Finite Group Databasebeta

← #30648926

Subgroups — #30648926

Order 16256.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550480C211yesmaximal, characteristic, complemented
2#22550355C211yesmaximal, characteristic, complemented
3#22550193C211yesmaximal, characteristic, complemented
4#22550188C211yesmaximal, characteristic, complemented
5#22550184C211yesmaximal, characteristic, complemented
6#22550176C211yesmaximal, characteristic, complemented
7#22550073C211yesmaximal, characteristic, complemented
8#22550018C211yesmaximal, characteristic, complemented
9#22550018C211yesmaximal, characteristic, complemented
10#22549988C211yesmaximal, characteristic, complemented
11#22549973C211yesmaximal, characteristic, complemented
12#22549967C211yesmaximal, characteristic, complemented
13D8×Dic127C211yesmaximal, characteristic, complemented
14#22549936C211yesmaximal, characteristic, complemented
15#459×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19#16861201C2211yescharacteristic
20#16861190—21no—
21#16861190—21no—
22#16861181—21no—
23#16861181—21no—
24#16861180C2211yescharacteristic
25#16861180C2211yescharacteristic
26C4×Dic254C2211yescharacteristic
27#16861174C2211yescharacteristic
28D4×Dic127C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31C4×C22⋊D127—21no—
32(C22⋊D127)⋊C4—21no—
33#16861166C2211yescharacteristic
34#16861165—21no—
35C2×Dic127⋊C4—21no—
36C4⋊C4×D127C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41C4⋊(C127⋊C8)C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861130C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48(C2×D127)⋊C8C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50#16861119C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54#127×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—22no—
61C2×Dic254C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C22×Dic127—21no—
66C22×Dic127—21no—
67C22×Dic127—21no—
68C22×Dic127—21no—
69C2×C4×D127—21no—
70C2×C4×D127C2311yescharacteristic
71#14523236—22no—
72#14523236—22no—
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4D411yescharacteristic
84Dic127⋊C4D411yescharacteristic
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90(C2×D127)⋊C4D411yescharacteristic
91(C2×D127)⋊C4D411yescharacteristic
92C4×Dic127—21no—
93C4×Dic127—21no—
94C4×Dic127C2311yescharacteristic
95C2×D4×C127—21no—
96C2×D4×C127—21no—
97C2×D4×C127C2311yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C22×C4×C127—21no—
101D8×C127—22no—
102D8×C127—22no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127C2×D411yescharacteristic
109Dic254—21no—
110Dic254#4112yes—
111C22⋊D127—24no—
112C22⋊D127—81no—
113C22⋊D127—81no—
114C2×Dic127—24no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—81no—
120C2×Dic127—81no—
121C2×Dic127—81no—
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C4×D127—21no—
126C4×D127—21no—
127C4×D127—81no—
128C127⋊C8—21no—
129C127⋊C8—21no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133D4×C127—21no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—81no—
139D4×C127—81no—
140D4×C127#4112yes—
141C2×C4×C127—21no—
142C2×C4×C127—81no—
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—22no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127C22×D411yescharacteristic
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—21no—
161C4×C127—21no—
162C4×C127—21no—
163C4×C127—41no—
164C4×C127C2×#4111yescharacteristic
165C4×C127C2×#4111yescharacteristic
166Dic127—21no—
167Dic127—22no—
168Dic127—21no—
169Dic127—21no—
170Dic127—41no—
171Dic127—41no—
172Dic127—41no—
173C2×C127—42no—
174C2×C127—81no—
175C2×C127—81no—
176C2×C127#52511yescharacteristic
177C2×C127#57611yescharacteristic
178C2×C127#57711yescharacteristic
179D127—41no—
180#3119—1271nomaximal, Sylow
