Finite Group Databasebeta

← #30648352

Subgroups — #30648352

Order 16256.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16861234C211yesmaximal, characteristic, complemented
2C2×M16×D127C211yesmaximal, characteristic, complemented
3#22550383C211yesmaximal, characteristic, complemented
4C2×#16861184C211yesmaximal, characteristic, complemented
5#22550128C212yesmaximal, complemented
6#22550125C212yesmaximal, complemented
7#22550111C212yesmaximal, complemented
8#22550108C212yesmaximal, complemented
9#22549916C211yesmaximal, characteristic, complemented
10#22549882C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C22×D508—21no—
13C22×D508C2211yescharacteristic
14C22×C4×D127C2211yescharacteristic
15#16861234—24no—
16C2×#14523235C2212yescomplemented
17C2×#14523233C2212yescomplemented
18C2×#14523232C2211yescharacteristic
19C2×#14523231C2212yescomplemented
20M16×D127—24no—
21C2×C8×D127C2212yescomplemented
22#16861190—21no—
23#16861185C2211yescharacteristic, complemented
24#16861185C2211yescharacteristic, complemented
25#16861184—21no—
26#16861184—21no—
27#16861184C2212yes—
28#16861184C2212yes—
29#16861180C2211yescharacteristic, complemented
30#16861180C2211yescharacteristic, complemented
31#16861177—21no—
32C4×C22⋊D127—21no—
33C4×C22⋊D127—21no—
34#16861168C2211yescharacteristic
35C2×(C2×D127)⋊C4—21no—
36#16861142C2212yescomplemented
37#16861140C2212yescomplemented
38#16861131C2212yescomplemented
39#16861131C2212yescomplemented
40C2×#41×C127C2211yescharacteristic
41C2×M16×C127C2211yescharacteristic
42C2×C4⋊C4×C127C2211yescharacteristic
43#103×C127C2212yescomplemented
44#102×C127C2212yescomplemented
45C23×D127—21no—
46C23×D127D411yescharacteristic
47C2×D508—22no—
48C2×D508—22no—
49C2×D508—22no—
50C2×D508—21no—
51C2×D508—22no—
52C2×D508—44no—
53C2×D508C2312yes—
54C2×C22⋊D127—21no—
55C2×C22⋊D127—21no—
56C22×Dic127D411yescharacteristic
57C2×C4×D127—22no—
58C2×C4×D127—22no—
59C2×C4×D127D411yescharacteristic
60C2×C4×D127D411yescharacteristic
61C2×C4×D127C2311yescharacteristic
62C2×C4×D127C2311yescharacteristic
63#14523235—24no—
64#14523233—24no—
65#14523232D414yescomplemented
66#14523231—24no—
67C2×C127⋊C8C2312yes—
68C8×D127—24no—
69C4⋊Dic127C2311yescharacteristic
70C4⋊Dic127C2311yescharacteristic
71Dic127⋊C4—21no—
72Dic127⋊C4—21no—
73C22⋊Dic127—21no—
74C22⋊Dic127—21no—
75C22⋊Dic127—21no—
76(C2×D127)⋊C4—21no—
77(C2×D127)⋊C4—21no—
78(C2×D127)⋊C4—22no—
79(C2×D127)⋊C4—22no—
80C4×Dic127—21no—
81#41×C127—22no—
82#41×C127—22no—
83C2×Q8×C127C2311yescharacteristic
84C2×D4×C127—21no—
85C2×D4×C127C2311yescharacteristic
86C22×C4×C127—21no—
87C22×C4×C127—21no—
88C22×C4×C127C2311yescharacteristic
89M16×C127—22no—
90C2×C8×C127C2312yes—
91C4⋊C4×C127—21no—
92C4⋊C4×C127C2312yes—
93C22×D127—22no—
94C22×D127—22no—
95C22×D127—22no—
96C22×D127—41no—
97C22×D127—42no—
98C22×D127—42no—
99C22×D127—42no—
100C22×D127C2×D411yescharacteristic
101C22×D127C2×D411yescharacteristic
102D508—22no—
103D508—22no—
104D508—44no—
105D508—44no—
106D508—42no—
107C22⋊D127—42no—
108C22⋊D127—42no—
109C2×Dic127—22no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127C2×D411yescharacteristic
113C2×Dic127C2×D411yescharacteristic
114C4×D127—22no—
115C4×D127—22no—
116C4×D127—21no—
117C4×D127—21no—
118C4×D127—21no—
119C4×D127—21no—
120C127⋊C8C2×D414yescomplemented
121C23×C127—21no—
122C23×C127C2×D411yescharacteristic
123Q8×C127—21no—
124Q8×C127—21no—
125D4×C127—21no—
126D4×C127—21no—
127D4×C127—42no—
128C2×C4×C127—22no—
129C2×C4×C127—21no—
130C2×C4×C127—42no—
131C2×C4×C127—42no—
132C2×C4×C127C2×D411yescharacteristic
133C2×C4×C127C2×D412yes—
134C2×C4×C127#4112yes—
135C2×C4×C127C2411yescharacteristic
136C8×C127—22no—
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—41no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127#12112yes—
143C22×C127C22×D411yescharacteristic
144C2×D127—21no—
145C2×D127—21no—
146C2×D127—21no—
147C2×D127—21no—
148C2×D127—21no—
149C2×D127—44no—
150C2×D127—42no—
151C2×D127—42no—
152C2×D127—42no—
153C2×D127—84no—
154C2×D127AGL(1,ℤ/8)12yescomplemented
155C4×C127—41no—
156C4×C127—42no—
157C4×C127#12112yes—
158C4×C127C22×D411yescharacteristic
