Finite Group Databasebeta

← #30650550

Subgroups — #30650550

Order 16256.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550581C211yesmaximal, characteristic, complemented
2D4×C22⋊D127C211yesmaximal, characteristic
3#22550221C211yesmaximal, characteristic, complemented
4#22550194C211yesmaximal, characteristic, complemented
5#22550193C211yesmaximal, characteristic, complemented
6#22550164C211yesmaximal, characteristic, complemented
7#22550105C211yesmaximal, characteristic, complemented
8#22550104C211yesmaximal, characteristic, complemented
9#22550075C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22550027C211yesmaximal, characteristic, complemented
12#22549967C211yesmaximal, characteristic, complemented
13C8×C22⋊D127C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15C2×#102×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127—21no—
17C2×D4×D127C2211yescharacteristic
18C2×#14523237C2211yescharacteristic, complemented
19C2×#14523236C2211yescharacteristic
20C2×#14523236C2211yescharacteristic, complemented
21C2×C8×D127C2211yescharacteristic, complemented
22#16861208C2211yescharacteristic, complemented
23#16861205C2211yescharacteristic
24#16861199—21no—
25#16861189C2211yescharacteristic, complemented
26#16861188—21no—
27#16861188C2211yescharacteristic, complemented
28#16861187—21no—
29#16861183C2211yescharacteristic
30#16861181—21no—
31#16861180C2211yescharacteristic
32#16861178—21no—
33#16861177—21no—
34#16861174C2211yescharacteristic, complemented
35D4×Dic127C2211yescharacteristic
36C4×C22⋊D127—21no—
37C4×C22⋊D127C2211yescharacteristic
38C4⋊C4×D127C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic, complemented
47#16861129C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic, complemented
49C22⋊(C127⋊C8)C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic, complemented
54C2×C4⋊C4×C127C2211yescharacteristic, complemented
55#102×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C2×Dic254—21no—
61C2×Dic254C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—44no—
69C2×C22⋊D127D411yescharacteristic
70D4×D127—22no—
71D4×D127—41no—
72C22×Dic127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2311yescharacteristic
75#14523237—41no—
76#14523236—22no—
77#14523236—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4D411yescharacteristic
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127D411yescharacteristic
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127—21no—
97C4×Dic127C2311yescharacteristic
98C24×C127—21no—
99C2×D4×C127—22no—
100C2×D4×C127—21no—
101C2×D4×C127—41no—
102C2×D4×C127C2311yescharacteristic
103C2×D4×C127C2311yescharacteristic
104C22×C4×C127—21no—
105C22×C4×C127C2311yescharacteristic
106C2×C8×C127—41no—
107C2×C8×C127C2311yescharacteristic
108C2×C8×C127C2311yescharacteristic
109C4⋊C4×C127—21no—
110C4⋊C4×C127C2311yescharacteristic
111C4⋊C4×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117Dic254—21no—
118Dic254—21no—
119Dic254—81no—
120D508—21no—
121D508—21no—
122C22⋊D127—42no—
123C22⋊D127—42no—
124C22⋊D127—42no—
125C22⋊D127—42no—
126C22⋊D127—81no—
127C22⋊D127SD1614yescomplemented
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127—42no—
132C2×Dic127C2×D411yescharacteristic
133C2×Dic127C2×D411yescharacteristic
134C2×Dic127C2×D411yescharacteristic
135C4×D127—21no—
136C4×D127—21no—
137C4×D127—81no—
138C127⋊C8—21no—
139C127⋊C8—21no—
140C23×C127—21no—
141C23×C127—21no—
142C23×C127—42no—
143C23×C127—42no—
144C23×C127—81no—
145C23×C127C2×D411yescharacteristic
146D4×C127—21no—
147D4×C127—21no—
148D4×C127—21no—
149D4×C127—42no—
150D4×C127#4112yes—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—81no—
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2411yescharacteristic
158C8×C127—21no—
159C8×C127—21no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—22no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—84no—
169C22×C127C2×SD1612yes—
170C22×C127C22×D411yescharacteristic
171C2×D127—21no—
172C2×D127—21no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—84no—
177C2×D127C2×SD1612yes—
178C4×C127—21no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127C2×#4111yescharacteristic
182C4×C127H(5,𝔽2)11yescharacteristic
183Dic127—21no—
184Dic127—21no—
185Dic127—41no—
186Dic127—41no—
187Dic127—41no—
188Dic127C2×SD1612yes—
189C2×C127—22no—
190C2×C127—42no—
191C2×C127—81no—
192C2×C127#54611yescharacteristic
193C2×C127C22×SD1611yescharacteristic
194C2×C127#57611yescharacteristic
195D127—22no—
196D127—81no—
197#3295—1271nomaximal, Sylow
198C127#329511yescharacteristic, complemented, Sylow
199#554—1271no—
200D42—1271no—
201#486—1271no—
202#481—1271no—
203#481—1271no—
204#476—1271no—
205#465—1271no—
206#465—1271no—
207#460—1271no—
208#450—1271no—
209#450—1271no—
210C4×SD16—1271no—
211C8×D4—1271no—
212C2×#102—1271no—
