Finite Group Databasebeta

← #30648985

Subgroups — #30648985

Order 16256.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550493C211yesmaximal, characteristic, complemented
2#22550359C211yesmaximal, characteristic, complemented
3#22550212C211yesmaximal, characteristic, complemented
4#22550187C211yesmaximal, characteristic, complemented
5#22550169C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550096C211yesmaximal, characteristic, complemented
8#22550089C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22549977C211yesmaximal, characteristic, complemented
11#22549973C211yesmaximal, characteristic, complemented
12#22549968C211yesmaximal, characteristic, complemented
13D8×Dic127C211yesmaximal, characteristic, complemented
14#22549934C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic
16C2×#14523238C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18#16861208C2211yescharacteristic
19#16861202—21no—
20#16861196—21no—
21#16861196C2211yescharacteristic
22#16861190—21no—
23#16861188C2211yescharacteristic
24#16861186—21no—
25Q8×Dic127C2211yescharacteristic
26Q8×Dic127C2211yescharacteristic
27#16861174C2211yescharacteristic
28D4×Dic127—21no—
29D4×Dic127—21no—
30D4×Dic127C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32D4×Dic127C2211yescharacteristic
33#16861168—21no—
34#16861168C2211yescharacteristic
35C2×C4⋊Dic127—21no—
36C2×C4⋊Dic127C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861142C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861133C2211yescharacteristic
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×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61C22×Dic127—21no—
62C22×Dic127—21no—
63C22×Dic127—21no—
64C22×Dic127—21no—
65#14523238—22no—
66#14523236—22no—
67C2×C127⋊C8C2311yescharacteristic
68C2×C127⋊C8C2311yescharacteristic
69C4⋊Dic127—21no—
70C4⋊Dic127—21no—
71C4⋊Dic127—21no—
72C4⋊Dic127D411yescharacteristic
73C4⋊Dic127D411yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80C22⋊Dic127—21no—
81C22⋊Dic127—21no—
82C22⋊Dic127—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127D411yescharacteristic
87C22⋊Dic127D411yescharacteristic
88C4×Dic127—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—
108Dic254—21no—
109Dic254—21no—
110Dic254—81no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—81no—
116C2×Dic127—81no—
117C2×Dic127—81no—
118C2×Dic127—81no—
119C2×Dic127C2×D411yescharacteristic
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C127⋊C8—21no—
124C127⋊C8—21no—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127C2×D411yescharacteristic
128Q8×C127—21no—
129Q8×C127#4112yes—
130D4×C127—24no—
131D4×C127—21no—
132D4×C127—21no—
133D4×C127—21no—
134D4×C127—81no—
135D4×C127#4112yes—
136C2×C4×C127—24no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—21no—
146C22×C127—22no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127C22×D411yescharacteristic
155C4×C127—21no—
156C4×C127—22no—
157C4×C127—41no—
158C4×C127C2×#4111yescharacteristic
159C4×C127C2×#4111yescharacteristic
160Dic127—21no—
161Dic127—21no—
162Dic127—21no—
163Dic127—21no—
164Dic127—41no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168C2×C127—41no—
169C2×C127—42no—
170C2×C127—81no—
171C2×C127#52511yescharacteristic
172C2×C127#57611yescharacteristic
173C2×C127#57811yescharacteristic
174#3124—1271nomaximal, Sylow
175C127#312411yescharacteristic, complemented, Sylow
176#542—1271no—
177#520—1271no—
178#484—1271no—
179#480—1271no—
180#477—1271no—
181#474—1271no—
182#464—1271no—
183#463—1271no—
184#452—1271no—
185#449D12711yescharacteristic
186#440—1271no—
187#440—1271no—
188C4×Q16—1271no—
189C4×D8—1271no—
190#433—1271no—
191C2×#41C2×D12711yescharacteristic
192C2×Q16—1271no—
193C2×SD16—1271no—
194C2×SD16C2×D12711yescharacteristic
195C2×D8C2×D12711yescharacteristic
196C4⋊Q8—1271no—
197#126—2541no—
198#124—1271no—
199#124—2541no—
200#123—2541no—
201C22⋊Q8—1271no—
202C22⋊Q8—2541no—
203#121C2×D12711yescharacteristic
204C4×Q8—1271no—
205C4×Q8—1271no—
206C4×Q8—1271no—
207C4×D4—1271no—
208C4×D4—1271no—
209C4×D4—1271no—
210C4×D4—2541no—
211C4×D4—2541no—
212#117—1271no—
213#117—2541no—
214C2×C4⋊C4—1271no—
215C2×C4⋊C4—2541no—
216#107—1271no—
217#107—1271no—
218#107—1271no—
219#107—1271no—
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222#103—1271no—
223#103—1271no—
224#103—1271no—
225#103C2×D12711yescharacteristic
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102C2×D12711yescharacteristic
230C22⋊C8—1271no—
231C22⋊C8C2×D12711yescharacteristic
232#97—1271no—
233C4×C8—1271no—
234#41—22no—
235#41—22no—
236C2×Q8—1271no—
237C2×Q8—2541no—
238C2×Q8C22×D12711yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C22×D12711yescharacteristic
242C2×D4C22×D12711yescharacteristic
243C22×C4—21no—
244C22×C4—2541no—
245C22×C4—2541no—
246C22×C4—2541no—
247C22×C4—2541no—
248C22×C4C22×D12711yescharacteristic
249Q16—2542no—
250SD16—22no—
251SD16—2542no—
252D8—22no—
253C2×C8—1271no—
254C2×C8—1271no—
255C2×C8C22×D12711yescharacteristic
256C2×C8C22×D12711yescharacteristic
257C4⋊C4—1271no—
258C4⋊C4—1271no—
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4C22×D12711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—1271no—
271C22⋊C4—1271no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C42—1271no—
279C42—1271no—
280C42—2541no—
281C42—2541no—
282C42—2541no—
283C23—21no—
284C23—21no—
285C23D4×D12711yescharacteristic
286Q8—21no—
287Q8—10161no—
288Q8—2541no—
289Q8—2541no—
290Q8#1452324712yes—
291D4—24no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4#1452324712yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—1271no—
303C2×C4—1271no—
304C2×C4—1271no—
305C2×C4—1271no—
306C2×C4—10161no—
307C2×C4—10161no—
308C2×C4—10161no—
309C2×C4—10161no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4D4×D12711yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2541no—
319C8—2541no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22C2×D4×D12711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—5081no—
334C4—5081no—
335C4—5081no—
336C4—5081no—
337C4—2541no—
338C4—2541no—
339C4—2541no—
340C4—2541no—
341C4C2×#1452324711yescharacteristic
342C4C2×#1452324711yescharacteristic
343C2—41no—
344C2—42no—
345C2—81no—
346C2#2255038211yescharacteristic
347C2#2255068011yescharacteristic
348C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3124
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. #22550493C2semidirect
2. #22550359C2semidirect
3. #22550212C2semidirect
4. #22550187C2semidirect
5. #22550169C2semidirect
6. #22550154C2semidirect
7. #22550096C2semidirect
8. #22550089C2semidirect
9. #22550038C2semidirect
10. #22549977C2semidirect
11. #22549973C2semidirect
12. #22549968C2semidirect
13. D8×Dic127C2semidirect
14. #22549934C2semidirect
175. C127#3124semidirect

Maximal subgroups

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

subgroupindex
#225504932
#225503592
#225502122
#225501872
#225501692
#225501542
#225500962
#225500892
#225500382
#225499772
#225499732
#225499682
D8×Dic1272
#225499342
#449×C1272
#3124127