Finite Group Databasebeta

← #30651441

Subgroups — #30651441

Order 16256.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#22550583C211yesmaximal, characteristic, complemented
2#22550546C211yesmaximal, characteristic, complemented
3#22550239C212yesmaximal, complemented
4#22550222C211yesmaximal, characteristic, complemented
5#22550178C212yesmaximal, complemented
6#22550167C211yesmaximal, characteristic, complemented
7#22550069C212yesmaximal, complemented
8#22550003C211yesmaximal, characteristic, complemented
9#22549988C212yesmaximal, complemented
10C4×#14523233C211yesmaximal, characteristic, complemented
11#459×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C2×#14523233C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861205C2212yes—
16#16861200C2212yes—
17#16861188—21no—
18#16861186—22no—
19#16861181—22no—
20#16861180—21no—
21C4×Dic254C2211yescharacteristic
22C4×D508C2211yescharacteristic
23D4×Dic127C2212yes—
24D4×Dic127C2212yes—
25C2×C4⋊Dic127—21no—
26#16861142C2211yescharacteristic
27#16861141C2212yes—
28Dic127⋊C8C2212yes—
29#16861131C2212yes—
30#16861130C2212yes—
31#16861130C2211yescharacteristic
32#16861130C2212yes—
33#16861129C2211yescharacteristic
34#16861119C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C2×D8×C127C2212yes—
37#127×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41#14523247—44no—
42C2×Dic254—21no—
43C2×Dic254D411yescharacteristic
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—22no—
46C2×C22⋊D127—22no—
47C22×Dic127—22no—
48C22×Dic127—22no—
49C2×C4×D127—21no—
50#14523233—22no—
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2311yescharacteristic
53C4⋊Dic127—22no—
54C4⋊Dic127—21no—
55C4⋊Dic127D411yescharacteristic
56C4⋊Dic127C2312yes—
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—22no—
59Dic127⋊C4D412yes—
60C22⋊Dic127—22no—
61C22⋊Dic127—22no—
62(C2×D127)⋊C4—21no—
63C4×Dic127C2312yes—
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127C2312yes—
67C2×D4×C127C2311yescharacteristic
68C22×C4×C127—21no—
69D8×C127—24no—
70C2×C8×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—21no—
73C42×C127C2311yescharacteristic
74C22×D127—21no—
75Dic254—22no—
76Dic254—81no—
77D508—21no—
78D508—21no—
79C22⋊D127—44no—
80C22⋊D127—82no—
81C2×Dic127—21no—
82C2×Dic127—22no—
83C2×Dic127—44no—
84C2×Dic127—82no—
85C2×Dic127C2×D412yes—
86C2×Dic127C2×D412yes—
87C4×D127—42no—
88C127⋊C8—21no—
89C127⋊C8—21no—
90C23×C127—21no—
91C23×C127—22no—
92D4×C127—22no—
93D4×C127—21no—
94D4×C127—22no—
95D4×C127—42no—
96D4×C127—81no—
97D4×C127#4112yescomplemented
98C2×C4×C127—21no—
99C2×C4×C127—42no—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2411yescharacteristic
103C8×C127—22no—
104C22×C127—22no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127C22×D411yescharacteristic
111C2×D127—41no—
112C2×D127—41no—
113C2×D127—41no—
114C4×C127—21no—
115C4×C127—41no—
116C4×C127AGL(1,ℤ/8)12yes—
117C4×C127C2×#4111yescharacteristic
118C4×C1272−1+411yescharacteristic
119Dic127—22no—
120Dic127—22no—
121Dic127—41no—
122Dic127—42no—
123C2×C127—42no—
124C2×C127—82no—
125C2×C127#54811yescharacteristic
126C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C127#57611yescharacteristic