213C2×#102D12711yescharacteristic, complemented
214C22×D4—1271no—
215C22×D4—2541no—
216C22×D4C2×D12711yescharacteristic
217C2×SD16—1271no—
218C2×SD16—1271no—
219C2×SD16—1271no—
220C22×C8—1271no—
221C22×C8C2×D12711yescharacteristic, complemented
222C4⋊Q8—1271no—
223#127—1271no—
224#125—2541no—
225C22⋊Q8—1271no—
226C22⋊Q8—1271no—
227C22⋊Q8—2541no—
228C22⋊Q8—2541no—
229#121—1271no—
230#121—1271no—
231#121—2541no—
232#120—2541no—
233#120—2541no—
234C4×Q8—1271no—
235C4×D4—1271no—
236C4×D4—1271no—
237C4×D4—2541no—
238C2×C4⋊C4—1271no—
239C2×C4⋊C4C2×D12711yescharacteristic, complemented
240#106—1271no—
241#106—1271no—
242#106—1271no—
243C4⋊C8—1271no—
244#103—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102C2×D12711yescharacteristic
251#102C2×D12711yescharacteristic
252#102C2×D12711yescharacteristic, complemented
253#102C2×D12711yescharacteristic, complemented
254C22⋊C8—1271no—
255C22⋊C8—1271no—
256C4×C8—1271no—
257C24—21no—
258C24—2541no—
259C2×Q8—1271no—
260C2×Q8—2541no—
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—1271no—
265C2×D4—1271no—
266C2×D4—5084no—
267C2×D4—5081no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4—2541no—
271C2×D4—2541no—
272C2×D4—2541no—
273C2×D4—2542no—
274C2×D4C22×D12711yescharacteristic
275C2×D4C22×D12711yescharacteristic
276C22×C4—21no—
277C22×C4—1271no—
278C22×C4—2541no—
279C22×C4—2541no—
280C22×C4C22×D12711yescharacteristic
281SD16—5081no—
282SD16—5081no—
283SD16—2542no—
284C2×C8—41no—
285C2×C8—1271no—
286C2×C8—1271no—
287C2×C8—5081no—
288C2×C8C22×D12711yescharacteristic
289C2×C8C22×D12711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—1271no—
292C4⋊C4—1271no—
293C4⋊C4—1271no—
294C4⋊C4—2541no—
295C4⋊C4—2541no—
296C4⋊C4—2541no—
297C4⋊C4—2541no—
298C4⋊C4—2541no—
299C4⋊C4C22×D12711yescharacteristic
300C4⋊C4C22×D12711yescharacteristic
301C22⋊C4—1271no—
302C22⋊C4—1271no—
303C22⋊C4—2541no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C22⋊C4—2541no—
307C22⋊C4—2541no—
308C42—1271no—
309C42—2541no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—1271no—
316C23—10161no—
317C23—5082no—
318C23—5082no—
319C23—2541no—
320C23D4×D12711yescharacteristic
321Q8—10161no—
322Q8—2541no—
323Q8—2541no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—1274no—
329D4—10161no—
330D4—5082no—
331D4—5082no—
332D4—5082no—
333D4—5082no—
334D4—2541no—
335D4—2541no—
336D4#1452324812yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—81no—
342C2×C4—1271no—
343C2×C4—1271no—
344C2×C4—1271no—
345C2×C4—10161no—
346C2×C4—5082no—
347C2×C4—2541no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4D4×D12711yescharacteristic
353C2×C4C23×D12711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2541no—
357C8—2541no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—1272no—
368C22—10164no—
369C22—5081no—
370C22—5081no—
371C22—5081no—
372C22—2541no—
373C22—2541no—
374C22SD16×D12712yes—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—1272no—
380C4—5081no—
381C4—5081no—
382C4—5081no—
383C4—2541no—
384C4—2541no—
385C4C2×#1452324811yescharacteristic
386C4#1686125411yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—10161no—
391C2—2542no—
392C2#2255051611yescharacteristic
393C2C2×SD16×D12711yescharacteristic
394C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC22×C127C22×D4
prime radicalD4×C22⋊D127C2
hypercenterC2×#102D127
hypocenterC127#3295
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #22550581C2semidirect
3. #22550221C2semidirect
4. #22550194C2semidirect
5. #22550193C2semidirect
6. #22550164C2semidirect
7. #22550105C2semidirect
8. #22550104C2semidirect
9. #22550075C2semidirect
10. #22550028C2semidirect
11. #22550027C2semidirect
12. #22549967C2semidirect
13. C8×C22⋊D127C2semidirect
14. #102×D127C2semidirect
15. C2×#102×C127C2semidirect
18. C2×#14523237C22semidirect
20. C2×#14523236C22semidirect
21. C2×C8×D127C22semidirect
22. #16861208C22semidirect
25. #16861189C22semidirect
27. #16861188C22semidirect
34. #16861174C22semidirect
38. C4⋊C4×D127C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. #16861140C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861131C22semidirect
46. #16861130C22semidirect
47. #16861129C22semidirect
48. #16861129C22semidirect
49. C22⋊(C127⋊C8)C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. C8×Dic127C22semidirect
53. C22×C8×C127C22semidirect
54. C2×C4⋊C4×C127C22semidirect
57. #102×C127C22semidirect
58. #102×C127C22semidirect
127. C22⋊D127SD16semidirect
198. C127#3295semidirect
213. C2×#102D127semidirect
221. C22×C8C2×D127semidirect
239. C2×C4⋊C4C2×D127semidirect
252. #102C2×D127semidirect
253. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505812
D4×C22⋊D1272
#225502212
#225501942
#225501932
#225501642
#225501052
#225501042
#225500752
#225500282
#225500272
#225499672
C8×C22⋊D1272
#102×D1272
C2×#102×C1272
#3295127