Finite Group Databasebeta

← #30648874

Subgroups — #30648874

Order 16256.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550421C211yesmaximal, characteristic, complemented
2#41×Dic127C211yesmaximal, characteristic, complemented
3#22550201C211yesmaximal, characteristic, complemented
4#22550197C211yesmaximal, characteristic, complemented
5#22550159C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550074C211yesmaximal, characteristic, complemented
8#22550069C211yesmaximal, characteristic, complemented
9#22550021C211yesmaximal, characteristic, complemented
10#22549988C211yesmaximal, characteristic, complemented
11#22549985C211yesmaximal, characteristic, complemented
12#22549979C211yesmaximal, characteristic, complemented
13#22549977C211yesmaximal, characteristic, complemented
14#22549930C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×#14523235C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19#16861208C2211yescharacteristic
20#16861205C2211yescharacteristic
21#16861195—21no—
22#16861183C2211yescharacteristic
23#16861181—21no—
24Q8×Dic127C2211yescharacteristic
25Q8×Dic127C2211yescharacteristic
26D4×Dic127—21no—
27D4×Dic127—21no—
28D4×Dic127C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31#16861171C2211yescharacteristic
32(C22⋊D127)⋊C4—21no—
33#16861168—21no—
34#16861168C2211yescharacteristic
35C2×C4×Dic127—21no—
36C2×C4×Dic127C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50#16861119C2211yescharacteristic
51#16861119C2211yescharacteristic
52C2×#41×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×D8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59#14523247—44no—
60C2×Dic254—21no—
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C22×Dic127—21no—
66C22×Dic127—21no—
67C22×Dic127—21no—
68C22×Dic127D411yescharacteristic
69C22×Dic127D411yescharacteristic
70C2×C4×D127—21no—
71#14523235—22no—
72#14523233—22no—
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C4⋊Dic127—21no—
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81C22⋊Dic127—21no—
82C22⋊Dic127—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87C4×Dic127—21no—
88C4×Dic127—21no—
89C4×Dic127—21no—
90C4×Dic127D411yescharacteristic
91C4×Dic127D411yescharacteristic
92C4×Dic127C2311yescharacteristic
93C4×Dic127C2311yescharacteristic
94#41×C127—22no—
95#41×C127—22no—
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127—21no—
98C2×D4×C127—21no—
99C2×D4×C127C2311yescharacteristic
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127—21no—
102C22×C4×C127C2311yescharacteristic
103SD16×C127—22no—
104D8×C127—22no—
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C22⋊C4×C127—21no—
109C22×D127—21no—
110Dic254—42no—
111D508—21no—
112D508—21no—
113C22⋊D127—42no—
114C22⋊D127—42no—
115C22⋊D127—42no—
116C2×Dic127—22no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—41no—
121C2×Dic127—42no—
122C2×Dic127—42no—
123C2×Dic127—42no—
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C4×D127—42no—
129C127⋊C8—21no—
130C127⋊C8—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127C2×D411yescharacteristic
134Q8×C127—21no—
135Q8×C127#4112yescomplemented
136D4×C127—24no—
137D4×C127—21no—
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—42no—
141D4×C127#4112yescomplemented
142C2×C4×C127—24no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
149C8×C127—21no—
150C8×C127—21no—
151C22×C127—21no—
152C22×C127—22no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127C22×D411yescharacteristic
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C4×C127—21no—
165C4×C127—22no—
166C4×C127—41no—
167C4×C127C2×#4111yescharacteristic
168C4×C127C2×#4111yescharacteristic
169Dic127—21no—
170Dic127—21no—
171Dic127—21no—
172Dic127—41no—
173Dic127—41no—
174Dic127—41no—
175Dic127AGL(1,ℤ/8)12yes—
176C2×C127—41no—
177C2×C127—42no—
178C2×C127—81no—
179C2×C127#52511yescharacteristic
180C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C127#57511yescharacteristic
