Finite Group Databasebeta

← #30650672

Subgroups — #30650672

Order 16256.

396 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550599C211yesmaximal, characteristic, complemented
2#22550546C211yesmaximal, characteristic
3#22550222C211yesmaximal, characteristic, complemented
4#22550199C212yesmaximal, complemented
5#22550162C211yesmaximal, characteristic, complemented
6#22550102C212yesmaximal, complemented
7#22550035C212yesmaximal, complemented
8C4×#14523238C211yesmaximal, characteristic, complemented
9#22549941C211yesmaximal, characteristic, complemented
10#22549869C212yesmaximal, complemented
11#460×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yes—
13C2×#14523238C2211yescharacteristic
14C2×C8×D127C2212yes—
15#16861209C2211yescharacteristic
16#16861205—21no—
17#16861204—22no—
18#16861197—21no—
19#16861185C2212yes—
20#16861181—22no—
21#16861180C2212yes—
22C4×Dic254—21no—
23C4×Dic254C2211yescharacteristic
24C4×D508C2211yescharacteristic
25D4×Dic127—21no—
26#16861167C2212yes—
27#16861142C2211yescharacteristic
28#16861141C2212yes—
29#16861134C2212yes—
30#16861133C2211yescharacteristic
31#16861132C2211yescharacteristic
32#16861130C2212yes—
33#16861130C2212yes—
34(C2×D127)⋊C8C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C2×SD16×C127C2212yes—
37#127×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41#14523247—24no—
42#14523247—42no—
43C2×Dic254—21no—
44C2×Dic254C2311yescharacteristic
45C2×D508D411yescharacteristic
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C22×Dic127—22no—
49C2×C4×D127C2312yes—
50#14523238—22no—
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2311yescharacteristic
53C8×D127—42no—
54C4⋊Dic127—21no—
55C4⋊Dic127D411yescharacteristic
56C4⋊Dic127C2312yes—
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—22no—
59Dic127⋊C4—21no—
60C22⋊Dic127—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4D412yes—
64C4×Dic127—22no—
65C4×Dic127—21no—
66C2×Q8×C127C2311yescharacteristic
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127C2312yes—
70SD16×C127—42no—
71C2×C8×C127C2312yes—
72C4⋊C4×C127—21no—
73C4⋊C4×C127C2311yescharacteristic
74C42×C127—21no—
75C42×C127C2311yescharacteristic
76C22×D127C2×D412yes—
77Dic254—21no—
78Dic254—42no—
79Dic254#4112yes—
80D508—22no—
81C22⋊D127—44no—
82C22⋊D127—82no—
83C2×Dic127—21no—
84C2×Dic127—22no—
85C2×Dic127—21no—
86C2×Dic127—44no—
87C2×Dic127C2×D412yes—
88C4×D127—22no—
89C4×D127—22no—
90C127⋊C8—21no—
91C127⋊C8—21no—
92C23×C127—22no—
93Q8×C127—21no—
94Q8×C127—21no—
95D4×C127—22no—
96D4×C127—22no—
97D4×C127—42no—
98D4×C127—81no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2411yescharacteristic
104C8×C127—22no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127C22×D411yescharacteristic
109C2×D127—22no—
110C2×D127—22no—
111C2×D127—22no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127#13512yes—
116C4×C127C2×#4111yescharacteristic
117C4×C127H(5,𝔽2)11yescharacteristic
118Dic127—22no—
119Dic127—22no—
120Dic127—42no—
121Dic127—41no—
122C2×C127—82no—
123C2×C127#54611yescharacteristic
124C2×C127C2×#13511yescharacteristic
125C2×C127#57711yescharacteristic
