Finite Group Databasebeta

← #30648089

Subgroups — #30648089

Order 16256.

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

normal subgroups only

This group has 762 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
1C2×#16861255C211yesmaximal, characteristic, complemented
2#22550676C211yesmaximal, characteristic, complemented
3#22550674C211yesmaximal, characteristic, complemented
4#22550445C211yesmaximal, characteristic, complemented
5#22550055C214yesmaximal, complemented
6#22550045C214yesmaximal, complemented
7#22549894C211yesmaximal, characteristic, complemented
8C2×#16861125C211yesmaximal, characteristic, complemented
9#421×C127C211yesmaximal, characteristic, complemented
10#16861255—22no—
11#16861255—22no—
12#16861255—22no—
13#16861255C2214yes—
14C2×#14523249—21no—
15C2×#14523248—21no—
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic, complemented
18C22×D508—21no—
19C22×D508C2211yescharacteristic
20C2×D4×D127—22no—
21C2×D4×D127—21no—
22#16861242—21no—
23#16861242C2212yescomplemented
24#16861240—21no—
25#16861239C2212yescomplemented
26#16861237—21no—
27#16861237C2212yescomplemented
28#16861235—21no—
29#16861234C2212yescomplemented
30#16861231—22no—
31#16861229—22no—
32C2×#14523232C2211yescharacteristic, complemented
33#16861206C2212yescomplemented
34#16861197C2212yescomplemented
35#16861184—22no—
36#16861179—22no—
37#16861137C2214yescomplemented
38#16861125C2214yescomplemented
39C2×#41×C127C2211yescharacteristic
40C2×M16×C127C2211yescharacteristic, complemented
41#117×C127C2211yescharacteristic, complemented
42#104×C127C2214yescomplemented
43C23×D127—22no—
44C23×D127D412yes—
45#14523249—24no—
46#14523249—24no—
47#14523248—24no—
48#14523248—24no—
49#14523248—21no—
50#14523248—21no—
51#14523248—21no—
52#14523248—21no—
53#14523248—21no—
54#14523248—21no—
55#14523248D412yescomplemented
56#14523248C2312yes—
57C2×Dic254D411yescharacteristic, complemented
58C2×Dic254C2311yescharacteristic, complemented
59C2×D508—22no—
60C2×D508—24no—
61C2×D508—24no—
62C2×D508—24no—
63C2×D508—21no—
64C2×D508—21no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508D412yes—
68C2×D508D411yescharacteristic, complemented
69C2×D508C2312yes—
70C2×D508C2311yescharacteristic, complemented
71C2×C22⋊D127—22no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74D4×D127—28no—
75D4×D127—24no—
76D4×D127—28no—
77D4×D127—24no—
78C2×C4×D127—22no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81Dic508—22no—
82#14523238—22no—
83#14523237—22no—
84#14523237—22no—
85#14523236—22no—
86#14523235—22no—
87D1016—22no—
88#14523233—22no—
89#14523232—21no—
90#14523232C2312yescomplemented
91C2×C127⋊C8—21no—
92(C2×D127)⋊C4—24no—
93#41×C127—21no—
94#41×C127—21no—
95#41×C127D412yescomplemented
96#41×C127C2312yes—
97C2×Q8×C127D411yescharacteristic, complemented
98C2×D4×C127—21no—
99C2×D4×C127D411yescharacteristic, complemented
100C22×C4×C127—21no—
101C22×C4×C127C2311yescharacteristic
102M16×C127—21no—
103M16×C127C2312yescomplemented
104C2×C8×C127—21no—
105C4⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127C2312yescomplemented
108C22×D127—22no—
109C22×D127—21no—
110C22×D127—28no—
111C22×D127—22no—
112C22×D127—21no—
113C22×D127—82no—
114C22×D127—84no—
115C22×D127C2×D414yes—
116Dic254—21no—
117Dic254—21no—
118Dic254—21no—
119Dic254C2×D412yes—
120D508—28no—