128D127—81no—
129#3366—1271nomaximal, Sylow
130C127#336611yescharacteristic, complemented, Sylow
131#554—1271no—
132#550—1271no—
133#490—1272no—
134#486—1271no—
135#478—1272no—
136#476—1271no—
137#459—1272no—
138#459D12711yescharacteristic, complemented
139#446—1271no—
140#442—1272no—
141C4×D8—1271no—
142C2×#41—2541no—
143C2×D8—1271no—
144C2×D8C2×D12712yes—
145C4⋊Q8—1271no—
146#127—1272no—
147#127C2×D12711yescharacteristic
148#125—1272no—
149C22⋊Q8—2542no—
150C22⋊Q8—2541no—
151#121—2541no—
152#121—2542no—
153C4×Q8—1271no—
154C4×D4—1272no—
155C4×D4—1272no—
156C4×D4—1271no—
157C4×D4C2×D12711yescharacteristic
158C2×C4⋊C4—2541no—
159#107—1271no—
160#106—1272no—
161C4⋊C8—1272no—
162C4⋊C8C2×D12711yescharacteristic
163#102—1271no—
164#102—1272no—
165#102—1272no—
166#102—1272no—
167#102—1271no—
168#102C2×D12712yes—
169#97—1272no—
170C4×C8—1271no—
171#41—5084no—
172C2×Q8—1271no—
173C2×Q8—2541no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—1271no—
177C2×D4—2542no—
178C2×D4—2542no—
179C2×D4C22×D12712yes—
180C2×D4C22×D12711yescharacteristic
181C22×C4—21no—
182C22×C4—2542no—
183C22×C4—2542no—
184C22×C4—2541no—
185D8—24no—
186D8—2542no—
187C2×C8—1271no—
188C2×C8—1271no—
189C2×C8C22×D12712yes—
190C4⋊C4—1272no—
191C4⋊C4—1271no—
192C4⋊C4—1272no—
193C4⋊C4—1271no—
194C4⋊C4—2542no—
195C4⋊C4—2542no—
196C4⋊C4—2541no—
197C4⋊C4C22×D12711yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—2542no—
200C22⋊C4—2542no—
201C22⋊C4—2541no—
202C42—1272no—
203C42C22×D12711yescharacteristic
204C23—21no—
205C23—22no—
206C23—2541no—
207Q8—10161no—
208Q8—2542no—
209D4—22no—
210D4—21no—
211D4—22no—
212D4—42no—
213D4—81no—
214D4—10162no—
215D4—5084no—
216D4—2541no—
217D4—2541no—
218D4#1452324712yescomplemented
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—1272no—
222C2×C4—1272no—
223C2×C4—10162no—
224C2×C4—5084no—
225C2×C4—5082no—
226C2×C4—2541no—
227C2×C4—2542no—
228C2×C4D4×D12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4C23×D12711yescharacteristic
231C8—22no—
232C8—2541no—
233C8—2541no—
234C22—22no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—5081no—
241C22—5081no—
242C22—5081no—
243C22C2×D4×D12711yescharacteristic
244C4—21no—
245C4—41no—
246C4—5081no—
247C4—5082no—
248C4—2542no—
249C4—2542no—
250C4#1686123312yes—
251C4C2×#1452324711yescharacteristic
252C4#1686125611yescharacteristic
253C2—42no—
254C2—82no—
255C2—10161no—
256C2#2255053211yescharacteristic
257C2C2×#1686123311yescharacteristic
258C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3366
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550583C2semidirect
2. #22550546C2semidirect
3. #22550239C2semidirect
4. #22550222C2semidirect
5. #22550178C2semidirect
6. #22550167C2semidirect
7. #22550069C2semidirect
8. #22550003C2semidirect
9. #22549988C2semidirect
10. C4×#14523233C2semidirect
11. #459×C127C2semidirect
97. D4×C127#41semidirect
130. C127#3366semidirect
138. #459D127semidirect
218. D4#14523247semidirect

Maximal subgroups

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

subgroupindex
#225505832
#225505462
#225502392
#225502222
#225501782
#225501672
#225500692
#225500032
#225499882
C4×#145232332
#459×C1272
#3366127