Finite Group Databasebeta

← #30648976

Subgroups — #30648976

Order 16256.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550486C211yesmaximal, characteristic, complemented
2#22550351C211yesmaximal, characteristic
3#22550218C211yesmaximal, characteristic, complemented
4#22550214C211yesmaximal, characteristic, complemented
5#22550178C211yesmaximal, characteristic, complemented
6#22550174C211yesmaximal, characteristic, complemented
7#22550079C211yesmaximal, characteristic, complemented
8#22550079C211yesmaximal, characteristic, complemented
9#22550027C211yesmaximal, characteristic
10#22549977C211yesmaximal, characteristic, complemented
11#22549973C211yesmaximal, characteristic, complemented
12#22549967C211yesmaximal, characteristic, complemented
13SD16×Dic127C211yesmaximal, characteristic, complemented
14#22549934C211yesmaximal, characteristic, complemented
15#450×C127C211yesmaximal, characteristic
16C2×#14523236C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18#16861207C2211yescharacteristic, complemented
19#16861202—21no—
20#16861202—21no—
21#16861200—21no—
22#16861200C2211yescharacteristic, complemented
23#16861199—21no—
24#16861188C2211yescharacteristic
25Q8×Dic127C2211yescharacteristic, complemented
26#16861174C2211yescharacteristic, complemented
27D4×Dic127—21no—
28D4×Dic127—21no—
29D4×Dic127C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32D4×Dic127C2211yescharacteristic
33#16861168C2211yescharacteristic
34#16861168C2211yescharacteristic
35C2×C22⋊Dic127—21no—
36C2×C22⋊Dic127—21no—
37#16861143C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861133C2211yescharacteristic, complemented
46#16861133C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic
55C22⋊Q8×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59C2×Dic254C2311yescharacteristic
60C22×Dic127—21no—
61C22×Dic127—21no—
62C22×Dic127—21no—
63C22×Dic127—21no—
64#14523236—22no—
65#14523236—22no—
66C2×C127⋊C8C2311yescharacteristic
67C2×C127⋊C8C2311yescharacteristic
68C4⋊Dic127—21no—
69C4⋊Dic127—21no—
70C4⋊Dic127D411yescharacteristic
71C4⋊Dic127D411yescharacteristic
72C4⋊Dic127C2311yescharacteristic
73C4⋊Dic127C2311yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75Dic127⋊C4—21no—
76Dic127⋊C4—21no—
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
93C24×C127—21no—
94C2×Q8×C127C2311yescharacteristic
95C2×D4×C127—22no—
96C2×D4×C127—22no—
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101SD16×C127—22no—
102SD16×C127—22no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127—21no—
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108Dic254—21no—
109Dic254—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—81no—
115C2×Dic127—81no—
116C2×Dic127—81no—
117C2×Dic127—81no—
118C2×Dic127C2×D411yescharacteristic
119C2×Dic127C2×D411yescharacteristic
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C127⋊C8—21no—
123C127⋊C8—21no—
124C23×C127—24no—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127—81no—
128C23×C127—81no—
129C23×C127C2×D411yescharacteristic
130Q8×C127—21no—
131Q8×C127—21no—
132D4×C127—24no—
133D4×C127—21no—
134D4×C127—21no—
135D4×C127#4112yes—
136D4×C127#4112yes—
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—22no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—84no—
155C22×C127C22×D411yescharacteristic
156C4×C127—21no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127C2×#4111yescharacteristic
160C4×C127C2×#4111yescharacteristic
161Dic127—21no—
162Dic127—21no—
163Dic127—21no—
164Dic127—21no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168Dic127—41no—
169C2×C127—41no—
170C2×C127—42no—
171C2×C127—42no—
172C2×C127#52511yescharacteristic
173C2×C127#57711yescharacteristic
174C2×C127#57711yescharacteristic
175#3123—1271nomaximal, Sylow
176C127#312311yescharacteristic, complemented, Sylow
177#541—1271no—
