Finite Group Databasebeta

← #30648351

Subgroups — #30648351

Order 16256.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861234C211yesmaximal, characteristic, complemented
2C22×#14523232C211yesmaximal, characteristic, complemented
3C2×#16861183C211yesmaximal, characteristic, complemented
4#22550127C212yesmaximal, complemented
5#22550124C212yesmaximal, complemented
6#22550110C212yesmaximal, complemented
7#22550107C212yesmaximal, complemented
8#22549919C211yesmaximal, characteristic, complemented
9#22549880C211yesmaximal, characteristic, complemented
10#22549877C211yesmaximal, characteristic, complemented
11#525×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C22⋊D127—21no—
14#16861234—24no—
15C2×#14523235C2212yescomplemented
16C2×#14523233C2212yescomplemented
17C2×#14523232—24no—
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20C2×#14523232C2212yescomplemented
21C22×C127⋊C8C2212yescomplemented
22#16861183—21no—
23#16861183—21no—
24#16861183C2212yes—
25#16861183C2212yes—
26C2×C4⋊Dic127C2211yescharacteristic
27C2×(C2×D127)⋊C4—21no—
28#16861142C2212yescomplemented
29#16861140C2212yescomplemented
30#16861133C2212yescomplemented
31#16861130C2212yescomplemented
32#16861129C2212yescomplemented
33#16861129C2212yescomplemented
34C2×#41×C127C2211yescharacteristic
35C23×C4×C127C2211yescharacteristic
36#123×C127—21no—
37C22⋊Q8×C127C2211yescharacteristic, complemented
38C22⋊Q8×C127C2211yescharacteristic, complemented
39#121×C127C2211yescharacteristic, complemented
40#121×C127C2211yescharacteristic, complemented
41#120×C127—21no—
42C4×D4×C127—21no—
43C4×D4×C127—21no—
44#117×C127C2211yescharacteristic
45C23×D127—21no—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—22no—
49C2×D508C2312yes—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—22no—
52C2×C22⋊D127—44no—
53C22×Dic127—21no—
54#14523235—24no—
55#14523233—24no—
56#14523232—24no—
57#14523232—22no—
58#14523232D414yescomplemented
59C2×C127⋊C8—24no—
60C2×C127⋊C8C2312yes—
61C2×C127⋊C8C2312yes—
62C4⋊Dic127—21no—
63C4⋊Dic127C2312yes—
64(C2×D127)⋊C4—22no—
65(C2×D127)⋊C4—22no—
66C24×C127D411yescharacteristic
67#41×C127—22no—
68#41×C127—22no—
69C2×Q8×C127C2311yescharacteristic
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—21no—
73C2×D4×C127C2311yescharacteristic
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—21no—
77C22×C4×C127D411yescharacteristic
78C22×C4×C127D411yescharacteristic
79C22×C4×C127D411yescharacteristic
80C22×C4×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2311yescharacteristic
83C4⋊C4×C127—21no—
84C4⋊C4×C127—21no—
85C4⋊C4×C127C2311yescharacteristic
86C4⋊C4×C127C2311yescharacteristic
87C22⋊C4×C127—21no—
88C22⋊C4×C127—21no—
89C22⋊C4×C127—21no—
90C22⋊C4×C127—21no—
91C22⋊C4×C127—21no—
92C42×C127—21no—
93C22×D127—22no—
94C22×D127—42no—
95C22×D127—42no—
96C22×D127—42no—
97D508—22no—
98D508—22no—
99D508—42no—
100C22⋊D127—44no—
101C22⋊D127—44no—
102C2×Dic127—22no—
103C2×Dic127—42no—
104C127⋊C8—22no—
105C127⋊C8C2×D414yescomplemented
106C23×C127—22no—
107C23×C127—22no—
108C23×C127—21no—
109C23×C127—41no—
110C23×C127C2×D411yescharacteristic
111C23×C127C2×D411yescharacteristic
112C23×C127C2×D411yescharacteristic
113Q8×C127—21no—
114Q8×C127—21no—
115D4×C127—21no—
116D4×C127—21no—
117D4×C127—42no—
118D4×C127—42no—
119D4×C127—42no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—42no—
131C2×C4×C127C2×D412yes—
132C2×C4×C127C2×D411yescharacteristic
133C2×C4×C127C2×D411yescharacteristic
134C2×C4×C127C2×D411yescharacteristic
135C2×C4×C127#4112yes—
136C2×C4×C127C2411yescharacteristic
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—44no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127#12112yes—
149C22×C127AGL(1,ℤ/8)12yescomplemented
150C22×C127C22×D411yescharacteristic
151C2×D127—42no—
152C2×D127—42no—
153C2×D127—42no—
154C2×D127—84no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127#12112yes—
161C4×C127C22×D411yescharacteristic
162C4×C127C2×#4111yescharacteristic
163Dic127—42no—
164C2×C127—22no—
165C2×C127—22no—
166C2×C127—41no—
167C2×C127—81no—
168C2×C127C2×#12111yescharacteristic
169C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C127#57511yescharacteristic
171D127—82no—
172#3055—1271nomaximal, Sylow
173C127#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1271no—
175C22×M16—1271no—
