—
114C22×D127—41no—
115C22×D127—82no—
116C22×D127C2×D412yes—
117Dic254—24no—
118Dic254—21no—
119Dic254—21no—
120Dic254—21no—
121Dic254C2×D412yes—
122D508—24no—
123D508—21no—
124D508—21no—
125D508—21no—
126D508—22no—
127D508—22no—
128D508—41no—
129D508—41no—
130D508—84no—
131D508C2×D412yes—
132C22⋊D127—24no—
133C22⋊D127—24no—
134C22⋊D127—22no—
135C22⋊D127—22no—
136C22⋊D127—22no—
137C22⋊D127—22no—
138C2×Dic127—24no—
139C2×Dic127—21no—
140C2×Dic127C2×D412yes—
141C4×D127—24no—
142C4×D127—24no—
143C4×D127—22no—
144C4×D127—22no—
145C4×D127—22no—
146C4×D127—22no—
147C127⋊C8—22no—
148C23×C127—21no—
149C23×C127C2×D411yescharacteristic
150Q8×C127—21no—
151Q8×C127C2×D412yes—
152D4×C127—22no—
153D4×C127—22no—
154D4×C127—21no—
155D4×C127C2×D412yes—
156C2×C4×C127—21no—
157C2×C4×C127—22no—
158C2×C4×C127—22no—
159C2×C4×C127—22no—
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2×D411yescharacteristic
162C2×C4×C127C2×D411yescharacteristic
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—22no—
167C22×C127—22no—
168C22×C127—41no—
169C22×C127—81no—
170C22×C127#12011yescharacteristic
171C22×C127#12011yescharacteristic
172C22×C127C22×D411yescharacteristic
173C2×D127—22no—
174C2×D127—22no—
175C2×D127—42no—
176C2×D127—44no—
177C2×D127—42no—
178C2×D127—81no—
179C2×D127—84no—
180C2×D127—82no—
181C2×D127—82no—
182C4×C127—22no—
183C4×C127—42no—
184C4×C127#12011yescharacteristic
185C4×C127#12011yescharacteristic
186C4×C127C22×D411yescharacteristic
187C4×C127C22×D411yescharacteristic
188Dic127—22no—
189Dic127—22no—
190C2×C127—21no—
191C2×C127—21no—
192C2×C127—21no—
193C2×C127—42no—
194C2×C127C2×#12011yescharacteristic
195D127—42no—
196D127—42no—
197D127—82no—
198#3021—1271nomaximal, Sylow
199C127#302111yescharacteristic, complemented, Sylow
200#585—1271no—
201C2×AGL(1,ℤ/8)—1271no—
202C2×AGL(1,ℤ/8)—1271no—
203#535—1271no—
204#454—1274no—
205#453—1274no—
206#421—1271no—
207#421D12711yescharacteristic, complemented
208#413—1271no—
2092−1+4—1272no—
2102−1+4—2541no—
2112−1+4—2541no—
212H(5,𝔽2)—1272no—
213H(5,𝔽2)—2542no—
214H(5,𝔽2)—2541no—
215H(5,𝔽2)—2541no—
216C2×#41—1271no—
217C2×#41—1271no—
218C2×#41—2542no—
219C2×#41—2542no—
220C2×#41—2542no—
221C2×#41C2×D12711yescharacteristic
222C22×D4—1271no—
223AGL(1,ℤ/8)—1274no—
224AGL(1,ℤ/8)—1274no—
225AGL(1,ℤ/8)—2542no—
226AGL(1,ℤ/8)—2542no—
227C2×SD16—2541no—
228C2×SD16—2541no—
229C2×D8—2541no—
230C2×D8—2541no—
231C2×M16—1271no—
232C2×M16C2×D12711yescharacteristic, complemented
233#127—1272no—
234#124—1272no—
235#121—2542no—
236#120—2542no—
237#117C2×D12711yescharacteristic, complemented
238#104—1274no—
239#104C2×D12714yescomplemented
240#101—1272no—
241#100—1272no—
242C24—2541no—
243#41—21no—
244#41—21no—
245#41—1272no—
246#41—1272no—
247#41—2542no—
248#41—2542no—
249#41—2544no—
250#41—2544no—
251#41—2544no—
252#41—2544no—
253#41—2542no—
254#41—2542no—
255#41—2542no—
256#41—2542no—
257#41—2541no—
258#41—2541no—
259#41—2541no—
260#41—2541no—
261#41—2541no—
262#41—2541no—
263#41D50812yescomplemented
264#41C22×D12712yes—
265C2×Q8—1271no—
266C2×Q8—1271no—
267C2×Q8—2542no—
268C2×Q8—2542no—
269C2×Q8—2542no—
270C2×Q8D50811yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—1272no—
273C2×D4—1271no—
274C2×D4—1271no—
275C2×D4—1272no—
276C2×D4—2542no—
277C2×D4—2544no—
278C2×D4—2544no—
279C2×D4—2544no—
280C2×D4—2542no—
281C2×D4—2544no—
282C2×D4—2541no—
