Finite Group Databasebeta

← #30651865

Subgroups — #30651865

Order 16256.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16861232C211yesmaximal, characteristic, complemented
2#135×D127C211yesmaximal, characteristic, complemented
3#22550318C212yesmaximal, complemented
4#22550317C211yesmaximal, characteristic, complemented
5#22550314C211yesmaximal, characteristic, complemented
6#22550313C212yesmaximal, complemented
7#22550311C211yesmaximal, characteristic, complemented
8#22550309C212yesmaximal, complemented
9#22550307C212yesmaximal, complemented
10M32×D127C211yesmaximal, characteristic, complemented
11#510×C127C211yesmaximal, characteristic, complemented
12C2×#14523249—21no—
13#41×D127—21no—
14#16861232—21no—
15#16861232—21no—
16#16861232C2212yes—
17#16861232C2212yescomplemented
18#16861232C2211yescharacteristic, complemented
19#16861231C2211yescharacteristic
20#16861230—21no—
21#16861229—21no—
22#16861228C2211yescharacteristic, complemented
23Q16×D127C2211yescharacteristic, complemented
24C2×#14523235—21no—
25SD16×D127—21no—
26C2×D1016C2211yescharacteristic, complemented
27D8×D127C2211yescharacteristic, complemented
28C2×C8×D127C2211yescharacteristic
29#16861155C2212yescomplemented
30#16861154C2212yescomplemented
31#16861152C2212yescomplemented
32#16861152C2212yescomplemented
33D2032C2212yescomplemented
34#16861150C2212yescomplemented
35#16861149C2211yescharacteristic, complemented
36#16861148C2212yescomplemented
37C16×D127C2212yescomplemented
38#135×C127C2211yescharacteristic
39C2×Q16×C127C2211yescharacteristic, complemented
40Q32×C127C2212yescomplemented
41SD32×C127C2212yescomplemented
42M32×C127C2211yescharacteristic, complemented
43#14523249—21no—
44#14523249—22no—
45#14523249—22no—
46#14523249—41no—
47#14523249—41no—
48#14523248—21no—
49#14523248—41no—
50#14523247—21no—
51#14523247—41no—
52Q8×D127—21no—
53C2×D508—21no—
54C2×D508—41no—
55D4×D127—21no—
56D4×D127—41no—
57C2×C4×D127—41no—
58C2×C4×D127—41no—
59C2×C4×D127D411yescharacteristic
60Dic508C2311yescharacteristic
61#14523238—21no—
62#14523237—21no—
63#14523236—21no—
64#14523235—21no—
65#14523235—22no—
66#14523235—22no—
67D1016—21no—
68D1016C2311yescharacteristic
69D1016C2312yes—
70#14523233—21no—
71C2×C127⋊C8D411yescharacteristic
72C8×D127D411yescharacteristic
73C8×D127D411yescharacteristic
74C8×D127C2311yescharacteristic
75C8×D127C2311yescharacteristic
76C127⋊C16C2312yes—
77#41×C127—21no—
78C2×Q8×C127—21no—
79Q16×C127—21no—
80Q16×C127C2311yescharacteristic
81Q16×C127C2312yes—
82SD16×C127—21no—
83D8×C127C2311yescharacteristic
84C2×C8×C127C2311yescharacteristic
85C16×C127C2312yes—
86C22×D127—41no—
87C22×D127—41no—
88C22×D127D811yescharacteristic
89Dic254—21no—
90Dic254—41no—
91D508—21no—
92D508—22no—
93D508—41no—
94D508—42no—
95D508—42no—
96D508—41no—
97C22⋊D127—41no—
98C22⋊D127—41no—
99C22⋊D127—41no—
100C2×Dic127—41no—
101C2×Dic127D811yescharacteristic
102C4×D127—41no—
103C4×D127—41no—
104C4×D127—41no—
105C4×D127—42no—
106C4×D127—42no—
107C4×D127D811yescharacteristic
108C4×D127D811yescharacteristic
109C4×D127C2×D411yescharacteristic
110C4×D127C2×D411yescharacteristic
111C127⋊C8C2×D411yescharacteristic
112C127⋊C8C2×D411yescharacteristic
113Q8×C127—21no—
114Q8×C127—22no—
115Q8×C127—41no—
116D4×C127—21no—
117D4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127C2×D411yescharacteristic
121C8×C127C2×D411yescharacteristic
122C8×C127C2411yescharacteristic
123C22×C127—41no—
124C22×C127C2×D811yescharacteristic
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—42no—
128C2×D127—82no—
129C2×D127—82no—
130C2×D127C2×D811yescharacteristic
131C2×D127C2×D811yescharacteristic
132C4×C127—41no—
133C4×C127—42no—
134C4×C127C2×D811yescharacteristic
135C4×C127C22×D411yescharacteristic
136Dic127—41no—
137Dic127C2×D811yescharacteristic
138Dic127C2×D811yescharacteristic
139C2×C127—21no—
140C2×C127—81no—
141C2×C127C22×D811yescharacteristic
142D127—21no—
143D127—21no—
144D127—81no—
145D127—82no—
146#3414—1271nomaximal, Sylow