159C4×C127C2×#4111yescharacteristic
160Dic127—21no—
161Dic127—21no—
162Dic127—41no—
163Dic127—41no—
164C2×C127—22no—
165C2×C127—81no—
166C2×C127C2×#12111yescharacteristic
167C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C127#57511yescharacteristic
169D127—22no—
170D127—41no—
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#428—1271no—
183#418—1271no—
184#417D12711yescharacteristic, complemented
185C2×#41C2×D12711yescharacteristic
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—1271no—
201#121—1271no—
202#121—1272no—
203#121—1272no—
204#121—2541no—
205#121—2541no—
206#120—2541no—
207C4×D4—2541no—
208C4×D4—2541no—
209#117—1271no—
210C2×C4⋊C4C2×D12711yescharacteristic
211C2×C22⋊C4—2541no—
212#107—1272no—
213#106—1272no—
214#103C2×D12712yescomplemented
215#102—1272no—
216#102—1272no—
217#102C2×D12712yescomplemented
218C24—1271no—
219C24—2541no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D12711yescharacteristic
223C2×D4—21no—
224C2×D4—1272no—
225C2×D4—5084no—
226C2×D4—2542no—
227C2×D4—2542no—
228C2×D4—2541no—
229C2×D4—2541no—
230C2×D4—2541no—
231C2×D4—2542no—
232C2×D4—2542no—
233C2×D4C22×D12711yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1271no—
237C22×C4—1271no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—1271no—
241C22×C4—2542no—
242C22×C4—2542no—
243C22×C4C22×D12711yescharacteristic
244SD16—2544no—
245D8—2544no—
246M16—22no—
247M16—1274no—
248M16—2544no—
249C2×C8—1272no—
250C2×C8—2544no—
251C2×C8C22×D12712yes—
252C4⋊C4—21no—
253C4⋊C4—1271no—
254C4⋊C4—1271no—
255C4⋊C4—2541no—
256C4⋊C4—2541no—
257C4⋊C4C22×D12712yes—
258C22⋊C4—2542no—
259C22⋊C4—2542no—
260C22⋊C4—2541no—
261C22⋊C4—2541no—
262C22⋊C4—2541no—
263C22⋊C4—2541no—
264C22⋊C4—2541no—
265C42—2541no—
266C23—21no—
267C23—1271no—
268C23—1271no—
269C23—5082no—
270C23—5082no—
271C23—5081no—
272C23—5082no—
273C23—2542no—
274C23—2542no—
275C23—2542no—
276C23D4×D12711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—5084no—
283D4—5084no—
284D4—5082no—
285D4—5082no—
286D4—5082no—
287D4—2542no—
288D4—2542no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—1271no—
294C2×C4—1271no—
295C2×C4—2541no—
296C2×C4—2541no—
297C2×C4—2542no—
298C2×C4—2541no—
299C2×C4—2541no—
300C2×C4—2541no—
301C2×C4—2541no—
302C2×C4—2542no—
303C2×C4—2542no—
304C2×C4D4×D12712yes—
305C2×C4D4×D12711yescharacteristic
306C2×C4#1452324812yes—
307C2×C4C23×D12711yescharacteristic
308C8—22no—
309C8—1274no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—1272no—
316C22—10164no—
317C22—5084no—
318C22—5082no—
319C22—5082no—
320C22—5082no—
321C22—2541no—
322C22—2541no—
323C22—2541no—
324C22—2541no—
325C22—2541no—
326C22#1686118212yes—
327C22C2×D4×D12711yescharacteristic
328C4—41no—
329C4—42no—
330C4—5081no—
331C4—5081no—
332C4—2541no—
333C4—2541no—
334C4#1686118212yes—
335C4C2×D4×D12711yescharacteristic
336C4C2×#1452324811yescharacteristic
337C2—22no—
338C2—81no—
339C2—10162no—
340C2—5081no—
341C2—2542no—
342C2C2×#1686118211yescharacteristic
343C2AGL(1,ℤ/8)×D12711yescharacteristic
344C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
hypercenter#417D127
hypocenterC127#3055
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×M16×D127C2semidirect
3. #22550383C2semidirect
4. C2×#16861184C2semidirect
5. #22550128C2semidirect
6. #22550125C2semidirect
7. #22550111C2semidirect
8. #22550108C2semidirect
9. #22549916C2semidirect
10. #22549882C2semidirect
11. #417×C127C2semidirect
16. C2×#14523235C22semidirect
17. C2×#14523233C22semidirect
19. C2×#14523231C22semidirect
21. C2×C8×D127C22semidirect
23. #16861185C22semidirect
24. #16861185C22semidirect
29. #16861180C22semidirect
30. #16861180C22semidirect
36. #16861142C22semidirect
37. #16861140C22semidirect
38. #16861131C22semidirect
39. #16861131C22semidirect
43. #103×C127C22semidirect
44. #102×C127C22semidirect
65. #14523232D4semidirect
120. C127⋊C8C2×D4semidirect
154. C2×D127AGL(1,ℤ/8)semidirect
173. C127#3055semidirect
184. #417D127semidirect
214. #103C2×D127semidirect
217. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×M16×D1272
#225503832
C2×#168611842
#225501282
#225501252
#225501112
#225501082
#225499162
#225498822
#417×C1272
#3055127