121D508—22no—
122D508—22no—
123D508—21no—
124D508—22no—
125D508—22no—
126D508—21no—
127D508—21no—
128D508—84no—
129D508C2×D412yes—
130C22⋊D127—28no—
131C22⋊D127—22no—
132C22⋊D127—22no—
133C22⋊D127—41no—
134C22⋊D127—41no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C4×D127—28no—
138C4×D127—22no—
139C4×D127—22no—
140C4×D127—41no—
141C4×D127—41no—
142C127⋊C8—22no—
143C23×C127—21no—
144C23×C127C2×D411yescharacteristic
145Q8×C127—21no—
146Q8×C127C2×D412yes—
147D4×C127—22no—
148D4×C127—22no—
149D4×C127—21no—
150D4×C127C2×D412yes—
151C2×C4×C127—21no—
152C2×C4×C127—22no—
153C2×C4×C127—22no—
154C2×C4×C127—22no—
155C2×C4×C127C2×D411yescharacteristic
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2×D411yescharacteristic
158C2×C4×C127C2×D411yescharacteristic
159C2×C4×C127C2×D411yescharacteristic
160C2×C4×C127C2411yescharacteristic
161C8×C127—22no—
162C22×C127—22no—
163C22×C127—41no—
164C22×C127—42no—
165C22×C127#12011yescharacteristic
166C22×C127#12011yescharacteristic
167C22×C127C22×D411yescharacteristic
168C2×D127—24no—
169C2×D127—22no—
170C2×D127—44no—
171C2×D127—42no—
172C2×D127—44no—
173C2×D127—42no—
174C2×D127—41no—
175C2×D127—82no—
176C2×D127—88no—
177C4×C127—22no—
178C4×C127—42no—
179C4×C127#12011yescharacteristic
180C4×C127#12011yescharacteristic
181C4×C127C22×D411yescharacteristic
182C4×C127C22×D411yescharacteristic
183Dic127—22no—
184Dic127—41no—
185C2×C127—21no—
186C2×C127—21no—
187C2×C127—21no—
188C2×C127—42no—
189C2×C127C2×#12011yescharacteristic
190D127—44no—
191D127—42no—
192D127—81no—
193#3019—1271nomaximal, Sylow
194C127#301911yescharacteristic, complemented, Sylow
195C2×H(5,𝔽2)—1271no—
196#575—1271no—
197#575—1271no—
198#535—1271no—
199#455—1274no—
200#453—1274no—
201#421—1271no—
202#421D12711yescharacteristic, complemented
203C2×#100—1271no—
204H(5,𝔽2)—1274no—
205H(5,𝔽2)—2542no—
206H(5,𝔽2)—2542no—
207H(5,𝔽2)—2542no—
208C2×#41—1271no—
209C2×#41—1271no—
210C2×#41—2541no—
211C2×#41—2541no—
212C2×#41C2×D12711yescharacteristic
213C22×D4—1271no—
214C22×D4—2541no—
215C22×D4—2542no—
216C22×D4—2541no—
217#137—1272no—
218#137—1272no—
219#137—2541no—
220#137—2541no—
221AGL(1,ℤ/8)—1272no—
222AGL(1,ℤ/8)—1272no—
223AGL(1,ℤ/8)—2541no—
224AGL(1,ℤ/8)—2541no—
225#135—2542no—
226#135—2542no—
227C2×M16—1271no—
228C2×M16C2×D12711yescharacteristic, complemented
229#127—1272no—
230#124—1272no—
231#121—2542no—
232#120—2542no—
233#117C2×D12711yescharacteristic, complemented
234#104—1274no—
235#104C2×D12714yescomplemented
236#100—1274no—
237C24—1272no—
238C24—2542no—
239#41—21no—
240#41—21no—
241#41—1272no—
242#41—1272no—
243#41—2544no—
244#41—2544no—
245#41—2541no—
246#41—2541no—
247#41—2541no—
248#41—2541no—
249#41—2544no—
250#41—2544no—
251#41—2541no—
252#41—2541no—
253#41D50812yescomplemented
254#41C22×D12712yes—
255C2×Q8—1271no—
256C2×Q8—1271no—
257C2×Q8D50811yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—1271no—
260C2×D4—1272no—
261C2×D4—1272no—
262C2×D4—1271no—
263C2×D4—2544no—
264C2×D4—2544no—
265C2×D4—2542no—
266C2×D4—2544no—
267C2×D4—2548no—
268C2×D4—2548no—
269C2×D4—2544no—
270C2×D4—2541no—
271C2×D4—2542no—
272C2×D4—2541no—
273C2×D4—2544no—
274C2×D4—2541no—
275C2×D4—2541no—
276C2×D4—2541no—
277C2×D4—2541no—
278C2×D4D50811yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—2542no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4C22×D12711yescharacteristic
