Finite Group Databasebeta

← #30648350

Subgroups — #30648350

Order 16256.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861237C211yesmaximal, characteristic, complemented
2C2×M16×D127C211yesmaximal, characteristic, complemented
3#22550388C211yesmaximal, characteristic, complemented
4C2×#16861184C211yesmaximal, characteristic, complemented
5#22550130C212yesmaximal, complemented
6#22550123C212yesmaximal, complemented
7#22550113C212yesmaximal, complemented
8#22550106C212yesmaximal, complemented
9#22549880C211yesmaximal, characteristic, complemented
10#22549875C211yesmaximal, characteristic, complemented
11#428×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508—21no—
14C22×D508C2211yescharacteristic
15C22×C4×D127C2211yescharacteristic
16#16861237—24no—
17C2×#14523237C2212yescomplemented
18C2×D1016C2212yescomplemented
19C2×#14523232C2211yescharacteristic
20C2×#14523231C2212yescomplemented
21M16×D127—24no—
22C2×C8×D127C2212yescomplemented
23#16861193—21no—
24#16861189C2211yescharacteristic, complemented
25#16861189C2211yescharacteristic, complemented
26#16861184—21no—
27#16861184—21no—
28#16861184C2212yes—
29#16861184C2212yes—
30#16861184C2211yescharacteristic, complemented
31#16861184C2211yescharacteristic, complemented
32#16861179—21no—
33C4×D508—21no—
34C4×D508—21no—
35C2×(C2×D127)⋊C4—21no—
36#16861132C2212yescomplemented
37#16861129C2212yescomplemented
38#16861129C2212yescomplemented
39#16861129C2212yescomplemented
40C2×M16×C127C2211yescharacteristic
41#117×C127C2211yescharacteristic
42C2×C4⋊C4×C127C2211yescharacteristic
43#107×C127C2212yescomplemented
44#106×C127C2212yescomplemented
45C23×D127—21no—
46C23×D127D411yescharacteristic
47#14523248—22no—
48#14523248—22no—
49C2×Dic254C2311yescharacteristic
50C2×D508—22no—
51C2×D508—22no—
52C2×D508—21no—
53C2×D508—21no—
54C2×D508—22no—
55C2×D508—21no—
56C2×D508—22no—
57C2×D508—44no—
58C2×D508C2312yes—
59C2×D508C2311yescharacteristic
60C2×C22⋊D127—21no—
61C22×Dic127D411yescharacteristic
62C2×C4×D127—22no—
63C2×C4×D127—21no—
64C2×C4×D127—22no—
65C2×C4×D127D411yescharacteristic
66C2×C4×D127D411yescharacteristic
67C2×C4×D127C2311yescharacteristic
68C2×C4×D127C2311yescharacteristic
69#14523237—24no—
70D1016—24no—
71#14523232—22no—
72#14523231—24no—
73C2×C127⋊C8C2312yes—
74C8×D127—24no—
75C4⋊Dic127—21no—
76C4⋊Dic127—21no—
77(C2×D127)⋊C4—21no—
78(C2×D127)⋊C4—21no—
79(C2×D127)⋊C4—22no—
80(C2×D127)⋊C4—22no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83C22×C4×C127—21no—
84C22×C4×C127C2311yescharacteristic
85M16×C127D414yescomplemented
86C2×C8×C127C2312yes—
87C4⋊C4×C127—21no—
88C4⋊C4×C127C2312yes—
89C4⋊C4×C127C2311yescharacteristic
90C4⋊C4×C127C2311yescharacteristic
91C22⋊C4×C127—21no—
92C42×C127—21no—
93C22×D127—22no—
94C22×D127—22no—
95C22×D127—21no—
96C22×D127—22no—
97C22×D127—41no—
98C22×D127—42no—
99C22×D127—42no—
100C22×D127—42no—
101C22×D127C2×D411yescharacteristic
102C22×D127C2×D411yescharacteristic
