Finite Group Databasebeta

← #30648273

Subgroups — #30648273

Order 16256.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861229C211yesmaximal, characteristic, complemented
2#22550443C211yesmaximal, characteristic, complemented
3#22550359C211yesmaximal, characteristic, complemented
4#22550096C212yesmaximal, complemented
5#22550089C212yesmaximal, complemented
6#22550074C212yesmaximal, complemented
7#22550069C212yesmaximal, complemented
8#22549904C211yesmaximal, characteristic, complemented
9#22549878C211yesmaximal, characteristic, complemented
10C2×#16861130C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861229—24no—
14C2×#14523238C2211yescharacteristic, complemented
15C2×#14523236C2211yescharacteristic, complemented
16C2×#14523235C2211yescharacteristic, complemented
17C2×#14523233C2211yescharacteristic, complemented
18C22×C127⋊C8C2211yescharacteristic
19#16861205C2212yes—
20#16861196C2212yes—
21#16861180—21no—
22#16861180—21no—
23#16861178—22no—
24Q8×Dic127C2212yescomplemented
25D4×Dic127—22no—
26D4×Dic127C2212yescomplemented
27#16861168—21no—
28#16861168C2211yescharacteristic
29C2×C4⋊Dic127—21no—
30C2×C4⋊Dic127C2211yescharacteristic
31Dic127⋊C8C2212yescomplemented
32Dic127⋊C8C2212yescomplemented
33#16861134C2212yescomplemented
34#16861131C2212yescomplemented
35#16861130C2212yes—
36#16861130C2212yes—
37C2×#41×C127C2211yescharacteristic
38C22×D4×C127C2211yescharacteristic
39AGL(1,ℤ/8)×C127—24no—
40C2×SD16×C127C2212yes—
41C2×D8×C127C2212yes—
42C2×M16×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254C2311yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—22no—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—21no—
50C22×Dic127—22no—
51C22×Dic127—21no—
52C2×C4×D127—21no—
53#14523238—22no—
54#14523236—22no—
55#14523235—22no—
56#14523233—22no—
57C2×C127⋊C8—22no—
58C2×C127⋊C8C2311yescharacteristic
59C2×C127⋊C8C2311yescharacteristic
60C4⋊Dic127—22no—
61C4⋊Dic127—21no—
62C4⋊Dic127D411yescharacteristic
63C4⋊Dic127D411yescharacteristic
64C4⋊Dic127C2312yes—
65C22⋊Dic127—22no—
66C22⋊Dic127—22no—
67C22⋊Dic127—22no—
68C22⋊Dic127D412yes—
69C4×Dic127—22no—
70C4×Dic127C2312yes—
71C24×C127—21no—
72#41×C127—22no—
73#41×C127D414yescomplemented
74C2×Q8×C127C2311yescharacteristic
75C2×D4×C127—21no—
76C2×D4×C127—22no—
77C2×D4×C127—21no—
78C2×D4×C127—22no—
79C2×D4×C127C2312yes—
80C2×D4×C127C2311yescharacteristic
81C22×C4×C127—21no—
82C22×C4×C127C2311yescharacteristic
83SD16×C127—24no—
84D8×C127—24no—
85M16×C127—22no—
86C2×C8×C127C2312yes—
87C22×D127—21no—
88Dic254—21no—
89Dic254—21no—
90D508—21no—
91D508—21no—
92C22⋊D127—42no—
93C22⋊D127—82no—
94C22⋊D127—82no—
95C2×Dic127—22no—
96C2×Dic127—21no—
97C2×Dic127—82no—
98C2×Dic127—81no—
99C2×Dic127C2×D412yes—
100C2×Dic127C2×D412yes—
101C4×D127—42no—
102C127⋊C8—21no—
103C127⋊C8—21no—
104C23×C127—21no—
105C23×C127—22no—
106C23×C127—42no—
107C23×C127—81no—
108C23×C127—81no—
109C23×C127C2×D411yescharacteristic
110Q8×C127—21no—
111Q8×C127C2×D412yes—
112D4×C127—24no—
113D4×C127—22no—
114D4×C127—21no—
115D4×C127—22no—
116D4×C127—42no—
117D4×C127C2×D412yes—
118C2×C4×C127—21no—
119C2×C4×C127—24no—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D412yes—
122C2×C4×C127#4112yes—
123C2×C4×C127C2411yescharacteristic
124C8×C127—22no—
125C22×C127—21no—
126C22×C127—21no—
127C22×C127—22no—
128C22×C127—41no—
129C22×C127—41no—
130C22×C127—42no—
131C22×C127—42no—
132C22×C127—42no—
133C22×C127—84no—
134C22×C127#12112yes—
135C22×C127C22×D411yescharacteristic
136C2×D127—41no—
137C2×D127—41no—
138C2×D127—41no—
139C4×C127—22no—
140C4×C127#12112yes—
141C4×C127C22×D411yescharacteristic
142C4×C127C2×#4111yescharacteristic
143Dic127—22no—
144Dic127—22no—
145Dic127—41no—
146Dic127—42no—
147C2×C127—22no—
148C2×C127—42no—
149C2×C127—82no—