284Q16—2542no—
285Q16—2542no—
286SD16—2542no—
287SD16—2542no—
288SD16—2542no—
289SD16—2542no—
290D8—2542no—
291D8—2542no—
292M16—21no—
293M16—1272no—
294M16—2541no—
295M16C22×D12712yescomplemented
296C2×C8—21no—
297C2×C8—2541no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—2544no—
301C42C22×D12712yescomplemented
302C23—21no—
303C23—1274no—
304C23—10164no—
305C23—10162no—
306C23—2548no—
307C23—2542no—
308C23—2541no—
309C23—2541no—
310C23—2542no—
311C23D4×D12711yescharacteristic
312Q8—21no—
313Q8—1272no—
314Q8—2541no—
315Q8—2541no—
316Q8—2541no—
317Q8C2×D50812yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—1272no—
322D4—10164no—
323D4—5081no—
324D4—5081no—
325D4—2548no—
326D4—2548no—
327D4—2542no—
328D4—2542no—
329D4—2541no—
330D4—2542no—
331D4—2541no—
332D4—2542no—
333D4—2541no—
334D4—2542no—
335D4—2542no—
336D4C2×D50812yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—5081no—
342C2×C4—5081no—
343C2×C4—2548no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2542no—
347C2×C4—2542no—
348C2×C4D4×D12711yescharacteristic
349C2×C4D4×D12711yescharacteristic
350C2×C4D4×D12711yescharacteristic
351C2×C4C2×D50811yescharacteristic
352C2×C4C2×D50811yescharacteristic
353C2×C4C23×D12711yescharacteristic
354C8—22no—
355C8—2542no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—10162no—
360C22—10168no—
361C22—5082no—
362C22—5084no—
363C22—5082no—
364C22—5084no—
365C22—5081no—
366C22—2542no—
367C22—2544no—
368C22#1686117911yescharacteristic
369C22#1686117911yescharacteristic
370C22C2×D4×D12711yescharacteristic
371C4—22no—
372C4—42no—
373C4—5081no—
374C4—2542no—
375C4#1686117911yescharacteristic
376C4#1686117911yescharacteristic
377C4C2×D4×D12711yescharacteristic
378C4C22×D50811yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—10161no—
384C2—5082no—
385C2—5084no—
386C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#421×C127C2
socleC2×C127C2×#120
hypercenter#421D127
hypocenterC127#3019
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. C2×#16861255C2semidirect
2. #22550676C2semidirect
3. #22550674C2semidirect
4. #22550445C2semidirect
5. #22550055C2semidirect
6. #22550045C2semidirect
7. #22549894C2semidirect
8. C2×#16861125C2semidirect
9. #421×C127C2semidirect
17. C2×#14523248C22semidirect
23. #16861242C22semidirect
25. #16861239C22semidirect
27. #16861237C22semidirect
29. #16861234C22semidirect
32. C2×#14523232C22semidirect
33. #16861206C22semidirect
34. #16861197C22semidirect
37. #16861137C22semidirect
38. #16861125C22semidirect
40. C2×M16×C127C22semidirect
41. #117×C127C22semidirect
42. #104×C127C22semidirect
55. #14523248D4semidirect
57. C2×Dic254D4semidirect
58. C2×Dic254C23semidirect
68. C2×D508D4semidirect
70. C2×D508C23semidirect
90. #14523232C23semidirect
95. #41×C127D4semidirect
97. C2×Q8×C127D4semidirect
99. C2×D4×C127D4semidirect
103. M16×C127C23semidirect
107. C42×C127C23semidirect
194. C127#3019semidirect
202. #421D127semidirect
228. C2×M16C2×D127semidirect
233. #117C2×D127semidirect
235. #104C2×D127semidirect
253. #41D508semidirect
257. C2×Q8D508semidirect
278. C2×D4D508semidirect
295. M16C22×D127semidirect
301. C42C22×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
#225506762
#225506742
#225504452
#225500552
#225500452
#225498942
C2×#168611252
#421×C1272
#3019127