103Dic254—21no—
104Dic254—21no—
105D508—21no—
106D508—22no—
107D508—22no—
108D508—21no—
109D508—42no—
110D508—44no—
111D508—42no—
112D508—44no—
113D508—42no—
114C22⋊D127—42no—
115C2×Dic127—22no—
116C2×Dic127—21no—
117C2×Dic127C2×D411yescharacteristic
118C2×Dic127C2×D411yescharacteristic
119C4×D127—22no—
120C4×D127—22no—
121C4×D127—21no—
122C4×D127—21no—
123C4×D127—21no—
124C4×D127—21no—
125C4×D127—42no—
126C127⋊C8—22no—
127C23×C127C2×D411yescharacteristic
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—22no—
131C2×C4×C127—42no—
132C2×C4×C127C2×D411yescharacteristic
133C2×C4×C127C2×D412yes—
134C2×C4×C127#4112yes—
135C2×C4×C127C2411yescharacteristic
136C8×C127C2×D414yescomplemented
137C22×C127—21no—
138C22×C127—21no—
139C22×C127#12112yes—
140C22×C127C22×D411yescharacteristic
141C2×D127—21no—
142C2×D127—21no—
143C2×D127—21no—
144C2×D127—21no—
145C2×D127—21no—
146C2×D127—44no—
147C2×D127—41no—
148C2×D127—42no—
149C2×D127—42no—
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—42no—
153C2×D127—84no—
154C2×D127AGL(1,ℤ/8)12yescomplemented
155C4×C127—42no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127#12112yes—
159C4×C127C22×D411yescharacteristic
160C4×C127C2×#4111yescharacteristic
161Dic127—21no—
162Dic127—21no—
163Dic127—41no—
164C2×C127—22no—
165C2×C127C2×#12111yescharacteristic
166C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C127#57511yescharacteristic
168D127—22no—
169D127—41no—
170D127—81no—
171D127—82no—
172#3055—1271nomaximal, Sylow
173C127#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1271no—
175C22×M16—1271no—
176#525—1271no—
177C2×#121—1271no—
178#469—1272no—
179#468—1272no—
180#466—1272no—
181#465—1272no—
182#428D12711yescharacteristic, complemented
183#418—1271no—
184#417—1271no—
185C2×#41—1271no—
186C22×D4—1271no—
187C22×D4—2541no—
188C23×C4—1271no—
189AGL(1,ℤ/8)—2544no—
190C2×SD16—1272no—
191C2×D8—1272no—
192C2×M16—1271no—
193C2×M16—1272no—
194C2×M16—2544no—
195C2×M16C2×D12711yescharacteristic
196C22×C8—1272no—
197#123—2541no—
198C22⋊Q8—1271no—
199C22⋊Q8—1271no—
200#121—1272no—
201#121—1272no—
202#121—1271no—
203#121—1271no—
204#121—2541no—
205#121—2541no—
206#120—2541no—
207C4×D4—2541no—
208C4×D4—2541no—
209#117C2×D12711yescharacteristic
210C2×C4⋊C4C2×D12711yescharacteristic
211C2×C22⋊C4—2541no—
212#107C2×D12712yescomplemented
213#106C2×D12712yescomplemented
214#103—1272no—
215#102—1272no—
216#102—1272no—
217#102—1272no—
218C24—1271no—
219C24—2541no—
220#41—2542no—
221#41—2542no—
222C2×Q8—1271no—
223C2×D4—1271no—
224C2×D4—1272no—
225C2×D4—5084no—
226C2×D4—2541no—
227C2×D4—2541no—
228C2×D4—2542no—
229C2×D4—2542no—
230C2×D4—2541no—
231C2×D4—2541no—
232C2×D4—2542no—
233C2×D4—2542no—
234C22×C4—21no—
235C22×C4—1271no—
236C22×C4—1271no—
237C22×C4—1271no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—2541no—
241C22×C4—2542no—
242C22×C4—2542no—
243C22×C4C22×D12711yescharacteristic
244SD16—2544no—
245D8—2544no—
246M16—2542no—