150C2×C127C2×#12111yescharacteristic
151C2×C127#57611yescharacteristic
152C2×C127#57711yescharacteristic
153D127—81no—
154#3045—1271nomaximal, Sylow
155C127#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
157C2×#135—1271no—
158#535—1271no—
159#520—1271no—
160#464—1272no—
161#463—1272no—
162#460—1272no—
163#459—1272no—
164#424—1271no—
165#417—1271no—
166C2×#102—1271no—
167C2×#41—1271no—
168C2×#41C2×D12711yescharacteristic
169C22×D4C2×D12711yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—2544no—
172C2×Q16—1271no—
173C2×SD16—1271no—
174C2×SD16—1271no—
175C2×SD16C2×D12712yes—
176C2×D8—1271no—
177C2×D8C2×D12712yes—
178C2×M16C2×D12711yescharacteristic
179C22×C8—1271no—
180#127—1272no—
181#124—1272no—
182#121—2541no—
183#121—2541no—
184#120—2542no—
185C4×Q8—1272no—
186C4×D4—1272no—
187C4×D4—2542no—
188#117—1271no—
189#117—2541no—
190C2×C4⋊C4—1271no—
191C2×C4⋊C4—2541no—
192C4⋊C8—1272no—
193C4⋊C8—1272no—
194#103—1272no—
195#102—1272no—
196#102—1272no—
197#102—1272no—
198C24—21no—
199#41—22no—
200#41—2542no—
201#41—2542no—
202#41C22⋊D12714yescomplemented
203C2×Q8—1271no—
204C2×Q8C22×D12711yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—1271no—
210C2×D4—2542no—
211C2×D4—2542no—
212C2×D4—2541no—
213C2×D4C22×D12712yes—
214C2×D4C22×D12711yescharacteristic
215C22×C4—21no—
216C22×C4—2542no—
217C22×C4—2541no—
218C22×C4—2541no—
219C22×C4C22×D12711yescharacteristic
220Q16—2542no—
221SD16—24no—
222SD16—2542no—
223SD16—2542no—
224D8—24no—
225D8—2542no—
226M16—22no—
227C2×C8—1271no—
228C2×C8—1271no—
229C2×C8—2542no—
230C2×C8C22×D12712yes—
231C4⋊C4—1271no—
232C4⋊C4—1271no—
233C4⋊C4—1272no—
234C4⋊C4—2542no—
235C4⋊C4—2541no—
236C22⋊C4—1272no—
237C22⋊C4—2542no—
238C22⋊C4—2542no—
239C22⋊C4—2542no—
240C42—1272no—
241C42—2542no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2541no—
248C23D4×D12711yescharacteristic
249Q8—21no—
250Q8—2541no—
251Q8—2541no—
252Q8C2×C22⋊D12712yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—10162no—
259D4—10162no—
260D4—5082no—
261D4—2541no—
262D4—2541no—
263D4C2×C22⋊D12712yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—1272no—
267C2×C4—1272no—
268C2×C4—10161no—
269C2×C4—10162no—
270C2×C4—5082no—
271C2×C4—2542no—
272C2×C4—2541no—
273C2×C4D4×D12711yescharacteristic
274C2×C4C2×C22⋊D12712yes—
275C2×C4#1452324712yes—
276C2×C4C23×D12711yescharacteristic
277C8—22no—
278C8—2541no—
279C8—2541no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—5081no—
290C22—5081no—
291C22—5081no—
292C22#1686118112yes—
293C22C2×D4×D12711yescharacteristic
294C4—22no—
295C4—5081no—
296C4—5082no—
297C4—2542no—
298C4—2542no—
299C4#1686118112yes—
300C4C22×C22⋊D12711yescharacteristic
301C4C2×#1452324711yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—10161no—
306C2C2×#1686118111yescharacteristic
307C2#2255068011yescharacteristic
308C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3045
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. #22550443C2semidirect
3. #22550359C2semidirect
4. #22550096C2semidirect
5. #22550089C2semidirect
6. #22550074C2semidirect
7. #22550069C2semidirect
8. #22549904C2semidirect
9. #22549878C2semidirect
10. C2×#16861130C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
14. C2×#14523238C22semidirect
15. C2×#14523236C22semidirect
16. C2×#14523235C22semidirect
17. C2×#14523233C22semidirect
24. Q8×Dic127C22semidirect
26. D4×Dic127C22semidirect
31. Dic127⋊C8C22semidirect
32. Dic127⋊C8C22semidirect
33. #16861134C22semidirect
34. #16861131C22semidirect
73. #41×C127D4semidirect
155. C127#3045semidirect
156. C2×AGL(1,ℤ/8)D127semidirect
202. #41C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
#225504432
#225503592
#225500962
#225500892
#225500742
#225500692
#225499042
#225498782
C2×#168611302
C2×AGL(1,ℤ/8)×C1272
#3045127