Finite Group Databasebeta

← #30648250

Subgroups — #30648250

Order 16256.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16861242C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3#22550443C211yesmaximal, characteristic
4#22550353C211yesmaximal, characteristic, complemented
5#22550097C214yesmaximal, complemented
6#22550075C214yesmaximal, complemented
7#22549904C211yesmaximal, characteristic, complemented
8#22549870C211yesmaximal, characteristic, complemented
9#418×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254C2211yescharacteristic
12#16861242—24no—
13C2×Dic508C2212yescomplemented
14C2×#14523237C2212yescomplemented
15C2×#14523236—24no—
16C2×#14523236C2214yes—
17C22×C127⋊C8C2211yescharacteristic
18#16861205C2212yes—
19#16861196C2212yes—
20#16861180—22no—
21#16861178—22no—
22C4×Dic254—22no—
23#16861174C2214yescomplemented
24#16861168C2211yescharacteristic
25#16861165—22no—
26Dic127⋊C8C2214yescomplemented
27#16861132C2212yescomplemented
28#16861129C2212yescomplemented
29C22×D4×C127C2211yescharacteristic
30C2×M16×C127C2211yescharacteristic
31#117×C127C2211yescharacteristic
32#102×C127C2214yes—
33#14523248—22no—
34#14523248—22no—
35C2×Dic254—21no—
36C2×Dic254—22no—
37C2×Dic254D414yescomplemented
38C2×Dic254C2311yescharacteristic
39C2×Dic254C2312yes—
40C2×D508C2311yescharacteristic
41C2×C22⋊D127—24no—
42C2×C22⋊D127—21no—
43C22×Dic127—21no—
44C2×C4×D127—21no—
45Dic508—24no—
46#14523237—24no—
47#14523236—28no—
48C2×C127⋊C8—22no—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127D412yes—
51Dic127⋊C4—24no—
52C22⋊Dic127—24no—
53C22⋊Dic127D412yes—
54C4×Dic127—24no—
55C4×Dic127C2312yes—
56C24×C127—21no—
57C2×D4×C127—22no—
58C2×D4×C127—21no—
59C2×D4×C127—22no—
60C2×D4×C127C2312yes—
61C22×C4×C127C2311yescharacteristic
62M16×C127—22no—
63C2×C8×C127C2312yes—
64C4⋊C4×C127C2312yes—
65C22⋊C4×C127—21no—
66C42×C127—21no—
67C22×D127—21no—
68Dic254—24no—
69Dic254—22no—
70Dic254—21no—
71Dic254—21no—
72Dic254C2×D414yes—
73D508—21no—
74D508—21no—
75C22⋊D127—42no—
76C22⋊D127—84no—
77C2×Dic127—24no—
78C2×Dic127—22no—
79C2×Dic127—21no—
80C2×Dic127C2×D414yes—
81C4×D127—42no—
82C127⋊C8—22no—
83C23×C127—22no—
84C23×C127—42no—
85C23×C127—82no—
86C23×C127C2×D411yescharacteristic
87D4×C127—22no—
88D4×C127—22no—
89D4×C127—42no—
90C2×C4×C127—22no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D412yes—
93C2×C4×C127#4112yes—
94C2×C4×C127C2411yescharacteristic
95C8×C127—22no—
96C22×C127—22no—
97C22×C127—42no—
98C22×C127—44no—
99C22×C127—84no—
100C22×C127#12112yes—
101C22×C127C22×D411yescharacteristic
102C2×D127—41no—
103C2×D127—42no—
104C4×C127—42no—
105C4×C127#12112yes—
106C4×C127C22×D411yescharacteristic
107C4×C127C2×#4111yescharacteristic
108Dic127—24no—
109Dic127—24no—
110Dic127—41no—
111C2×C127—22no—
112C2×C127—82no—
113C2×C127C2×#12111yescharacteristic
114C2×C127#57712yes—
115D127—81no—
116#3043—1271nomaximal, Sylow