126D127—42no—
127#3304—1271nomaximal, Sylow
128C127#330411yescharacteristic, complemented, Sylow
129#556—1271no—
130#550—1271no—
131#486—1271no—
132#482—1272no—
133#475—1271no—
134#465—1272no—
135#460D12711yescharacteristic, complemented
136#452—1272no—
137C4×Q16—1271no—
138C8×D4—1271no—
139#416—1272no—
140C2×#41—1272no—
141C2×Q16—1271no—
142C2×SD16C2×D12712yes—
143C22×C8—1272no—
144C4⋊Q8—1271no—
145#127—2541no—
146#127C2×D12711yescharacteristic
147#126—2542no—
148#124—2541no—
149C22⋊Q8—1272no—
150#121—1272no—
151#121—2542no—
152C4×Q8—1271no—
153C4×Q8—2541no—
154C4×Q8C2×D12711yescharacteristic
155C4×D4—1271no—
156C4×D4—2541no—
157#117—1272no—
158#107—1271no—
159#106—1272no—
160C4⋊C8C2×D12711yescharacteristic
161#103—1271no—
162#103—1272no—
163#103—1271no—
164#102—1272no—
165#102—1272no—
166#102C2×D12712yes—
167C22⋊C8—1272no—
168C4×C8—1271no—
169#41—5082no—
170#41—2544no—
171C2×Q8—1271no—
172C2×Q8—2541no—
173C2×Q8C22×D12711yescharacteristic
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—1271no—
177C2×D4—2542no—
178C2×D4—2542no—
179C2×D4C22×D12712yes—
180C22×C4—1272no—
181C22×C4—2542no—
182Q16—2542no—
183SD16—42no—
184C2×C8—1271no—
185C2×C8—1271no—
186C2×C8—5082no—
187C2×C8C22×D12712yes—
188C4⋊C4—21no—
189C4⋊C4—1271no—
190C4⋊C4—1272no—
191C4⋊C4—1271no—
192C4⋊C4—2542no—
193C4⋊C4—2541no—
194C4⋊C4—2541no—
195C4⋊C4C22×D12711yescharacteristic
196C22⋊C4—1272no—
197C22⋊C4—2542no—
198C22⋊C4—2542no—
199C22⋊C4—2542no—
200C42—21no—
201C42—2542no—
202C42—2541no—
203C42C22×D12711yescharacteristic
204C23—22no—
205C23—1272no—
206Q8—21no—
207Q8—21no—
208Q8—1272no—
209Q8—5082no—
210Q8—2541no—
211D4—22no—
212D4—22no—
213D4—42no—
214D4—81no—
215D4—10162no—
216D4—5084no—
217D4—2542no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—1272no—
221C2×C4—5084no—
222C2×C4—2542no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2542no—
226C2×C4—2542no—
227C2×C4D4×D12711yescharacteristic
228C2×C4D4×D12711yescharacteristic
229C2×C4C23×D12711yescharacteristic
230C8—22no—
231C8—2541no—
232C8—2541no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—2542no—
237C22—2542no—
238C22—2542no—
239C22C2×D4×D12711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—5082no—
244C4—5081no—
245C4—2542no—
246C4—2542no—
247C4#1686123012yes—
248C4#41×D12711yescharacteristic
249C4#1686125411yescharacteristic
250C2—82no—
251C2—5082no—
252C2#2255052311yescharacteristic
253C2C2×#1686123011yescharacteristic
254C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
prime radical#22550546C2
hypercenter#460D127
hypocenterC127#3304
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #22550599C2semidirect
3. #22550222C2semidirect
4. #22550199C2semidirect
5. #22550162C2semidirect
6. #22550102C2semidirect
7. #22550035C2semidirect
8. C4×#14523238C2semidirect
9. #22549941C2semidirect
10. #22549869C2semidirect
11. #460×C127C2semidirect
128. C127#3304semidirect
135. #460D127semidirect

Maximal subgroups

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

subgroupindex
#225505992
#225505462
#225502222
#225501992
#225501622
#225501022
#225500352
C4×#145232382
#225499412
#225498692
#460×C1272
#3304127