178#518—1271no—
179#485—1271no—
180#485—1271no—
181#478—1271no—
182#478—1271no—
183#461—1271no—
184#461—1271no—
185#450—1271no—
186#450D12711yescharacteristic
187#440—1271no—
188#440—1271no—
189C4×SD16—1271no—
190C4×SD16—1271no—
191#433—1271no—
192C22×D4C2×D12711yescharacteristic
193C2×SD16—1271no—
194C2×SD16—1271no—
195C2×SD16C2×D12711yescharacteristic
196C2×SD16C2×D12711yescharacteristic
197C4⋊Q8—1271no—
198#126—2541no—
199#126—2541no—
200#125—1271no—
201#125—2541no—
202#125—2541no—
203C22⋊Q8—1271no—
204C22⋊Q8C2×D12711yescharacteristic
205C4×Q8—1271no—
206C4×Q8—1271no—
207C4×D4—1271no—
208C4×D4—1271no—
209C4×D4—1271no—
210C4×D4—1271no—
211C4×D4—2541no—
212C4×D4—2541no—
213#117—1271no—
214#117—1271no—
215C2×C22⋊C4—2541no—
216C2×C22⋊C4—2541no—
217#107—1271no—
218#107—1271no—
219#106—1271no—
220#106—1271no—
221C4⋊C8—1271no—
222C4⋊C8—1271no—
223#103—1271no—
224#103—1271no—
225#103—1271no—
226#103—1271no—
227#102—1271no—
228#102—1271no—
229#102C2×D12711yescharacteristic
230#102C2×D12711yescharacteristic
231C22⋊C8—1271no—
232C22⋊C8C2×D12711yescharacteristic
233#97—1271no—
234C4×C8—1271no—
235C24—21no—
236C2×Q8—1271no—
237C2×Q8C22×D12711yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D12711yescharacteristic
242C2×D4C22×D12711yescharacteristic
243C22×C4—2541no—
244C22×C4—2541no—
245C22×C4—2541no—
246C22×C4—2541no—
247C22×C4C22×D12711yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—2542no—
251SD16—2542no—
252C2×C8—1271no—
253C2×C8—1271no—
254C2×C8C22×D12711yescharacteristic
255C2×C8C22×D12711yescharacteristic
256C4⋊C4—21no—
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⋊C4—2541no—
269C4⋊C4C22×D12711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1271no—
272C22⋊C4—1271no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C42—1271no—
280C42—1271no—
281C42—2541no—
282C42—2541no—
283C42—2541no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D12711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—2541no—
293Q8—2541no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1452324712yes—
298D4#1452324712yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—1271no—
304C2×C4—1271no—
305C2×C4—1271no—
306C2×C4—1271no—
307C2×C4—10161no—
308C2×C4—10161no—
309C2×C4—10161no—
310C2×C4—10161no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4D4×D12711yescharacteristic
316C2×C4C23×D12711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2541no—
320C8—2541no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D12711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—5081no—
336C4—5081no—
337C4—5081no—
338C4—5081no—
339C4—2541no—
340C4—2541no—
341C4—2541no—
342C4—2541no—
343C4C2×#1452324711yescharacteristic
344C4C2×#1452324711yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2255038211yescharacteristic
349C2#2255068211yescharacteristic
350C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
hypercenter#450D127
hypocenterC127#3123
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22550486C2semidirect
3. #22550218C2semidirect
4. #22550214C2semidirect
5. #22550178C2semidirect
6. #22550174C2semidirect
7. #22550079C2semidirect
8. #22550079C2semidirect
10. #22549977C2semidirect
11. #22549973C2semidirect
12. #22549967C2semidirect
13. SD16×Dic127C2semidirect
14. #22549934C2semidirect
18. #16861207C22semidirect
22. #16861200C22semidirect
25. Q8×Dic127C22semidirect
26. #16861174C22semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861134C22semidirect
45. #16861133C22semidirect
46. #16861133C22semidirect
50. #16861119C22semidirect
51. C8×Dic127C22semidirect
176. C127#3123semidirect

Maximal subgroups

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

subgroupindex
#225504862
#225503512
#225502182
#225502142
#225501782
#225501742
#225500792
#225500272
#225499772
#225499732
#225499672
SD16×Dic1272
#225499342
#450×C1272
#3123127