176#525D12711yescharacteristic, complemented
177C2×#121—1271no—
178#469—1272no—
179#468—1272no—
180#466—1272no—
181#465—1272no—
182#428—1271no—
183#418—1271no—
184#417—1271no—
185C2×#41C2×D12711yescharacteristic
186C22×D4—1271no—
187C22×D4—2541no—
188C23×C4C2×D12711yescharacteristic
189AGL(1,ℤ/8)—2544no—
190C2×SD16—1272no—
191C2×D8—1272no—
192C2×M16—1271no—
193C2×M16—1271no—
194C2×M16—1272no—
195C2×M16—2544no—
196C22×C8—1272no—
197#123—21no—
198C22⋊Q8C2×D12711yescharacteristic, complemented
199C22⋊Q8C2×D12711yescharacteristic, complemented
200#121—1272no—
201#121—1272no—
202#121—2541no—
203#121—2541no—
204#121C2×D12711yescharacteristic, complemented
205#121C2×D12711yescharacteristic, complemented
206#120—21no—
207C4×D4—21no—
208C4×D4—21no—
209#117C2×D12711yescharacteristic
210C2×C4⋊C4—1271no—
211C2×C22⋊C4—2541no—
212#107—1272no—
213#106—1272no—
214#103—1272no—
215#102—1272no—
216#102—1272no—
217#102—1272no—
218C24—2541no—
219C24C22⋊D12711yescharacteristic
220#41—22no—
221#41—22no—
222C2×Q8C22×D12711yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—1272no—
227C2×D4—5084no—
228C2×D4—2542no—
229C2×D4—2542no—
230C2×D4—2542no—
231C2×D4—2541no—
232C2×D4—2542no—
233C2×D4C22×D12711yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—2541no—
238C22×C4C22⋊D12711yescharacteristic
239C22×C4C22⋊D12711yescharacteristic
240C22×C4C22⋊D12711yescharacteristic
241C22×C4C22×D12711yescharacteristic
242C22×C4C22×D12711yescharacteristic
243C22×C4C22×D12711yescharacteristic
244SD16—2544no—
245D8—2544no—
246M16—1274no—
247M16—2544no—
248M16—2542no—
249C2×C8—1272no—
250C2×C8—1272no—
251C2×C8—2544no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—1272no—
255C4⋊C4—2541no—
256C4⋊C4C22×D12711yescharacteristic
257C4⋊C4C22×D12711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—2542no—
264C22⋊C4—2542no—
265C42—21no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—41no—
270C23—5082no—
271C23—5082no—
272C23—5082no—
273C23—2542no—
274C23C2×C22⋊D12711yescharacteristic
275C23C2×C22⋊D12711yescharacteristic
276C23C2×C22⋊D12711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—42no—
283D4—42no—
284D4—5084no—
285D4—5084no—
286D4—5082no—
287D4—2542no—
288D4—2542no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—5082no—
301C2×C4—2542no—
302C2×C4D4×D12712yes—
303C2×C4C2×C22⋊D12711yescharacteristic
304C2×C4C2×C22⋊D12711yescharacteristic
305C2×C4C2×C22⋊D12711yescharacteristic
306C2×C4#1452324712yes—
307C2×C4C23×D12711yescharacteristic
308C8—1274no—
309C8—2542no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—44no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—10164no—
322C22—5082no—
323C22—5082no—
324C22—5082no—
325C22#1686118112yes—
326C22#1686123412yescomplemented
327C22C22×C22⋊D12711yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—5082no—
334C4#1686118112yes—
335C4C2×D4×D12711yescharacteristic
336C4C2×#1452324711yescharacteristic
337C2—22no—
338C2—22no—
339C2—41no—
340C2—81no—
341C2—10162no—
342C2C2×#1686118111yescharacteristic
343C2C2×#1686123411yescharacteristic
344C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#525×C127C2
socleC22×C127C22×D4
hypercenter#525D127
hypocenterC127#3055
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C22×#14523232C2semidirect
3. C2×#16861183C2semidirect
4. #22550127C2semidirect
5. #22550124C2semidirect
6. #22550110C2semidirect
7. #22550107C2semidirect
8. #22549919C2semidirect
9. #22549880C2semidirect
10. #22549877C2semidirect
11. #525×C127C2semidirect
15. C2×#14523235C22semidirect
16. C2×#14523233C22semidirect
20. C2×#14523232C22semidirect
21. C22×C127⋊C8C22semidirect
28. #16861142C22semidirect
29. #16861140C22semidirect
30. #16861133C22semidirect
31. #16861130C22semidirect
32. #16861129C22semidirect
33. #16861129C22semidirect
37. C22⋊Q8×C127C22semidirect
38. C22⋊Q8×C127C22semidirect
39. #121×C127C22semidirect
40. #121×C127C22semidirect
58. #14523232D4semidirect
105. C127⋊C8C2×D4semidirect
149. C22×C127AGL(1,ℤ/8)semidirect
173. C127#3055semidirect
176. #525D127semidirect
198. C22⋊Q8C2×D127semidirect
199. C22⋊Q8C2×D127semidirect
204. #121C2×D127semidirect
205. #121C2×D127semidirect
326. C22#16861234semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×#145232322
C2×#168611832
#225501272
#225501242
#225501102
#225501072
#225499192
#225498802
#225498772
#525×C1272
#3055127