Finite Group Databasebeta

← #30650246

Subgroups — #30650246

Order 16256.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861238C211yesmaximal, characteristic, complemented
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550541C211yesmaximal, characteristic, complemented
4#22550521C211yesmaximal, characteristic
5#22550268C211yesmaximal, characteristic, complemented
6#22550125C211yesmaximal, characteristic, complemented
7#22550116C211yesmaximal, characteristic, complemented
8#22550099C211yesmaximal, characteristic, complemented
9#22550035C211yesmaximal, characteristic, complemented
10#22550033C211yesmaximal, characteristic, complemented
11#22550025C211yesmaximal, characteristic, complemented
12#22550016C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#461×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic, complemented
19C2×D4×D127C2211yescharacteristic
20#16861238—24no—
21C2×Dic508C2211yescharacteristic, complemented
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523238C2211yescharacteristic, complemented
24C2×#14523237C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523235C2211yescharacteristic
28SD16×D127—24no—
29C2×#14523231C2211yescharacteristic, complemented
30C2×C8×D127C2211yescharacteristic
31#16861208—21no—
32#16861197C2211yescharacteristic
33#16861193—21no—
34#16861189C2211yescharacteristic
35#16861187—21no—
36#16861187—21no—
37#16861185C2211yescharacteristic
38#16861185C2211yescharacteristic, complemented
39#16861184C2211yescharacteristic
40#16861183—21no—
41#16861179—21no—
42C4×Dic254C2211yescharacteristic, complemented
43C4×D508C2211yescharacteristic
44#16861171—21no—
45C2×(C2×D127)⋊C4—21no—
46#16861142C2211yescharacteristic, complemented
47#16861134C2211yescharacteristic, complemented
48#16861132C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic
50#16861130C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861120C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56C4⋊Q8×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#103×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523247—22no—
63#14523247—41no—
64C2×Dic254—21no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—22no—
70Q8×D127—41no—
71Q8×D127—41no—
72C2×D508—21no—
73C2×D508D411yescharacteristic
74C2×D508C2311yescharacteristic
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77D4×D127—22no—
78D4×D127—41no—
79C22×Dic127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127C2311yescharacteristic
83C2×C4×D127C2311yescharacteristic
84Dic508—22no—
85#14523238—22no—
86#14523238—41no—
87#14523237—22no—
88#14523236D414yescomplemented
89#14523235—22no—
90#14523235—41no—
91#14523231—22no—
92C2×C127⋊C8C2311yescharacteristic
93C2×C127⋊C8C2311yescharacteristic
94C8×D127—22no—
95C4⋊Dic127—21no—
96C4⋊Dic127D411yescharacteristic
97C4⋊Dic127C2311yescharacteristic
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108(C2×D127)⋊C4D411yescharacteristic
109C4×Dic127—21no—
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114SD16×C127—22no—
115SD16×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127—21no—
119C4⋊C4×C127—21no—
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—42no—
126C22×D127—81no—
127C22×D127C2×D411yescharacteristic
128C22×D127C2×D411yescharacteristic
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—42no—
133Dic254—42no—
134Dic254C2×D412yes—
135D508—22no—
136D508—21no—
137D508—21no—
138D508—81no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—42no—
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C4×D127—21no—
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—21no—
151C4×D127—42no—
152C4×D127—42no—
153C127⋊C8—21no—
154C127⋊C8C2×D412yes—
155C23×C127—21no—
156Q8×C127—21no—
157Q8×C127—21no—
158Q8×C127—21no—
159Q8×C127—21no—
160D4×C127—21no—
161D4×C127C2×D412yes—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—81no—
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2411yescharacteristic
169C8×C127—21no—
170C8×C127—21no—
171C22×C127—22no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127C22×D411yescharacteristic
175C2×D127—21no—
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—21no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—84no—
184C2×D127#13712yescomplemented
185C4×C127—21no—
186C4×C127—21no—
187C4×C127—41no—
188C4×C127—41no—
189C4×C127—41no—
190C4×C127C22×D411yescharacteristic
191C4×C127H(5,𝔽2)11yescharacteristic
192Dic127—21no—
193Dic127—21no—
194Dic127—22no—
195Dic127—41no—
196Dic127—41no—
197C2×C127—42no—
198C2×C127D4211yescharacteristic
199C2×C127C2×#13711yescharacteristic
200C2×C127#57711yescharacteristic
201D127—22no—
202D127—41no—
203D127—81no—
204#3266—1271nomaximal, Sylow
205C127#326611yescharacteristic, complemented, Sylow
206C2×#137—1271no—
207C22×SD16—1271no—
208D4×Q8—1271no—
209#546—1271no—
210#497—1271no—
211#468—1271no—
212#467—1271no—
213#464—1271no—