182D127—81no—
183#3115—1271nomaximal, Sylow
184C127#311511yescharacteristic, complemented, Sylow
185#532—1271no—
186C4×#41—1271no—
187#482—1271no—
188#482—1271no—
189#475—1271no—
190#474—1271no—
191#460—1271no—
192#459—1271no—
193#449—1271no—
194#449D12711yescharacteristic, complemented
195#442—1271no—
196#442—1271no—
197#440—1271no—
198#440—1271no—
199#432—1271no—
200C2×#41—2541no—
201C2×#41C2×D12711yescharacteristic
202C2×SD16—1271no—
203C2×SD16C2×D12711yescharacteristic
204C2×D8—1271no—
205C2×D8C2×D12711yescharacteristic
206C4⋊Q8—1271no—
207#127—1271no—
208#124—2541no—
209#121—1271no—
210#121—2541no—
211#121C2×D12711yescharacteristic
212C4×Q8—1271no—
213C4×Q8—1271no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—1271no—
217C4×D4—1271no—
218C4×D4—2541no—
219C4×D4—2541no—
220C4×D4—2541no—
221#117—1271no—
222#117—2541no—
223C2×C42—1271no—
224C2×C42—2541no—
225#107—1271no—
226#107—1271no—
227#106—1271no—
228#106—1271no—
229C4⋊C8—1271no—
230C4⋊C8—1271no—
231#103—1271no—
232#103C2×D12711yescharacteristic
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102—1271no—
238#102C2×D12711yescharacteristic
239C22⋊C8—1271no—
240C22⋊C8C2×D12711yescharacteristic
241#97—1271no—
242#97—1271no—
243#41—22no—
244#41—22no—
245#41—5084no—
246C2×Q8—2541no—
247C2×Q8C22×D12711yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—1271no—
251C2×D4—2541no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4C22×D12711yescharacteristic
255C2×D4C22×D12711yescharacteristic
256C22×C4—21no—
257C22×C4—1271no—
258C22×C4—1271no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4C22×D12711yescharacteristic
264SD16—22no—
265SD16—2542no—
266D8—22no—
267D8—2542no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4C22×D12711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C42—1271no—
287C42—1271no—
288C42—1271no—
289C42—1271no—
290C42—2541no—
291C42—2541no—
292C42—2541no—
293C23—21no—
294C23—21no—
295C23—2541no—
296C23D4×D12711yescharacteristic
297Q8—21no—
298Q8—5082no—
299Q8#1452324712yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—5082no—
306D4—5082no—
307D4—5082no—
308D4—2541no—
309D4—2541no—
310D4#1452324712yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—1271no—
317C2×C4—1271no—
318C2×C4—1271no—
319C2×C4—1271no—
320C2×C4—5082no—
321C2×C4—5082no—
322C2×C4—5082no—
323C2×C4—5082no—
324C2×C4—5081no—
325C2×C4—2542no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4D4×D12711yescharacteristic
330C2×C4C23×D12711yescharacteristic
331C8—21no—
332C8—21no—
333C8—2541no—
334C8—2541no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—5081no—
345C22—5081no—
346C22—5081no—
347C22C2×D4×D12711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—1272no—
352C4—5081no—
353C4—5081no—
354C4—5081no—
355C4—2541no—
356C4—2541no—
357C4—2541no—
358C4C2×#1452324711yescharacteristic
359C4C2×#1452324711yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—10161no—
364C2#2255038211yescharacteristic
365C2AGL(1,ℤ/8)×D12711yescharacteristic
366C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3115
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #22550421C2semidirect
2. #41×Dic127C2semidirect
3. #22550201C2semidirect
4. #22550197C2semidirect
5. #22550159C2semidirect
6. #22550154C2semidirect
7. #22550074C2semidirect
8. #22550069C2semidirect
9. #22550021C2semidirect
10. #22549988C2semidirect
11. #22549985C2semidirect
12. #22549979C2semidirect
13. #22549977C2semidirect
14. #22549930C2semidirect
15. #449×C127C2semidirect
135. Q8×C127#41semidirect
141. D4×C127#41semidirect
184. C127#3115semidirect
194. #449D127semidirect
299. Q8#14523247semidirect
310. D4#14523247semidirect

Maximal subgroups

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

subgroupindex
#225504212
#41×Dic1272
#225502012
#225501972
#225501592
#225501542
#225500742
#225500692
#225500212
#225499882
#225499852
#225499792
#225499772
#225499302
#449×C1272
#3115127