247M16—2544no—
248M16D50814yescomplemented
249C2×C8—1272no—
250C2×C8—2544no—
251C2×C8C22×D12712yes—
252C4⋊C4—21no—
253C4⋊C4—2541no—
254C4⋊C4—2541no—
255C4⋊C4C22×D12711yescharacteristic
256C4⋊C4C22×D12711yescharacteristic
257C4⋊C4C22×D12712yes—
258C22⋊C4—21no—
259C22⋊C4—2542no—
260C22⋊C4—2542no—
261C22⋊C4—2541no—
262C22⋊C4—2541no—
263C22⋊C4—2541no—
264C22⋊C4—2541no—
265C42—21no—
266C23—1271no—
267C23—1271no—
268C23—5082no—
269C23—5082no—
270C23—5081no—
271C23—5082no—
272C23—2542no—
273C23—2541no—
274C23—2542no—
275C23—2542no—
276C23D4×D12711yescharacteristic
277Q8—2541no—
278Q8—2541no—
279D4—5084no—
280D4—5082no—
281D4—5084no—
282D4—5082no—
283D4—5082no—
284D4—5082no—
285D4—2541no—
286D4—2542no—
287D4—2542no—
288D4—2541no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—1271no—
294C2×C4—1271no—
295C2×C4—5082no—
296C2×C4—2541no—
297C2×C4—2542no—
298C2×C4—2542no—
299C2×C4—2542no—
300C2×C4—2541no—
301C2×C4—2541no—
302C2×C4—2541no—
303C2×C4—2541no—
304C2×C4D4×D12711yescharacteristic
305C2×C4C2×D50812yes—
306C2×C4#1452324912yes—
307C2×C4C23×D12711yescharacteristic
308C8—2542no—
309C8C2×D50814yescomplemented
310C22—21no—
311C22—21no—
312C22—1272no—
313C22—10164no—
314C22—5084no—
315C22—5081no—
316C22—5082no—
317C22—5082no—
318C22—5081no—
319C22—5081no—
320C22—5082no—
321C22—2541no—
322C22—2541no—
323C22—2541no—
324C22—2541no—
325C22—2541no—
326C22#1686118412yes—
327C22C2×D4×D12711yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—5081no—
332C4—2541no—
333C4—2541no—
334C4#1686118412yes—
335C4C22×D50811yescharacteristic
336C4C2×#1452324911yescharacteristic
337C2—22no—
338C2—10162no—
339C2—10161no—
340C2—5081no—
341C2—2542no—
342C2C2×#1686118411yescharacteristic
343C2AGL(1,ℤ/8)×D12711yescharacteristic
344C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
hypercenter#428D127
hypocenterC127#3055
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C2×M16×D127C2semidirect
3. #22550388C2semidirect
4. C2×#16861184C2semidirect
5. #22550130C2semidirect
6. #22550123C2semidirect
7. #22550113C2semidirect
8. #22550106C2semidirect
9. #22549880C2semidirect
10. #22549875C2semidirect
11. #428×C127C2semidirect
17. C2×#14523237C22semidirect
18. C2×D1016C22semidirect
20. C2×#14523231C22semidirect
22. C2×C8×D127C22semidirect
24. #16861189C22semidirect
25. #16861189C22semidirect
30. #16861184C22semidirect
31. #16861184C22semidirect
36. #16861132C22semidirect
37. #16861129C22semidirect
38. #16861129C22semidirect
39. #16861129C22semidirect
43. #107×C127C22semidirect
44. #106×C127C22semidirect
85. M16×C127D4semidirect
136. C8×C127C2×D4semidirect
154. C2×D127AGL(1,ℤ/8)semidirect
173. C127#3055semidirect
182. #428D127semidirect
212. #107C2×D127semidirect
213. #106C2×D127semidirect
248. M16D508semidirect
309. C8C2×D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×M16×D1272
#225503882
C2×#168611842
#225501302
#225501232
#225501132
#225501062
#225498802
#225498752
#428×C1272
#3055127