214#461D12711yescharacteristic, complemented
215#452—1271no—
216#451—1271no—
217#450—1271no—
218#448—1271no—
219#440—1271no—
220#435—1271no—
221C2×#41—1271no—
222C22×Q8—1271no—
223C22×Q8—1271no—
224C22×D4—1271no—
225#137—2544no—
226C2×Q16—1271no—
227C2×Q16—1271no—
228C2×Q16—1271no—
229C2×SD16—1271no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×SD16—1271no—
233C2×SD16—2544no—
234C2×SD16C2×D12711yescharacteristic
235C2×SD16C2×D12711yescharacteristic
236C2×M16—1271no—
237C22×C8—1271no—
238C4⋊Q8—2541no—
239C4⋊Q8C2×D12711yescharacteristic, complemented
240#124—1271no—
241#123—2541no—
242C22⋊Q8—1271no—
243C22⋊Q8—1271no—
244C22⋊Q8—1271no—
245C22⋊Q8—2541no—
246C22⋊Q8—2541no—
247#121—1271no—
248#121—2541no—
249#120—2541no—
250C4×Q8—1271no—
251C4×D4—1271no—
252C4×D4—2541no—
253C4×D4C2×D12711yescharacteristic
254C2×C22⋊C4—2541no—
255#107—1271no—
256C4⋊C8C2×D12711yescharacteristic, complemented
257#103—1271no—
258#103—1271no—
259#103C2×D12711yescharacteristic, complemented
260#103C2×D12711yescharacteristic, complemented
261#102—1271no—
262#102—1271no—
263C22⋊C8—1271no—
264C22⋊C8—1271no—
265#97—1271no—
266C24—2541no—
267#41—5081no—
268#41—2542no—
269C2×Q8—1271no—
270C2×Q8—1271no—
271C2×Q8—5081no—
272C2×Q8—5081no—
273C2×Q8—2541no—
274C2×Q8—2541no—
275C2×Q8—2542no—
276C2×Q8—2542no—
277C2×Q8C22×D12711yescharacteristic
278C2×Q8C22×D12711yescharacteristic
279C2×D4—1271no—
280C2×D4—1271no—
281C2×D4—5081no—
282C2×D4—2541no—
283C2×D4—2541no—
284C2×D4—2541no—
285C2×D4—2542no—
286C2×D4C22×D12711yescharacteristic
287C22×C4—21no—
288C22×C4—1271no—
289C22×C4—1271no—
290C22×C4—2541no—
291C22×C4—2541no—
292C22×C4—2541no—
293Q16—5081no—
294Q16—2542no—
295Q16—2542no—
296SD16—22no—
297SD16—22no—
298SD16—1274no—
299SD16—5081no—
300SD16—2542no—
301SD16—2542no—
302M16—2542no—
303C2×C8—1271no—
304C2×C8—1271no—
305C2×C8—2542no—
306C2×C8C22×D12711yescharacteristic
307C2×C8C22×D12711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—1271no—
311C4⋊C4—1271no—
312C4⋊C4—2541no—
313C4⋊C4—2541no—
314C4⋊C4—2541no—
315C4⋊C4—2541no—
316C4⋊C4C22×D12711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—1271no—
319C22⋊C4—1271no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C42—2541no—
327C42C22×D12711yescharacteristic
328C23—21no—
329C23—1271no—
330C23—1271no—
331C23—10161no—
332C23—5082no—
333C23—5082no—
334C23—2541no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—1272no—
340Q8—5082no—
341Q8—5082no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345D4—21no—
346D4—10161no—
347D4—5082no—
348D4—5082no—
349D4—2542no—
350D4—2541no—
351D4—2541no—
352D4D4×D12712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—1271no—
358C2×C4—1271no—
359C2×C4—5082no—
360C2×C4—5082no—
361C2×C4—5082no—
362C2×C4—2541no—
363C2×C4—2541no—
364C2×C4—2541no—
365C2×C4—2541no—
366C2×C4—2541no—
367C2×C4—2541no—
368C2×C4—2541no—
369C2×C4D4×D12711yescharacteristic
370C2×C4D4×D12711yescharacteristic
371C2×C4C23×D12711yescharacteristic
372C8—21no—
373C8—21no—
374C8—1272no—
375C8—2541no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—1272no—
380C22—10164no—
381C22—5081no—
382C22—5081no—
383C22—5081no—
384C22—2541no—
385C22—2541no—
386C22—2541no—
387C22—2541no—
388C22—2541no—
389C22C2×D4×D12711yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—5081no—
396C4—5081no—
397C4—2541no—
398C4—2541no—
399C4—2542no—
400C4C2×D4×D12711yescharacteristic
401C4#1686125511yescharacteristic
402C2—42no—
403C2—10161no—
404C2—5081no—
405C2—2542no—
406C2#2255051011yescharacteristic
407C2#137×D12711yescharacteristic
408C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
prime radical#22550521C2
hypercenter#461D127
hypocenterC127#3266
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. C2×SD16×D127C2semidirect
3. #22550541C2semidirect
5. #22550268C2semidirect
6. #22550125C2semidirect
7. #22550116C2semidirect
8. #22550099C2semidirect
9. #22550035C2semidirect
10. #22550033C2semidirect
11. #22550025C2semidirect
12. #22550016C2semidirect
13. #22549976C2semidirect
14. #22549946C2semidirect
15. #461×C127C2semidirect
18. C2×Q8×D127C22semidirect
21. C2×Dic508C22semidirect
22. C2×#14523238C22semidirect
23. C2×#14523238C22semidirect
29. C2×#14523231C22semidirect
38. #16861185C22semidirect
42. C4×Dic254C22semidirect
46. #16861142C22semidirect
47. #16861134C22semidirect
48. #16861132C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861120C22semidirect
56. C4⋊Q8×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
60. #103×C127C22semidirect
88. #14523236D4semidirect
184. C2×D127#137semidirect
205. C127#3266semidirect
214. #461D127semidirect
239. C4⋊Q8C2×D127semidirect
256. C4⋊C8C2×D127semidirect
259. #103C2×D127semidirect
260. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C2×SD16×D1272
#225505412
#225505212
#225502682
#225501252
#225501162
#225500992
#225500352
#225500332
#225500252
#225500162
#225499762
#225499462
#461×C1272
#3266127