283C2×D4—2542no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4D50811yescharacteristic, complemented
290C22×C4—21no—
291C22×C4—1272no—
292C22×C4—2544no—
293C22×C4—2541no—
294C22×C4C22×D12711yescharacteristic
295SD16—2544no—
296SD16—2544no—
297D8—2544no—
298D8—2544no—
299M16—21no—
300M16—1272no—
301M16—2541no—
302M16C22×D12712yes—
303C2×C8—21no—
304C2×C8—2541no—
305C4⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—2544no—
308C42C22×D12712yes—
309C23—21no—
310C23—1272no—
311C23—10162no—
312C23—5081no—
313C23—5081no—
314C23—2544no—
315C23—2541no—
316C23—2542no—
317C23D4×D12711yescharacteristic
318Q8—21no—
319Q8—1272no—
320Q8—2544no—
321Q8—2541no—
322Q8—2541no—
323Q8—2541no—
324Q8C2×D50812yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—1272no—
329D4—10164no—
330D4—5081no—
331D4—5081no—
332D4—2544no—
333D4—2544no—
334D4—2544no—
335D4—2542no—
336D4—2542no—
337D4—2542no—
338D4—2542no—
339D4—2542no—
340D4—2541no—
341D4—2542no—
342D4—2541no—
343D4—2541no—
344D4C2×D50812yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—1272no—
350C2×C4—2544no—
351C2×C4—2544no—
352C2×C4—2544no—
353C2×C4—2541no—
354C2×C4—2542no—
355C2×C4—2542no—
356C2×C4—2542no—
357C2×C4—2542no—
358C2×C4D4×D12711yescharacteristic
359C2×C4D4×D12711yescharacteristic
360C2×C4D4×D12711yescharacteristic
361C2×C4C2×D50811yescharacteristic
362C2×C4C2×D50811yescharacteristic
363C2×C4C23×D12711yescharacteristic
364C8—22no—
365C8—2542no—
366C22—22no—
367C22—41no—
368C22—81no—
369C22—10162no—
370C22—10162no—
371C22—10161no—
372C22—10164no—
373C22—5084no—
374C22—5082no—
375C22—5082no—
376C22—2542no—
377C22—2542no—
378C22#1686117911yescharacteristic
379C22#1686117911yescharacteristic
380C22C2×D4×D12711yescharacteristic
381C4—22no—
382C4—42no—
383C4—2542no—
384C4—2542no—
385C4#1686117911yescharacteristic
386C4#1686117911yescharacteristic
387C4C2×D4×D12711yescharacteristic
388C4C22×D50811yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—10162no—
394C2—5082no—
395C2—5082no—
396C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#421×C127C2
socleC2×C127C2×#120
hypercenter#421D127
hypocenterC127#3021
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. C2×#16861237C2semidirect
3. C2×#16861234C2semidirect
4. #22550445C2semidirect
5. #22550050C2semidirect
6. #22550045C2semidirect
7. #22549894C2semidirect
8. #22549858C2semidirect
9. #421×C127C2semidirect
12. #16861256C22semidirect
14. #16861255C22semidirect
22. C22×D508C22semidirect
24. #16861237C22semidirect
26. #16861234C22semidirect
31. C2×#14523232C22semidirect
32. #16861206C22semidirect
33. #16861197C22semidirect
36. #16861137C22semidirect
37. #16861127C22semidirect
38. #16861125C22semidirect
40. C2×M16×C127C22semidirect
41. #117×C127C22semidirect
42. #104×C127C22semidirect
56. #14523248D4semidirect
63. C2×Dic254D4semidirect
74. C2×D508D4semidirect
97. #41×C127D4semidirect
99. C2×Q8×C127D4semidirect
101. C2×D4×C127D4semidirect
199. C127#3021semidirect
207. #421D127semidirect
232. C2×M16C2×D127semidirect
237. #117C2×D127semidirect
239. #104C2×D127semidirect
263. #41D508semidirect
270. C2×Q8D508semidirect
289. C2×D4D508semidirect

Maximal subgroups

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

subgroupindex
#225507122
C2×#168612372
C2×#168612342
#225504452
#225500502
#225500452
#225498942
#225498582
#421×C1272
#3021127