147C127#341411yescharacteristic, complemented, Sylow
148C2×#135—1271no—
149C2×#135—1271no—
150#510—1271no—
151#510—1272no—
152#510D12711yescharacteristic, complemented
153#509—1271no—
154#509—1272no—
155#509—1271no—
156#508—1272no—
157#508—1272no—
158C2×M32—1271no—
159C2×#41—2541no—
160C2×#41—2541no—
161#135—1272no—
162#135—1271no—
163#135—1272no—
164#135—1271no—
165#135—1271no—
166#135—2541no—
167#135—2541no—
168#135—2541no—
169#135—2541no—
170#135C2×D12711yescharacteristic
171C2×Q16—1271no—
172C2×Q16C2×D12711yescharacteristic, complemented
173C2×SD16—2541no—
174C2×SD16—2541no—
175C2×D8—1271no—
176C2×D8—1271no—
177C22×C8—1271no—
178Q32—1272no—
179Q32C2×D12712yescomplemented
180SD32—1272no—
181SD32—1272no—
182SD32—1272no—
183SD32C2×D12712yescomplemented
184D16—1272no—
185D16—1272no—
186M32—1271no—
187M32—1272no—
188M32C2×D12711yescharacteristic, complemented
189C2×C16—1272no—
190#41—21no—
191#41—5081no—
192#41—5081no—
193#41—5081no—
194#41—5081no—
195#41—2541no—
196#41—2542no—
197#41—2541no—
198#41—2542no—
199#41—2541no—
200C2×Q8—21no—
201C2×Q8—2541no—
202C2×D4—5081no—
203C2×D4—5081no—
204C2×D4—2541no—
205C2×D4—2541no—
206C22×C4—1271no—
207C22×C4—5081no—
208C22×C4—5081no—
209Q16—21no—
210Q16—1271no—
211Q16—2541no—
212Q16C22×D12711yescharacteristic
213Q16C22×D12712yes—
214SD16—21no—
215SD16—2541no—
216SD16—2542no—
217SD16—2541no—
218SD16—2542no—
219SD16—2541no—
220D8—1271no—
221D8—1272no—
222D8—2541no—
223D8—2541no—
224D8C22×D12711yescharacteristic
225C2×C8—1271no—
226C2×C8—1271no—
227C2×C8—1271no—
228C2×C8—1271no—
229C2×C8—1271no—
230C2×C8C22×D12711yescharacteristic
231C16—1272no—
232C16C22×D12712yes—
233C23—1271no—
234C23—5081no—
235C23—5081no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—5081no—
240Q8—2541no—
241D4—21no—
242D4—41no—
243D4—5081no—
244D4—5082no—
245D4—5082no—
246D4—5081no—
247D4—5081no—
248D4—5081no—
249D4—5081no—
250D4—2541no—
251D4—2542no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1271no—
255C2×C4—1271no—
256C2×C4—1271no—
257C2×C4—1271no—
258C2×C4—1271no—
259C2×C4—5081no—
260C2×C4—5082no—
261C2×C4—5082no—
262C2×C4—5081no—
263C2×C4—5081no—
264C2×C4—5081no—
265C2×C4D4×D12711yescharacteristic
266C8—1271no—
267C8—1271no—
268C8D4×D12711yescharacteristic
269C8C23×D12711yescharacteristic
270C22—41no—
271C22—1271no—
272C22—1271no—
273C22—10162no—
274C22—10162no—
275C22—5081no—
276C22—5082no—
277C22—5081no—
278C22D8×D12711yescharacteristic
279C4—41no—
280C4—42no—
281C4—1271no—
282C4—1271no—
283C4—5081no—
284C4D8×D12711yescharacteristic
285C4C2×D4×D12711yescharacteristic
286C2—21no—
287C2—81no—
288C2—10162no—
289C2—10161no—
290C2—2541no—
291C2—2541no—
292C2C2×D8×D12711yescharacteristic
subgroupG/H
centerC2C2×D8×D127
commutator, cosocleC8×C127C24
FrattiniC8C23×D127
Fitting#510×C127C2
socleC2×C127C22×D8
hypercenter#510D127
hypocenterC127#3414
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. #135×D127C2semidirect
3. #22550318C2semidirect
4. #22550317C2semidirect
5. #22550314C2semidirect
6. #22550313C2semidirect
7. #22550311C2semidirect
8. #22550309C2semidirect
9. #22550307C2semidirect
10. M32×D127C2semidirect
11. #510×C127C2semidirect
17. #16861232C22semidirect
18. #16861232C22semidirect
22. #16861228C22semidirect
23. Q16×D127C22semidirect
26. C2×D1016C22semidirect
27. D8×D127C22semidirect
29. #16861155C22semidirect
30. #16861154C22semidirect
31. #16861152C22semidirect
32. #16861152C22semidirect
33. D2032C22semidirect
34. #16861150C22semidirect
35. #16861149C22semidirect
36. #16861148C22semidirect
37. C16×D127C22semidirect
39. C2×Q16×C127C22semidirect
40. Q32×C127C22semidirect
41. SD32×C127C22semidirect
42. M32×C127C22semidirect
147. C127#3414semidirect
152. #510D127semidirect
172. C2×Q16C2×D127semidirect
179. Q32C2×D127semidirect
183. SD32C2×D127semidirect
188. M32C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612322
#135×D1272
#225503182
#225503172
#225503142
#225503132
#225503112
#225503092
#225503072
M32×D1272
#510×C1272
#3414127