181C127#311911yescharacteristic, complemented, Sylow
182#540—1271no—
183#520—1271no—
184#481—1271no—
185#480—1271no—
186#479—1271no—
187#478—1271no—
188#460—1271no—
189#459D12711yescharacteristic, complemented
190#449—1271no—
191#449—1271no—
192#442—1271no—
193#440—1271no—
194C4×SD16—1271no—
195C4×D8—1271no—
196#433—1271no—
197C2×#41—1271no—
198C2×SD16—1271no—
199C2×SD16—1271no—
200C2×D8C2×D12711yescharacteristic
201C2×D8C2×D12711yescharacteristic
202#127C2×D12711yescharacteristic
203#125—1271no—
204#123—2541no—
205#123—2541no—
206#121—1271no—
207#121—1271no—
208#121—2541no—
209#121—2541no—
210C4×Q8—1271no—
211C4×Q8—1271no—
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215C4×D4—2541no—
216C4×D4—2541no—
217C4×D4C2×D12711yescharacteristic
218#117—1271no—
219#117—2541no—
220C2×C4⋊C4—1271no—
221C2×C4⋊C4—2541no—
222#107—1271no—
223#107—1271no—
224#106—1271no—
225#106—1271no—
226C4⋊C8—1271no—
227C4⋊C8C2×D12711yescharacteristic
228#103—1271no—
229#103—1271no—
230#102—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102C2×D12711yescharacteristic
235#102C2×D12711yescharacteristic
236C22⋊C8—1271no—
237C22⋊C8—1271no—
238#97—1271no—
239C4×C8—1271no—
240#41—2542no—
241#41—2542no—
242C2×Q8—1271no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4—2541no—
248C2×D4C22×D12711yescharacteristic
249C2×D4C22×D12711yescharacteristic
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—1271no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4—2541no—
258SD16—2542no—
259SD16—2542no—
260D8—22no—
261D8—22no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8C22×D12711yescharacteristic
265C2×C8C22×D12711yescharacteristic
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—1271no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1271no—
279C22⋊C4—1271no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C42—1271no—
286C42—2541no—
287C42—2541no—
288C42C22×D12711yescharacteristic
289C23—21no—
290C23—21no—
291C23—21no—
292C23—1271no—
293Q8—1272no—
294Q8—2541no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—10161no—
303D4—10161no—
304D4—2544no—
305D4#1452324712yes—
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—1271no—
309C2×C4—1271no—
310C2×C4—1271no—
311C2×C4—10161no—
312C2×C4—10161no—
313C2×C4—10161no—
314C2×C4—10161no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2544no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4D4×D12711yescharacteristic
323C2×C4D4×D12711yescharacteristic
324C2×C4C23×D12711yescharacteristic
325C8—21no—
326C8—21no—
327C8—2541no—
328C8—2541no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—2541no—
339C22—2541no—
340C22—2541no—
341C22C2×D4×D12711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—5081no—
346C4—5081no—
347C4—5081no—
348C4—2541no—
349C4—2542no—
350C4—2541no—
351C4—2541no—
352C4#41×D12711yescharacteristic
353C4C2×#1452324711yescharacteristic
354C2—42no—
355C2—81no—
356C2—81no—
357C2—5081no—
358C2#2255038611yescharacteristic
359C2#2255067711yescharacteristic
360C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3119
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550480C2semidirect
2. #22550355C2semidirect
3. #22550193C2semidirect
4. #22550188C2semidirect
5. #22550184C2semidirect
6. #22550176C2semidirect
7. #22550073C2semidirect
8. #22550018C2semidirect
9. #22550018C2semidirect
10. #22549988C2semidirect
11. #22549973C2semidirect
12. #22549967C2semidirect
13. D8×Dic127C2semidirect
14. #22549936C2semidirect
15. #459×C127C2semidirect
181. C127#3119semidirect
189. #459D127semidirect

Maximal subgroups

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

subgroupindex
#225504802
#225503552
#225501932
#225501882
#225501842
#225501762
#225500732
#225500182
#225499882
#225499732
#225499672
D8×Dic1272
#225499362
#459×C1272
#3119127