117C127#304311yescharacteristic, complemented, Sylow
118C2×#137—1271no—
119C22×SD16—1271no—
120#535—1271no—
121#519—1271no—
122#464—1274no—
123#460—1274no—
124#424—1271no—
125#418D12711yescharacteristic, complemented
126#416—1271no—
127C2×#41—1271no—
128C22×Q8—1271no—
129C22×D4C2×D12711yescharacteristic
130#137—2544no—
131C2×Q16—1272no—
132C2×SD16—1274no—
133C2×SD16—1272no—
134C2×SD16—2544no—
135C2×M16C2×D12711yescharacteristic
136C22×C8—1271no—
137#127—1272no—
138#124—1272no—
139#121—2542no—
140#120—2542no—
141C4×Q8—1274no—
142C4×Q8—2542no—
143#117—1271no—
144#117—2542no—
145#117C2×D12711yescharacteristic
146C4⋊C8—1274no—
147#103—1272no—
148#102—1272no—
149#102C2×D12714yes—
150C24—21no—
151#41—2542no—
152#41—2542no—
153C2×Q8—1274no—
154C2×Q8—1272no—
155C2×Q8—1271no—
156C2×Q8—2541no—
157C2×Q8—2542no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—1271no—
162C2×D4—2544no—
163C2×D4—2541no—
164C2×D4C22×D12712yes—
165C22×C4—2541no—
166C22×C4—2541no—
167C22×C4C22×D12711yescharacteristic
168Q16—2544no—
169SD16—2544no—
170SD16—2548no—
171M16—22no—
172C2×C8—1272no—
173C2×C8—2542no—
174C2×C8C22×D12712yes—
175C4⋊C4—1272no—
176C4⋊C4—2544no—
177C4⋊C4C22×D12712yes—
178C22⋊C4—21no—
179C22⋊C4—1272no—
180C22⋊C4—2544no—
181C42—21no—
182C42—1272no—
183C42—2544no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—2541no—
188C23D4×D12711yescharacteristic
189Q8—1274no—
190Q8—2541no—
191Q8—2541no—
192Q8—2542no—
193Q8—2544no—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—10164no—
198D4—5082no—
199D4—2541no—
200D4—2541no—
201C2×C4—22no—
202C2×C4—1274no—
203C2×C4—5082no—
204C2×C4—2541no—
205C2×C4—2542no—
206C2×C4—2544no—
207C2×C4D4×D12712yes—
208C2×C4D4×D12711yescharacteristic
209C2×C4#1452324812yes—
210C2×C4C23×D12711yescharacteristic
211C8—22no—
212C8—2542no—
213C22—22no—
214C22—42no—
215C22—44no—
216C22—84no—
217C22—5081no—
218C22—5082no—
219C22#1686118212yes—
220C22C2×D4×D12711yescharacteristic
221C4—42no—
222C4—5081no—
223C4—2544no—
224C4—2544no—
225C4#1686118212yes—
226C4C2×D4×D12711yescharacteristic
227C4C2×#1452324811yescharacteristic
228C2—22no—
229C2—82no—
230C2—10161no—
231C2C2×#1686118211yescharacteristic
232C2#2255068612yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
prime radical#22550443C2
hypercenter#418D127
hypocenterC127#3043
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C22×#14523236C2semidirect
4. #22550353C2semidirect
5. #22550097C2semidirect
6. #22550075C2semidirect
7. #22549904C2semidirect
8. #22549870C2semidirect
9. #418×C127C2semidirect
13. C2×Dic508C22semidirect
14. C2×#14523237C22semidirect
23. #16861174C22semidirect
26. Dic127⋊C8C22semidirect
27. #16861132C22semidirect
28. #16861129C22semidirect
37. C2×Dic254D4semidirect
117. C127#3043semidirect
125. #418D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C22×#145232362
#225504432
#225503532
#225500972
#225500752
#225499042
#225498702
#418×C1272
#3043127