Finite Group Databasebeta

← #30648922

Subgroups — #30648922

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#22550479C211yesmaximal, characteristic, complemented
2#22550359C211yesmaximal, characteristic, complemented
3#22550192C211yesmaximal, characteristic, complemented
4#22550190C211yesmaximal, characteristic, complemented
5#22550185C211yesmaximal, characteristic, complemented
6#22550174C211yesmaximal, characteristic, complemented
7#22550074C211yesmaximal, characteristic, complemented
8#22550069C211yesmaximal, characteristic, complemented
9#22550021C211yesmaximal, characteristic, complemented
10#22549985C211yesmaximal, characteristic, complemented
11#22549977C211yesmaximal, characteristic, complemented
12#22549961C211yesmaximal, characteristic, complemented
13D8×Dic127C211yesmaximal, characteristic, complemented
14#22549934C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523235C2211yescharacteristic
17C2×#14523233C2211yescharacteristic
18#16861205C2211yescharacteristic
19#16861200C2211yescharacteristic
20#16861193—21no—
21#16861191—21no—
22#16861183C2211yescharacteristic
23#16861181—21no—
24#16861180—21no—
25Q8×Dic127C2211yescharacteristic
26Q8×Dic127C2211yescharacteristic
27D4×Dic127—21no—
28D4×Dic127—21no—
29D4×Dic127C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32#16861171C2211yescharacteristic
33#16861168—21no—
34#16861168C2211yescharacteristic
35C2×C4⋊Dic127—21no—
36C2×C4⋊Dic127C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861140C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50#16861119C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×#41×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×D8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59C2×D508C2311yescharacteristic
60C2×C22⋊D127—21no—
61C2×C22⋊D127—21no—
62C2×C22⋊D127—21no—
63C22×Dic127—21no—
64C22×Dic127—21no—
65C22×Dic127—21no—
66C22×Dic127—21no—
67C2×C4×D127—21no—
68#14523235—22no—
69#14523233—22no—
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C4⋊Dic127—21no—
73C4⋊Dic127—21no—
74C4⋊Dic127—21no—
75C4⋊Dic127—21no—
76C4⋊Dic127D411yescharacteristic
77C4⋊Dic127D411yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82C22⋊Dic127—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127D411yescharacteristic
86C22⋊Dic127D411yescharacteristic
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89C4×Dic127—21no—
90C4×Dic127—21no—
91C4×Dic127C2311yescharacteristic
92C4×Dic127C2311yescharacteristic
93#41×C127—22no—
94#41×C127—22no—
95C2×Q8×C127C2311yescharacteristic
96C2×D4×C127—21no—
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C22×C4×C127—21no—
101C22×C4×C127C2311yescharacteristic
102SD16×C127—22no—
103D8×C127—22no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22×D127—21no—
109D508—21no—
110D508—21no—
111C22⋊D127—81no—
112C22⋊D127—81no—
113C22⋊D127—81no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—81no—
118C2×Dic127—81no—
119C2×Dic127—81no—
120C2×Dic127—81no—
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C4×D127—81no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127C2×D411yescharacteristic
131Q8×C127—21no—
132Q8×C127#4112yes—
133D4×C127—24no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—81no—
138D4×C127#4112yes—
139C2×C4×C127—24no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—21no—
149C22×C127—22no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127C22×D411yescharacteristic
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C4×C127—21no—
162C4×C127—22no—
163C4×C127—41no—
164C4×C127C2×#4111yescharacteristic
165C4×C127C2×#4111yescharacteristic
166Dic127—21no—
167Dic127—21no—
168Dic127—21no—
169Dic127—21no—
170Dic127—41no—
171Dic127—41no—
172Dic127—41no—
173C2×C127—41no—
174C2×C127—42no—
175C2×C127—81no—
176C2×C127#52511yescharacteristic
177C2×C127#57611yescharacteristic
178C2×C127#57711yescharacteristic
179D127—81no—
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#459—1271no—
190#449—1271no—
191#449D12711yescharacteristic, complemented
192#442—1271no—
193#440—1271no—
194C4×SD16—1271no—
195C4×D8—1271no—
196#433—1271no—
197C2×#41C2×D12711yescharacteristic
198C2×SD16—1271no—
199C2×SD16C2×D12711yescharacteristic
200C2×D8—1271no—
201C2×D8C2×D12711yescharacteristic
202#127—1271no—
203#125—1271no—
204#123—2541no—
205#123—2541no—
206#121—1271no—
207#121—2541no—
208#121—2541no—
209#121C2×D12711yescharacteristic
210C4×Q8—1271no—
211C4×Q8—1271no—
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—2541no—
217C4×D4—2541no—
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⋊C8—1271no—
228#103—1271no—
229#103C2×D12711yescharacteristic
230#102—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102C2×D12711yescharacteristic
236C22⋊C8—1271no—
237C22⋊C8C2×D12711yescharacteristic
238#97—1271no—
239C4×C8—1271no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D12711yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—1271no—
246C2×D4—2541no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4C22×D12711yescharacteristic
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—2541no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4C22×D12711yescharacteristic
258SD16—22no—
259SD16—2542no—
260D8—22no—
261D8—2542no—
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—1271no—
287C42—2541no—
288C42—2541no—
289C23—21no—
290C23—21no—
291C23—2541no—
292C23D4×D12711yescharacteristic
293Q8—21no—
294Q8#1452324712yes—
295D4—24no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—10161no—
301D4—10161no—
302D4—10161no—
303D4—2541no—
304D4—2541no—
305D4#1452324712yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—1271no—
314C2×C4—1271no—
315C2×C4—10161no—
316C2×C4—10161no—
317C2×C4—10161no—
318C2×C4—10161no—
319C2×C4—10161no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4D4×D12711yescharacteristic
324C2×C4C23×D12711yescharacteristic
325C8—21no—
326C8—21no—
327C8—2541no—
328C8—2541no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—5081no—
339C22—5081no—
340C22—5081no—
341C22C2×D4×D12711yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—5081no—
346C4—5081no—
347C4—5081no—
348C4—2541no—
349C4—2541no—
350C4—2541no—
351C4—2541no—
352C4C2×#1452324711yescharacteristic
353C4C2×#1452324711yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—10161no—
358C2#2255038211yescharacteristic
359C2#2255067811yescharacteristic
360C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3119
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. #22550479C2semidirect
2. #22550359C2semidirect
3. #22550192C2semidirect
4. #22550190C2semidirect
5. #22550185C2semidirect
6. #22550174C2semidirect
7. #22550074C2semidirect
8. #22550069C2semidirect
9. #22550021C2semidirect
10. #22549985C2semidirect
11. #22549977C2semidirect
12. #22549961C2semidirect
13. D8×Dic127C2semidirect
14. #22549934C2semidirect
15. #449×C127C2semidirect
181. C127#3119semidirect
191. #449D127semidirect

Maximal subgroups

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

subgroupindex
#225504792
#225503592
#225501922
#225501902
#225501852
#225501742
#225500742
#225500692
#225500212
#225499852
#225499772
#225499612
D8×Dic1272
#225499342
#449×C1272
#3119127